ALGEBRAS NOTAVEIS em ALGEBRA COMUTATIVA

Transcrição

ALGEBRAS NOTAVEIS
em
ALGEBRA COMUTATIVA
A. Simis, Universidade Federal de Pernambuco (Brasil)
(Roteiro
para um ciclo de conferências)
Universidade do Porto, Portugal, Primavera de 2004
Anfitriões: Jorge Almeida e Antonio Machiavelo
Resumo
Estas notas constituem um roteiro para um ciclo de conferências proferidas no Departamento de Matemática da FCUP. Como tal, não inclui as demonstrações da grande
maioria dos resultados apresentados. Em compensação, este roteiro fornece mais material do que foi efetivamente transmitido em sala, dando um quadro mais claro e completo
daquilo que, por força do tempo exı́guo, não foi possı́vel explicar. Enfim, a lista de referências é suficientemente ampla (80 itens!) de modo a permitir ao interessado uma
consulta adequada aos tópicos apenas esboçados em aula.
Espero a condescendência do leitor pela salada idiomática inadvertida, resultante
do estilo jornalı́stico de citações diversas.
Quero expressar meu reconhecimento aos meus prezados anfitriões pelo esforço em
me convidar para este ciclo de palestras e aos demais participantes pela sua grande
simpatia ao me acolher nesta bela casa.
Em 6 de maio, 2004
Conteúdo
1 Divagação: origens e precursores
1.1 Dedekind e a teoria de ideais [11] . . . . . . . . . . .
1.1.1 Lagrange . . . . . . . . . . . . . . . . . . . .
1.1.2 Kronecker . . . . . . . . . . . . . . . . . . . .
1.2 Hilbert e a teoria dos invariantes [24], [25] . . . . . .
1.3 Contribuições de Lasker [38], Hurwitz [29], Macaulay
1.4 E. Noether [48], [49], [50], [51], e contemporâneos . .
1.5 O perı́odo “pós-noetheriano” . . . . . . . . . . . . .
1.6 A fase de ouro . . . . . . . . . . . . . . . . . . . . .
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[41], [42] .
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2
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2 O legado de D. Rees
2.1 Álgebra simétrica versus álgebra de Rees . . . . . . . . .
2.1.1 Primeiras manifestações . . . . . . . . . . . . . .
2.1.2 O caso de um ideal . . . . . . . . . . . . . . . . .
2.2 Álgebras de Rees e simetrizações do complexo de Koszul
2.2.1 Complexos simetrizados e de aproximação . . . .
2.2.2 Aplicações recentes . . . . . . . . . . . . . . . . .
2.3 Avanços recentes no caso de módulos . . . . . . . . . . .
2.3.1 Variações iniciais . . . . . . . . . . . . . . . . . .
2.3.2 O método do ideal genérico de Bourbaki . . . . .
2.3.3 Aplicações em baixas dimensões . . . . . . . . .
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3 Miscelânia
3.1 Álgebras graduadas direcionais . . . . . .
3.2 Ω-álgebras . . . . . . . . . . . . . . . . . .
3.2.1 A álgebra das estrelas tangentes .
3.2.2 Selecta . . . . . . . . . . . . . . . .
3.2.3 O problema de van Gastel . . . . .
3.2.4 A álgebra de Gauss de um módulo
3.2.5 A álgebra tangencial . . . . . . . .
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23
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4 Alguns problemas
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44
Divagação: origens e precursores
A posição da Álgebra Comutativa no seio das outras diciplinas matemáticas é singular. Por
um lado, é admirada e respeitada pela sua estética, auto-suficiência e sofisticação. Por outro
lado, como as regiões polares da terra, é uma região curiosa mas pode-se viver sem ela sem
maiores consequências. As teorias mais recentes da estrutura do universo (espaço-tempo)
são essencialmente não-comutativas (ou metade), passando por cima da comutatividade
(A. Connes). É claro que na “velha” relatividade riemanniana, C ∞ (M ) é comutativa e
desempenha papel significativo. Mas, um algebrista comutativo dificilmente consideraria
uma tal álgebra (mesmo quando M é compacto) como o centro nervoso da área.
2
Assim sendo, nesta primeira parte, gostaria de salientar, pela enésima vez, as raı́zes
sólidas da área, dentro do antigo sonho de Hilbert na sua introdução aos Problemas: “A
unidade orgânica da matemática é inerente à sua natureza, uma vez que a matemática se
constitui no fundamento de todo conhecimento exato dos fenômenos naturais.”
1.1
1.1.1
Dedekind e a teoria de ideais [11]
Lagrange
O primeiro a elaborar uma estrutura algébrica abstrata parece ter sido Lagrange. Senão,
vejamos:
Lagrange and the Solution of Numerical Equations
Reinhard Laubenbacher and David Pengelley
Lagrange developed the theory of numerical resolution of polynomial equations to a
sophisticated degree. After the Refléxions (1770), he had a first version of the “numerical”.
Much later (first published respectively in 1795 and 1798) he developed two methods to find
a lower bound more efficiently (i.e., of the so called “differences equation”) without actually
computing the coefficients of the equation of differences. Our interest is particularly in the
historical significance of the final version of his technique. From a modern point of view
he clearly decides to work, to great practical advantage, in the quotient ring of R[x] by the
ideal generated by the original polynomial, one of the earliest instances of this idea. It is
not surprising that Lagrange should employ a theoretical algebraic approach in order to
improve his numerical algorithms; much has been written about his belief that algebra is
the central analytical tool in mathematical investigations, and this philosophical approach
to mathematics is used by Lagrange to clear advantage here.
In the process he implicitly exploited algebraic congruences, one of the central concepts
of abstract algebra, 50 years before it made an explicit appearance in the work of Kummer
and Cauchy.
As Lagrange observes in Note IV, p. 146, of the Traité (Lagrange, J.-L., Traité de la
résolution des equations numériques de tous les degrés, 1798, revised in 1808, in Oeuvres de
Lagrange, J.-A. Serret, Ed. GauthierVillars, Paris, 18671892, VIII.), either method above
for finding the equation D(v) = 0 - the so called “equation of differences” - is quite laborious,
since in general its degree is quite high relative to the degree of the given polynomial whose
roots one wishes to “isolate”.
He developed two methods for finding an upper bound for the roots of D(v) without
explicitly computing it. The second builds from the first and the final version is impressive,
allowing him always to work with polynomials of degree less than the degree of f , and thus
to find an upper bound much more effectively.
Lagrange was pleased with this final approach, as one can tell from his comment, added
to the Introduction of the second edition of the Traité in 1808: “Since the first edition of
this work in 1798, different methods for resolving numerical equations have appeared; but
the rigorous solution of the problem has remained at the same place to which I carried it,
and up to now nothing has been found that could dispense with the search for a bound less
3
than the smallest difference between the roots, or which would be preferable to the means
given in Note IV for facilitating this search.” (ibid., p. 18)
The idea behind his final approach deserves our attention, since it amounts to working in
the quotient ring obtained from the polynomial ring R[x]/(f ) . Even though the techniques
of modular arithmetic were in the air at the time Note IV was published, this method is a
surprisingly early instance of implicit use of such algebraic congruences.
Algebraic congruences appear explicitly in two 1847 memoirs by A.-L. Cauchy (Mémoire
sur la théorie des équivalences algébriques, in Oeuvres complétes. Académie des Sciences,
18821981, II, 14: 93120 and Mémoire sur une nouvelle théorie des imaginaires, et sur les
racines symboliques des équations et des équivalences, in Oeuvres complétes. Académie des
Sciences, 18821981, I, 10, 312323). One goal in [Cauchy 1847b] is to construct the field of
complex numbers algebraically, as R[x]/(x2 + 1). There Cauchy builds on Kummers 1846
extension of Gauss modular arithmetic. Kummer extends the notion to congruences with
respect to an integer modulus only between polynomial forms having integer coefficients
(i.e., worked in the ring Z[x]/mZ[x] ' (Z/mZ)[x]). Cauchys construction is based on
extending these polynomial congruences to polynomial moduli, thus making explicit an
algebraic concept that Lagrange had employed implicitly almost 50 years earlier. It would
be interesting to make a more systematic study of the evolution of this central algebraic
concept during this half-century.
Here is what Lagrange says (ibid., p. 149): “Since then, it has occurred to me that one
could always eliminate the unknown x of the polynomial Y by multiplying it by a suitable
polynomial of the same degree m − 1, and make all the powers of x higher than xm−1
disappear, by means of the equation f = 0.”
1.1.2
Kronecker
Kronecker parece ter sido um dos primeiros a notar estruturas mais sofisticadas, antes da
fulgurante intervenção de Dedekind:
ELEMENTARY COMMUTATIVE ALGEBRA
(with an angle for effectiveness)
(Em preparação)
A. Simis
Further abstract structures seemed to be introduced by Kronecker, even before Dedekind
(cf. the Festchrift in honour of Kummer’s Fünfzigjahr, in Kronecker’s Gesammeltewerke,
where he mentions he long before (1850 circa) had suggested the concept to others, Dedekind
included)
Another paper by Kronecker, where he already introduces the idea of the sum of two
ideals and talks about “decomposable” ideals in the sense of being the ideal product of two
others
Kronecker foresaw the concept of prime ideal (called Primodule) - have to check carefully
whether his definition is exactly that the product of two ideals contained implies one of them
contained
4
However, though Kronecker considered only finitely generated ideals, he seemed to have
completely missed the relevance of the noetherian assumptions only later clarified
It seems that Kronecker already considered the some kind of primary or prime decomposition, but it is not clear he had the correct notion
Kronecker already uses the field concept and the integral domain concept (Rationalitätbericht and Ganzhaliggebericht, resp.). In this respect, he uses the respective notation
(S1 , S2 , . . .) and [S1 , S2 , . . .], our modern notation being reminiscent
1.2
Hilbert e a teoria dos invariantes [24], [25]
1.3
Contribuições de Lasker [38], Hurwitz [29], Macaulay [41], [42]
1.4
E. Noether [48], [49], [50], [51], e contemporâneos
(B. L. van der Waerden [79], W. Krull [36], [37] E. Artin, K. Henzelt [17])
1.5
O perı́odo “pós-noetheriano”
Noether emigrou aos EUA, voltou-se para representações de álgebras não comutativas e
para a teoria dos corpos de classes pouco antes de falecer (1935 - mesmo ano em que foi
publicado o “Idealtheorie” de Krull)
Van der Waerden tornou-se crescentemente geométrico e, mais adiante, dedicado a
questões de computação e de relatividade (com sua célebre descrição matemática dos “semivetores” de Einstein–Mayer).
Artin, também emigrado aos EUA, mergulhou na teoria dos corpos de funções de uma
variável e na teoria analı́tica dos números – legando uma das conjecturas mais profundas
sobre uma vasta generalização para L-séries da hipótese de Riemann. Suas contribuições à
álgebra comutativa, apesar de marcantes, foram esporádicas.
Henzelt há muito desaparecido (dado por morto na Primeira Guerra, em combate (?)),
mal publicara sua tese (de fato, rearrumada e publicada por Noether)
Krull após uma sequência de artigos realçando a centralidade de ideais primos em álgebra
comutativa, desviou-se no final para o estudo detalhado das valorações de Ostrowski.
O esplêndido brotar da álgebra comutativa parecia estar caminhando para uma estado
de paralização para não dizer decadência.
1.6
A fase de ouro
Esta fase consistiu, a grosso modo, das seguintes vertentes:
1. Aprofundamento da teoria de módulos (em contraposição à sua consideração como
entes abstratos satisfazendo propriedades gerais, imitando ideais inteiros ou fracionários)
2. Métodos homológicos (colocando módulos como objeto central de uma teoria homológica e não só como linguagem cômoda para enunciar construções homológicas)
3. Invariantes de ideais individuais (em contraposição à consideração de todos os ideais
de um anel, sem distinção de sua especificidade com objeto em si)
5
Bourbaki (Cartan, Serre)
Cartan–Eilenberg
Auslander–Buchsbaum
2
O legado de D. Rees
2.1
Álgebra simétrica versus álgebra de Rees
Rees used the algebra that bears his name for the purpose of studying analytic properties of
a set of generators of an ideal in a local ring. Actually, Rees used the nowcalled “extended”
Rees algebra (sometimes called the Rees ring), namely, R[It, t−1 ]. The reason was that this
algebra is a deformation of the associated graded ring of the ideal (originally introduced by
Krull).
Nearly at the same time, the French school (enhanced by the Grothendieck program)
was using the “ordinary” Rees algebra as the corresponding object of the geometric blow-up
process used in resolution of singularities.
It is possible that even earlier versions were made available. It would be interesting to
find out who was the first to clearly pin-point these algebras.
2.1.1
Primeiras manifestações
Review by M. Nagata of one of the first results on symmetric algebras by A. Micali:
“Let A be a commutative ring having 1, and let M be an A-module. The author considers
SA (M ), the symmetric algebra of M (algèbre symétrique de M ) (No definition of this is
given, but it appears to be the most general commutative ring generated by M over A. The
reviewer takes exception to the use of the term, because SA (M ) has no special relationship
with the usual notion of a symmetric algebra, which is a generalization of a group algebra of
a finite group.) Let ϕ be a homomorphism defined on M into a second A-module M 0 . Then
ϕ induces a homomorphism from SA (M ) into SA (M 0 ), and this homomorphism is denoted
by SA (ϕ). The author gives some conditions for SA (ϕ) to be injective, in the case where A
is an integral domain and ϕ is a linear map from M into A or a direct sum of such linear
maps (which maps M into a direct sum of a finite number of copies of A). At the end of
the second paper, the author proves that a local ring A with maximal ideal m is regular if
and only if SA (m) is an integral domain.”
The definition of the Rees algebra of a module we will use below was first given by
Micali ([46], [47]). From a review by P. Samuel of [loc. cit.]:
“Etant donnés un anneau commutatif A et un A-module M , soit S l’ensemble des
non-diviseurs de zéro de A; le noyau t(M ) de M → S −1 M s’appelle le sous-module de
torsion de M . On appelle algèbre de Rees R(M ) de M la solution du problème universel
relatif aux applications A-linéaires de M dans les A-algèbres (commutatives) E telles que
t(E) = 0. On construit cette algèbre comme quotient de l’algèbre symétrique S(M ); elle
est munie d’une structure graduée. C’est un foncteur en M ; si M → M 0 est injective,
alors R(M ) → R(M 0 ) est injective. On tudie le comportement du foncteur R par limites
inductives, somme directe, et changement d’anneau. ...L’auteur compare ensuite l’algèbre
de Rees ici dfinie avec l’algèbre de Rees relative au systme de toutes les formes linéaires sur
M .”
6
From a recent paper ([68]):
“ Let R be a Noetherian ring and let E a finitely generated R–module having a rank.
The Rees algebra R(E) of E is S(E) modulo its R-torsion submodule.
If E is a submodule of a free R-module G, some authors define the Rees algebra of
E to be the image of the natural map S(E) → S(G). The two definitions coincide if the
assumptions overlap, i.e., for a finitely generated torsionfree module having a rank (i.e., for
a finitely generated module E such that K ⊗R E is K-free and the R-map E → K ⊗R E
is injective) as in this situation the kernel of S(E) → S(G) is the R-torsion submodule
of S(E). The present definition depends solely on E, not on any particular embedding
of E. Moreover, R(E) thus defined has the obvious universal property with respect to
R-homomorphisms E → B into a torsionfree R-algebra B.”
From another recent paper ([12]):
“In this paper we show that the Rees algebra can be made into a functor on modules
over a ring in a way that extends its classical definition for ideals. The Rees algebra of a
module M may be computed in terms of a “maximal” map f from M to a free module as
the image of the map induced by f on symmetric algebras.”
The latter is somehow related, unadvertedly, to the first definitions in the sixties. None of
the proposed definitions work all the way through all questions of uniqueness or invariance.
Adopting one definition or another is a question of taste and initial assumptions or final
objectives.
2.1.2
O caso de um ideal
We first consider the case of an ideal I ⊂ R. In this case, there is the ordinary blowup
algebra ⊕n≥0 I n ' R[It] ⊂ R[t] and a canonical surjective R-algebra homomorphism
α
S(I) −→ R[It].
(1)
The ideal I is said to be linear type if the map (1) is injective. The basic model of an ideal
of linear type is given by an ideal generated by a regular sequence.
Proposition 1 If I is an ideal of linear type then µ(IP ) ≤ ht P for every P ∈ Spec (R/I).
Proof. Since both algebras commute with localization we may assume that R, m is local
and that I ⊂ m. But then
µR (I) = µR/m (I/mI) = dim(SR (I)/mSR (I)) ≤ dim(SR (I)/ISR (I)) = dim grI (R),
the latter equality since I is of linear type. But, in the local case dim grI (R) = dim R.
The following proposition is an easy consequence of the general dimension formula of
Huneke–Rossi [28]. However, we give an independent simple proof.
Proposition 2 If grade I ≥ 1 then dim S(I) = max {dim R + 1, dim S(I/I 2 )}.
7
Proof. Consider the canonical map (1): for every P ∈ Spec (R) such that I 6⊂ P , the
induced localization αP : S(I)P → RP [IP t] is clearly an isomorphism. This shows that
I t ker α = (0) for t >> 0, hence every prime ideal of the ring S(I) contains either IS(I) or
ker α. Therefore,
dim S(I) = max {dim R[It], dim SR/I (I/I 2 )}
= max {dim R + 1, dim SR/I (I/I 2 )},
since dim R[It] = dim R + 1 for ideals containing regular elements.
Corollary 3 Let R, m be a local ring. If I ⊂ R is an m-primary ideal then dim S(I) =
max {dim R + 1, µ(I)}.
Proof. Since Sup(S(I/I 2 )) = Sup(S(I/mI)), the result follows from Proposition 2.
Ideals satisfying the formula of Corollary 3 were referred to by this author as being of
Valla type (cf. [57]).
Exercise 4 Let A = k[X, Y, Z](X,Y,Z) , n = (X, Y, Z)(X,Y,Z) and let J ⊂ A be n-primary
with 5 generators. Let T be an indeterminate over A, let R = A[T ](n,T ) , m = (n, T )R and
let I = JR. Then dim SR (I) = dim SA (J) + 1, thus showing that the ideal I is not of Valla
type.
Determinantal ideals also fail to be of Valla type, except for special row and column
sizes [28].
Ideals of linear type are also important because of the next seemingly innocent result.
Proposition 5 Let A be a noetherian ring and let I ⊂ A be an ideal of linear type. Then
dim A ≥ sup {dim A/P + µ(IP )}.
P ⊇I
Proof. It follows straightforwardly from Proposition 1.
Of course, one has to impose further conditions in order to obtain equality above. We
say that an ideal I ⊂ A is a junction ideal if there is a P ∈ Min(A/I) such that a maximal
chain of primes of A passes through P . The notion is perhaps only interesting if dim A < ∞.
Theorem 6 Let A be a noetherian ring of finite dimension admitting a junction ideal I of
linear type. Then
dim A = sup {dim A/P + µ(IP )}.
P ⊇I
Proof. By Proposition 5, it suffices to show that dim A = dim A/P +µ(IP ) for some P ⊇ I.
Let P ∈ Min(A/I) be such that dim A = dim A/P + ht P . Then ht P = ht IP ≤ µ(IP ) by
general reasons. Since I is of linear type, one must have ht P = µ(IP ).
We are now ready to give a proof of the Huneke–Rossi formula.
8
Theorem 7 ([28]) Let R be a noetherian ring and let E be a finitely generated R-module.
Then
dim S(E) =
sup {dim R/P + µ(EP )}.
P ∈ Spec (R)
Proof. Since S(E) is graded with R as its zero part, we won’t affect its dimension by passing
to the ring of fractions A := S(E)1+S(E)+ . The extended ideal I := (S(E)+ )S(E)1+S(E)+ is
clearly contained in the (graded) Jacobson radical of A. One can verify that I is a junction
ideal.
On the other hand, S(E)+ is an ideal of linear type [20, Example 2.3], hence so is I.
Thus, we can apply Theorem 6 by further noticing that
1. Any Q ∈ Spec (S)(E) containing S(E)+ is of the form (P, S(E)+ ) for some P ∈
Spec (R)
2. S(E)/S(E)+ ' E.
The rest is standard.
Exercise 8 Show that, conversely, the Huneke–Rossi formula implies the formula stated
in Theorem 6 for ideals contained in the Jacobson radical.
2.2
2.2.1
Álgebras de Rees e simetrizações do complexo de Koszul
Complexos simetrizados e de aproximação
Let R be a commutative ring, let ψ : G → F be a map of R-modules and let M be a
third R-module. The basic construction of the theory is the so-called Koszul complex of ψ
with coefficients in M , denoted K(ψ, M ): the components of the complex are the modules
∧r G ⊗R St (F ) ⊗R M and the differential is
∧r G ⊗R St (F ) ⊗ M
→ ∧r−1 G ⊗R St+1 (F ) ⊗ M
P
g1 ∧ . . . ∧ gr ⊗ f ⊗ m 7→
(−1)i g1 ∧ . . . ∧ gˆi ∧ . . . ∧ gr ⊗ ψ(gi ) · f ⊗ m.
Now, one reason of the ubiquity of this complex is the following.
Lemma 9 Let F, G be free modules of finite rank and let E := coker ψ (i.e., ψ is a presentation map of E). Then:
(i) K(ψ, R) has a natural structure of graded complex over S(F ) and, as such, it is a
direct sum of R-complexes
Kt : 0 → ∧q G ⊗ St−q (F ) → . . . → ∧1 G ⊗ St−1 (F ) → St (F ),
with q = min {t, rank G} and H0 (Kt ) ' St (E).
P
(ii) As a graded complex defined over the polynomial ring S(F ) = t St (F ), K(ψ, R) is
isomorphic with the (ordinary) Koszul complex attached to a set of generators of the
presentation ideal J(ψ) ⊂ S(F ) of S(E) coming from the presentation map ψ. As
such, K(ψ, R) is a complex such that H0 (K(ψ, R)) ' S(E).
9
Proof. (i) As a graded complex over the polynomial ring S(F ), K(ψ, R) is the direct sum
of the complexes
0
0
→ G ⊗ S0 (F )
0
→ ∧2 G ⊗ S0 (F ) → G ⊗ S1 (F )
3
0 → ∧ G ⊗ S0 (F ) → ∧2 G ⊗ S1 (F ) → G ⊗ S2 (F )
..
..
..
.
.
.
→
→
→
→
S0 (F )
S1 (F )
S2 (F )
S3 (F )
..
.
→
→
→
→
0
0
0
0
Therefore, K(ψ, R) is of the form
0 → ∧m G ⊗ S(F )(−m) → . . . → ∧2 G ⊗ S(F )(−2) → G ⊗ S(F )(−1) → S(F ) → 0.
P
It is not difficult to see that H0 (Kt ) ' St (E). Clearly, then H0 (K(ψ, R)) ' t≥0 St (E) =
S(E).
(ii) Let {g1 , . . . , gm } be a basis of G. Then G⊗R S(F ) is a free S(F )-module on the generators
{g1 ⊗ 1, . . . , gm ⊗ 1}. On the other hand, J(ψ) ⊂ S(F ) is, by definition, generated by the
polynomials
S1 (F ) · ψ(g1 ), . . . , S1 (F ) · ψ(gm ),
where · denotes matrix product. Therefore, we have a map of free S(F )-modules
Ψ
G ⊗R S(F ) → S(F ), gj ⊗ 1 7→ S1 (F ) · ψ(gj ), 1 ≤ j ≤ m.
If we now consider the ordinary Koszul complex K(Ψ, S(F )) associated to the map ψ, then
there is an isomorphism of complexes over S(F )
K(ψ, R) :
k
K(Ψ, S(F )) :
0
→
0
→
(∧m G) ⊗ S(F )(−m)
k
∧m (G ⊗ S(F ))(−m)
→
...
→
→
...
→
G ⊗ S(F )(−1)
k
(G ⊗ S(F ))(−1)
→
→
S(F )
k
S(F )
→
0
→
0
We leave the details as a rewarding exercise.
We now specialize a bit our settting, to wit, we assume that F = G and that the given
map ψ : F → F is the identity map. Clearly, in this case K(id, R) is isomorphic to the
ordinary Koszul complex on the variables of S(F ) (i.e., the generators of the irrelevant ideal
S(F )+ ), which is then obviously acyclic.
But now assume, moreover, one is given a second map ϕ : F → R. Then ∧F ⊗ S(F ) is
a double complex
..
.
↓
. . . → ∧r F ⊗ St (F ) → ∧r−1 F
↓
r−1
. . . → ∧ F ⊗ St (F ) → ∧r−2 F
↓
..
.
10
..
.
↓
⊗ St+1 (F ) → . . .
↓
⊗ St+1 (F ) → . . .
↓
..
.
where the horizontal subcomplexes are acyclic and the vertical ones come from the ordinary
Koszul complex K(ϕ, R) by “symmetrization” (i.e., by extending coeffcients to S(F )).
Here, a typical square composed of horizontal and vertical differentials is skew-commutative. If we denote the cycles of K(ϕ, R) by Z(ϕ, R) then, from the (skew-) commutativity
of the maps and the fact that extending coefficients by a flat map is left-exact, we see that
the horizontal complexes induce subcomplexes
Zr+t : . . . → Zr (ϕ, R) ⊗ St (F ) → Zr−1 (ϕ, R) ⊗ St+1 (F ) → . . . .
Therefore, we obtain a graded complex Z(ϕ; R) of the form
0 → Zm (ϕ, R) ⊗ S(F )(−m) → . . . → Z1 (ϕ, R) ⊗ S(F )(−1) → S(F ) → 0.
This complex has been dubbed the Z-complex attached to ϕ (or to its image in R). There
are similar versions using the boundaries and the homology of the complex K(ϕ, R), as we
now indicate.
Proposition 10 Let Br = Br (ϕ, R) (resp. Hr = Hr (ϕ, R)) denote the boundaries (resp.
the homology) in degree r of the complex K(ϕ, R).
(i) There are graded complexes B(ϕ; R) and M(ϕ; R) of the form, respectively
0 → Bm (ϕ, R) ⊗ S(F )(−m) → . . . → B1 (ϕ, R) ⊗ S(F )(−1) → (ϕ(F ))S(F ) → 0
and
0 → Hm (ϕ, R) ⊗ S(F )(−m) → . . . → H1 (ϕ, R) ⊗ S(F )(−1) → S(F )/(ϕ(F )S(F ) → 0.
(ii) The length of the M-complex, as defined in (i), is usually much shorter.
(iii) Let I = (ϕ(F )) ⊂ R. Then
H0 (Z(ϕ, R)) ' S(I),
H0 (B(ϕ, R)) ' IS(I),
H0 (M(ϕ, R)) ' S(I/I 2 ).
Remark 11 All these complexes can be defined with coefficients in an R-module. We leave
the details to the reader.
The main relation between the complexes Z = Z(I; R) and M = M(I; R) is as follows.
Proposition 12 For every integer r ≥ 0 there is a long exact sequence in homology
. . . −→ Ht (Zr+1 ) −→ Ht (Zr ) −→ Ht (Mr ) −→ Ht−1 (Zr+1 ) −→ . . .
Corollary 13 The following assertions are equivalent:
(i) M is acyclic.
(ii) Z is acyclic and λ is injective.
11
Theorem 14 Let I ⊂ A be an ideal and let M and Z denote the corresponding associated
complexes as above. Consider the following comumtative diagram with exact rows:
σ
λ
H1 (Mt ) −→ H0 (Zt+1 ) −→ H0 (Zt ) −→ H0 (Mt ) −→ 0
.
↓αt+1
↓αt
↓β t
It
t+1
t
−→ 0
0
−→
I
−→
I
−→
I t+1
The following conditions are equivalent:
(i) σ is the zero map.
(ii) λ is injective.
(iii) α is an isomorphism.
In particular, if the M-complex is acyclic then I is an ideal of linear type. Huneke has
shown that M-complex is acyclic if and only if I is generated by a d-sequence.
curiously, we also have:
Proposition 15 With the previous notation, α is an isomorphism if and only if β is an
isomorphism.
The following condition on the depth of the Koszul homology has been known as sliding
depth [19] (but cf. [71] and [20] for earlier appearances).
Thus, one is given a noetherian ring R and a map ϕ : F → R, where F is a free
R-module. Let a be a set of generators of the ideal (ϕ(F )).
Definition 16 The set a (or the map ϕ) is said to satisfy the sliding depth condition if
depth Hi (K(a, R)) ≥ dim R − rank F + i,
(2)
for all i ≥ 0.
Sliding depth with coefficients is defined in the same manner if the complex has coefficients in a module, with no change in the right hand side of (2).
Remark 17 (i) (Invariance [19]) Although the homology of the Koszul complex varies while
the set of generators of I = (ϕ(F )) ⊂ R changes, it does so in a stable fashion sufficient to
imply that the sliding depth condition is independent of the chosen generators. This means
that, if R is local then rank F may be replaced by µ(I) in the right hand side of (2) as soon
as we replace a by a minimal generating set of I.
(ii) (Localization [19]) If R is a Cohen–Macaulay local ring then the sliding depth condition
implies the same condition locally at every prime of R. Thus, taking in account (i), one
would have similarly
depth Hi (K(b, R))P ≥ dim RP − µ(IP ) + i,
for every P ∈ Spec (R), where b is a minimal generating set of the ideal I.
12
Put together in a single statement, the facts in Remark 17 amount to saying that if
sliding depth holds locally at every maximal ideal of a locally Cohen–Macaulay ring R then
it holds locally at every prime ideal of R.
Next is the main application of the M-complex. It explains why it is called an “approximation ” complex - its homology measures the failure of the natural map S(I) → R(I) to
be injective.
Theorem 18 ([20, Theorem 9.1]) Let R be a locally Cohen–Macaulay noetherian ring, let
ϕ : F → R be a map and let I = (ϕ(F )). Suppose the following conditions hold:
(i) I satisfies sliding depth locally at every maximal ideal of R.
(ii) µ(I℘ ) ≤ dim R℘ for every ℘ ∈ Spec (R)).
Then
(a) The M-complex M(ϕ, R) is acyclic (hence, so is the Z-complex Z(ϕ, R)).
(b) H0 (M(ϕ, R)(' SR/I (I/I 2 )) ' gr I (C) (hence also H0 (Z(ϕ, R)(' SR (I) ) ' RR (I)).
(c) If R/I is a Cohen–Macaulay ring then gr I (C) is a Cohen–Macaulay ring.
The theorem has a more general version with coefficients on an R-module ([58, Theorem
4.7]). The theory also extends, non-trivially, to the case of a certain Z-complex Z(E)
associated to an R-module - “on the nose”, not as coefficients! This was studied to a large
extent in [21] (cf. also [77] for a more updated account and complete references).
2.2.2
Aplicações recentes
From the abstract of a recent paper ([5]; cf. also [6]):
“In this paper, we investigate some topics around the closed image S of a rational map
λ : Proj(A) → Pn−1 given by homogeneous elements {f1 , . . . , fn } of the same degree in a
graded algebra A. We first compute the degree of this closed image in case λ is generically
finite and {f1 , . . . , fn } define isolated base points in Proj(A). We then relate the definition
ideal of S to the symmetric and the Rees algebras of the ideal I = (f1 , . . . , fn ) ⊂ A, and
prove some new acyclicity criterions for the associated approximation complexes. Finally, we
use these results to obtain the implicit equation of S in case S is a hypersurface, Proj(A) =
Pn−2 over a field and base points are either absent or local complete intersection isolated
points.”
The first task the above authors set to themselves is to obtain a formula for the degree
of the hypersurface in question. This is the main result of Chapter 2 in [loc. cit.]. This sort
of formula actually comes naturally from some more general inequalities by imposing some
further constraints, according to some of the results obtained in [67, Section 6] by different
methods. Namely, one has the following
13
Theorem 19 Let R be a reduced and equidimensional standard graded Noetherian ring of
dimension d > 0 with R0 = k a field and let I be an R–ideal of height g generated by forms
of degree s > 0. Introduce the following ingredients:
• P denotes the set of primes p in k[Is ] ⊂ k[Rs ] ⊂ R with dim k[Is ]/p = dim k[Is ],
• R denotes the set of primes p in V (I) ∩ Proj(R) with dim R/p = dim R/I,
• An integer r ≥ 1 is given such that rank k[Is ]p k[Rs ]p ≥ r for every p ∈ P.
The following hold:
(a) If `(I) = d then


e(k[Is ]) ≤ e(R)sg −
X
p∈R
This inequality is an equality if rank
in case R is a domain.
sd−g−1
eIp (Rp ) e(R/p)
.
r
k[Is ] k[Rs ]
= r and d ≤ g + 1. The converse holds
(b) If `(I) = d 6= g and for every p ∈ R, either Ip is prime or Rp is Cohen–Macaulay,
then
sd−g−1
e(k[Is ]) ≤ (e(R)sg − e(R/I))
.
r
This inequality is an equality if rank k[Is ] k[Rs ] = r, d = g + 1 and Ip is a complete
intersection for every p ∈ R. The converse holds in case R is a domain.
(c) If `(I) 6= g (i.e., I is not equimultiple), R is a domain and, again, for every p ∈ R,
either Ip is prime or Rp is Cohen–Macaulay, then the same inequality as in (b) holds
with r replaced by degK (K ⊗ k[Is ] k[Rs ]), where K is the field of fractions of k[Is ].
At first sight, the theorem does not look like any of the geometry above, but at a closer look
it does. Thus, one ought to think about R as the coordinate ring of an equidimensional
projective algebraic set X ⊂ Pn and k[Is ] ⊂ R as the homogeneous coordinate ring (up to
degree normalization) of the image of the rational map λ : X 99K Pm defined by the forms
generating I. The locus of base points of λ is given by Proj(R/I) whose codimension in X
is denoted g. The analytic spread `/I) is equal to dim Y + 1, where Y is the (closure) of
the image of λ.
The sets P and R correspond, respectively, to the components of Y and of Proj(R/I)
of maximal dimension. Finally, rank k[Is ]p k[Rs ]p corresponds to the degree of the map λ at
a generic point of Y . Of course, one could always take r = 1 but the inequality improves as
r gets bigger. Note that equality both in (a) and in (b) requires that r be the sum of these
generic degrees. This is indeed the usual degree of the map in case (c), where X (hence
also Y ) is integral.
Note that the equality both in (a) and (b) require that g ≥ d − 1 which means that
the base point locus Proj(R/I) has dimension zero (finite set of points) or is empty (λ is
regular). To this moment, there are no clearcut theoretical results in case the base locus
has positive dimension (we state this at the end as one of the open problems).
A good application of the above formulas is to the so-called Teissier–Plücker formula.
Namely, let k be an algebraically closed field and let X ⊂ Pnk (n ≥ 2) be a reduced and
14
0
irreducible hypersurface of degree e > 1. Assume that the dual variety X 0 ⊂ Pnk is again a
hypersurface (i.e., X 0 is non-deficient). Notice that X 0 is also the image of the Gauss map
of X ⊂ Pnk . Let r be the degree of this map and let R denote the (possibly empty) set of
irreducible components of Sing(X) of maximal dimension. Let f be the irreducible homogeneous polynomial defining X. Set R = k[X0 , . . . , Xn ]/(f ), I = (∂f /∂X0 , . . . , ∂f /∂Xn )R
(the jacobian ideal of R) and I for the ideal sheaf corresponding to the homogeneous ideal
I. Finally, let g = ht I (the codimension of Sing(X) in case X is singular). Now, since
0
the special fiber k ⊗R R(I) is the homogeneous coordinate ring of X 0 ⊂ Pnk , Theorem 19.a
yields the following inequality
!
X
1
0
n−1
n−g−1
deg(X ) ≤
e(e − 1)
−
eIx (OX,x ) deg(x) (e − 1)
,
r
x∈R
with equality holding if and only if X has at most isolated singularities.
We next allow X to be reducible. It is known that in characteristic zero the Gauss map
of a hypersurface with non-deficient dual has degree one ([32, Theorem 4], cf. also [52,
Proposition 3.3]). If in addition X has at most isolated singularities then the formula of
Theorem 19(a) specializes to the formula proved by Teissier ([33], [39], [52], [74]):
X
deg(X 0 ) = e(e − 1)n−1 −
eIx (OX,x ).
x∈Sing(X)
2.3
2.3.1
Avanços recentes no caso de módulos
Variações iniciais
The main object of this portion is defined as follows. Let E be an R-module, where R is
an arbitrary ring. We assume throughout that M has a (generic) rank. (Recall that the
module E is said to have (generic) rank, say, r, if Ep is Rp -free of rank r for every prime
p ∈ Ass(R)/I. If this is the case, we denote rank E = r.)
The Rees algebra of E is the residue algebra of the symmetric algebra S(E) by its Rtorsion A. We denote it by R(E). The module E is said to be of linear type if A = 0, i.e.,
if the canonical map S(E) R(E) is an isomorphism.
One of the reason to look at such objects is that they give a handy algebraic way of
considering certain incidence correspondences in biprojective space.
We start with the following basic result.
Proposition 20 Let R be a noetherian ring of finite Krull dimension. Let E be a finitely
generated module R-module with rank. Then dim R(E) = dim R + rank E.
Proof. (1) dim R(E) ≥ dim R + rank E.
First observe that R(E) is a graded R-algebra with a grading naturally inherited from
S(E). Now, R(E)/R(E)+ ' R implies the inequality dim R(E) ≥ dim R + ht R(E)+ , so it
suffices to show that ht R(E)+ = rank E. Thus, let P ⊃ R(E)+ be a prime in R(E) such
that ht P = ht R(E)+ . We have P = (℘, R(E)+ ) where ℘ = P ∩ R ∈ Min R. Therefore, by
the rank hypothesis,
R(E)P = (RR℘ (E℘ ))P ' R℘ [T](℘℘ ,T) ,
15
where #T = rank E. This shows that
ht R(E)+ (= ht P = ht PP = dim R(E)P ) = dim R℘ + rank E = rank E,
as required.
(2) dim R(E) ≤ dim R + rank E.
Let P ⊂ R(E) be a minimal prime with dim R(E) = dim R(E)/P. Set P ∩R = ℘. Applying
the dimension inequality ([44, Theorem 15.5]) to the extension of domains R/℘ ⊂ R(E)/P,
we obtain:
dim R(E) = dim R(E)/P ≤ dim R/℘ + trdeg R/℘ R(E)/P
≤ dim R/℘ + dim(R(E)/P) ⊗R R℘ /℘℘ ≤ dim R/℘ + dim R(E) ⊗R R℘ /℘℘
≤ dim R/℘ + dim R(E) ⊗R R℘ = dim R/℘ + dim RR℘ (E℘ )
(3)
= dim R/℘ + dim SR℘ (E℘ ) = dim R/℘ + dim R℘ [T]
(4)
= dim R/℘ + ht ℘ + rank E ≤ dim R + rank E.
Note that to pass from (3) to (4) one uses that ℘ ⊂ q for some q ∈ Ass(R) since R(E) is
R-torsionfree and hence E℘ = (Eq )℘q is R℘ -free.
We specify the following settings throughout this section: (R, m) is a local ring and its
maximal ideal or a standard graded ring R = k[R1 ] and its irrelevant ideal m = (R1 ) = (R+ ).
All R-modules will be finitely generated (and graded if R is graded).
Definition 21 The analytic spread of E is `(E) = dim R(E)/mR(E).
Germs of this notion appear in [59, §2.4] and [34]. A more thorough account appears in
[68]. If E happens to be a torsionfree module of rank one and an embedding of E is given
as an ideal I ⊂ B containing a nonzerodivisor then the present notion coincides with the
usual notion of analytic spread `(I).
Since R(E)/mR(E) is a residue ring of S(E)/mS(E), we get immediately the upper
bound `(E) ≤ ν(E). In the tradition of ideals we search for yet another upper bound.
Corollary 22 If E has a rank then `(E) ≤ dim R + rank E − 1 (resp. `(E) = rank E) if
dim R ≥ 1 (resp. if dim R = 0).
Proof. The case where dim R = 0 is clear since E is then automatically free. Let dim R ≥ 1.
The result is immediate if E is free since then R(E) is a polynomial ring over R in rank E
variables and the extension of the maximal ideal has height at least one. Thus, assume
that E is not free. We still claim that mR(E) has height at least one. Indeed, a minimal
prime of R(E) containing m would have to contract to m. But since R(E) is torsionfree,
m would necessarily be an associated prime of R, hence E = Em would be free. In any
case, the result follows immediately from Proposition 20.
R(E) has a kind of universal mapping property parallel to that of S(E), to wit, given a
torsionfree R-algebra T and an R-module homomorphism E → T there exists a unique Ralgebra map R(E) → T making the obvious diagram commutative. Alas, torsion behaves
quite badly with respect to base change R → R0 . As a result, a drawback of the Rees
algebra, in contrast to the symmetric algebra, is that it is not quite functorial (i.e., it does
not commute) with respect to base change. To remedy this situation, we prove the following
result.
16
Proposition 23 Let R be a noetherian ring, let E be a finitely generated R-module and
let I ⊂ R be an ideal such that Ep is of linear type for every p ∈ Ass(R/I). Then
the canonical surjection SR (E) SR/I (E/IE) descends to a map SR (E)/τR (SR (E)) SR/I (E/IE)/τR/I (SR/I (E/IE)).
Proof. Let A = τR (SR (E)). We show that the image of A under the map SR (E) SR/I (E/IE) is contained in the R/I-torsion submodule of SR/I (E/IE). Let then f ∈ A.
It suffices to show that (0 : f ) ∩ R 6⊂ ℘ for every ℘ ∈ Ass(R/I), where the annihilator is
taken in SR (E). By assumption, A℘ = (0) for every ℘ ∈ Ass(R/I), i.e., ℘ 6∈ Sup() R(A) =
Sup(()R/(0 : A) ∩ R) for every ℘ ∈ Ass(R/I). Therefore, (0 : A) ∩ R 6⊂ ℘, hence also
(0 : f ) ∩ R 6⊂ ℘ for every ℘ ∈ Ass(R/I).
For us the preferred situation is when both E and E/IE have ranks over R and
R/I, respectively. In this case, under the assumption of the proposition, there is a map
RR (E)/IRR (E) RR/I (E/IE). Note that if both E and E/IE have ranks over R and
R/I, respectively, one has rkR/I E/IE ≥ rkR E, since if ℘ ∈ Ass(R/I) then
rkR/I E/IE = νR℘ /I℘ (E℘ /I℘ E℘ ) = νR℘ (E℘ ) ≥ rank E.
A typical such case is when E = I ⊂ R is an ideal of positive grade which is generically
a complete intersection of fixed codimension.
This in turn allows us to express the Rees algebra of the conormal module I/I 2 .
Corollary 24 Let R be a noetherian ring and let I ⊂ R be an ideal of positive grade which
is generically a complete intersection of fixed codimension. Then there is an isomorphism
of R/I-algebras
gr I (R)
R(I/I 2 ) '
.
R/I − torsion
Proof. Applying Lemma 23, along with the fact that both I and I/I 2 have ranks over
their respective rings, we get a surjective map gr I (R) R(I/I 2 ). Since the Rees algebra
is torsionfree, this map induces yet another surjective map
gr I (R)/(R/I − torsion) R(I/I 2 ).
(5)
Now the canonical map SR/I (I/I 2 ) gr I (R) induces naturally a map
SR/I (I/I 2 )/(R/I − torsion) gr I (R)/(R/I − torsion).
Therefore, we get a map going in the opposite direction of (5) which is clearly an inverse.
Corollary 25 Let (R, m) be a noetherian local ring or a positively graded ring and its
graded maximal ideal and let I ⊂ R be an ideal of positive grade which is generically a
complete intersection of fixed codimension. Set G = gr I (R). Then `(I) ≥ `(I/I 2 ) =
dim G/(mG + (R/I − torsion)).
17
Proof. By Corollary 24, we get an isomorphism
G
R(I/I 2 )
'
.
mR(I/I 2 )
mG + (R/I − torsion)
Since `(I) = dim G/mG, everything is clear.
In a special situation one can nearly get a hold of R(E). Namely, let EP
be a torsionfree
r
A-module with a rank along with a given embedding E ⊂ Rr . Let E P
= m
j=1 Rvj ⊂ R .
Let t = t1 , . . . , tt stand for the canonical basis of Rr , so that vj =
i aij ti ∈ R[t] for
suitable aij ∈ R. This choice of generators of E and of a basis of Rr induces a surjective
R-algebra map S(E) R[v1 , . . . , vm ] ⊂ R[t]. By assumption this map is an isomorphism
generically on R, hence its kernel is contained in the R-torsion of S(E) (since the R-torsion
is the kernel of the canonical map S(E) → S(U −1 E), where U = R \ ∪℘∈Ass(R) ℘). This
shows that there is an R-algebra isomorphism R(E) ' R[v1 , . . . , vm ].
We wish to dig further into the structure of the algebr R(E)/mR(E). Sticking to the
graded case, but relaxing the proviso on R, we assume that R = R0 [R1 ], where R0 is a
noetherian ring. Set m = (R1 ) = (R+ ). Let E stand for a graded submodule of Rr , the
latter being equipped with the usual grading inherited from R.
The following definition is basic throughout.
Definition 26 The special fiber (or the fiber cone) of the Rees algebra R(E) is the residue
algebra F(E) := R(E)/mR(E).
Proposition 27 If E is, moreover, generated by homogeneous vectors of a fixed degree
then F(E) has a structure of graded R0 -algebra and is graded-isomorphic to a graded R0 subalgebra of R(E) making F(E) into a direct summand of R(E) as F(E)-module.
Proof. Let v1 , . . . , vm be homogeneous vectors of fixed degree d of Rr generating E. As
above, letting t denote the canonical basis of Rr , R(E) gets identified with R[v1 , . . . , vm ] ⊂
R[t]. Clearly, R(E) has a natural bigrading as R0 -algebra. From this, R(E) has a structure
of graded R0 -algebra (by taking “total” degrees). Since (R+ )R(E) is a homogeneous ideal
for this grading, R(E)/(R+ )R(E) is a graded R0 -algebra as well.
Now, one has
X
R(E) = R ⊕ (Rv1 + . . . + Rvm ) ⊕ (
R(vj vk ) ⊕ . . . .
j,k
Using the bigrading and the fact that the R-algebra generators are of fixed bidegree (d, 1),
one sees that
R[v1 , . . . , vm ]/(R+ )R[v1 , . . . , vm ] ' R0 [v1 , . . . , vm ]
as graded R0 -algebras.
Finally, the composite map
R0 [v1 , . . . , vm ] → R(E) → R(E)/(R+ )R(E)
is the identity map. The contention follows from this by general considerations.
18
Remark 28 With the above definition of analytic sprea one cannot retrieve the case where
R is local. A variation of the definition is to assume that R0 is itself local, with maximal
ideal n, and take m to mean the maximal graded ideal (n, R1 ) ⊂ R. This has the advantage
of including the local case (with R1 = (0)). Moreover, the definitions are the same in the
standard case R = k[R1 ] over a field k.
Corollary 29 Under the hypothesis of Proposition 27, if R is a domain then F(E) is a
domain. Moreover, if R(E) is a normal domain then so is F(E).
Proof. The domain part of the assertion follows immediately from proposition 27, (2),
while the normality argument is a consequence of the “Reynolds operator” in part (3) of
that same proposition.
2.3.2
O método do ideal genérico de Bourbaki
The material of this and the subsequent subsections is taken from [68].
By a Bourbaki ideal of a module E we mean an ideal I fitting into a Bourbaki sequence
0 → F −→ E −→ I → 0,
with F a free module. This concept was originally considered in the context of torsionfree
modules over a normal domain. Here we develop a sort of generic such ideal, nearly uniquely
defined.
Thus, let (R, m) be a Noetherian local ring. If Z is a set of indeterminates over R, we
denote by R(Z) the localization R[Z]mR[Z] . Let E a finitely generated R–module having a
rank e > 0 and let a1 , . . . , an be a set of generators of E. Further let zij , 1 ≤ i ≤ n, 1 ≤
j ≤ e − 1 , be indeterminates
R00 = R({zij }),
E 00 = R00 ⊗R E,
xj =
n
X
zij ai ∈ E 00 ,
i=1
Pe−1
and F = j=1 R00 xj , a free R00 –module of rank e − 1. Assume that E 00 /F is torsionfree
over R00 (which holds, e.g., if E is a torsionfree module which is free locally in depth one).
In this case E 00 /F ∼
= I for some R00 –ideal I with grade I > 0.
Definition 30 We call an R00 –ideal I with I ∼
= E 00 /F a generic Bourbaki ideal of E and
denote it by I(E).
Remark 31 (a) A generic Bourbaki ideal of E with respect to U is essentially unique.
Indeed, suppose I ⊂ R(Z) and K ⊂ R(Y ) are two such ideals defined using generating
sequences a1 , . . . , an and b1 , . . . , bm of E, and sets of variables Z = {zij | 1 ≤ i ≤ n, 1 ≤
j ≤ e − 1} and Y = {yij | 1 ≤ i ≤ m, 1 ≤ j ≤ e − 1}, respectively; then there exists an
automorphism ϕ of the R–algebra S = R(Y, Z) and a unit u of Quot(S) so that ϕ(IS) =
uKS. Furthermore, u = 1 if I and K have grade ≥ 2.
(b) If E ∼
= Re−1 ⊕ L, for some R–ideal L, then LR00 = I(E).
(c) Conversely, if I = I(E) has grade ≥ 3, then E ∼
= Re−1 ⊕ L for some R–ideal L.
(d) I(E) can be obtained explicitly from a presentation matrix of E. Let R be a
Noetherian local ring, let E a finitely generated R-module having a rank e > 0, and let
19
Pe−1 00
F =
j=1 R xj be defined as above. Extend the basis of F to a generating sequence
x1 , . . . , xm of E 00 and let ϕ be a matrix presenting E 00 with respect to these generators.
Finally let ψ be the m − e + 1 by m − e submatrix of ϕ consisting of the last rows and
columns of ϕ, and write I = Im−e (ψ). After elementary column operations on ϕ one may
assume that grade I > 0. Now I is indeed a generic Bourbaki ideal of the torsionfree module
E/τR (E), provided this module is locally free in depth one.
Let U ⊂ E be a submodule. One says that U is a reduction of E or, equivalently, E is
integral over U if R(E) is integral over the R-subalgebra generated by U .
Alternatively, the integrality condition is expressed by R(E)r+1 = U · R(E)r , for r
sufficiently large. The least integer r ≥ 0 for which this equality holds is called the reduction
number of E with respect to U and denoted by rU (E). For any reduction U of E the module
E/U is torsion, hence U has a rank and rank U = rank E.
If R is moreover local or positively graded, with residue field k, then we defined earlier
the special fiber F(E) = k ⊗R R(E); its Krull dimension is what we then called the analytic
spread of E, denoted by `(E).
Next we introduce, copying the ideal case after Rees, the notion of reduction and reduction number of a module. Thus, assume in addition that k is infinite. A reduction of E is
said to be minimal if it is minimal with respect to inclusion. For any reduction U of E one
has ν(U ) ≥ `(E) (ν(·) denotes the minimal number of generators function), and equality
holds if and only if U is minimal. Minimal reductions arise from the following construction:
The algebra F(E) is a standard graded algebra of dimension ` = `(E) over the infinite field
k. Thus it admits a Noether normalization k[y1 , . . . , y` ] generated by linear forms; lift these
linear forms to elements x1 , . . . , x` in R(E)1 = E, and denote by U the submodule generated by x1 , . . . , x` . By Nakayama’s Lemma, for all large r we have R(E)r+1 = U · R(E)r ,
making U a minimal reduction of E.
Having established the existence of minimal reductions, we can define the reduction
number r(E) of E to be the minimum of rU (E), where U ranges over all minimal reductions
of E (N.B. These definitions make sense even if the module does not have a rank if one
follows the definition of Rees algebra of a module as given in [12]).
The basic frame for the Bourbaki method is as follows.
Theorem 32 Let R be a Noetherian local ring and E a finitely generated R–module having
a rank e > 0 and let I(E) be a generic Bourbaki ideal of E.
(i) R(E) is Cohen–Macaulay if and only if R(I(E)) is Cohen–Macaulay.
(ii) (In case R is universally catenary) R(E) is normal with depth R(E) ⊗R Rp ≥ e + 1
for every nonzero prime p of R if and only if R(I(E)) is normal.
(iii) E is of linear type with grade R(E)+ ≥ e if and only if I(E) is of linear type.
(iv) If any of the previous conditions hold, then R(E 00 )/(F ) ∼
= R(I(E)) and the generators x1 , . . . , xe−1 of F form a regular sequence on R(E 00 ). Moreover, whenever
R(E 00 )/(F ) ∼
= R(I(E)) holds and the residue field of R is infinite then r(E) = r(I(E))
20
2.3.3
Aplicações em baixas dimensões
In this portion we collect some consequences of Theorem 32 under especial circumstances.
V
Recall that a module E is called orientable if it has a positive rank e and ( e E)∗∗ ' R.
Proposition 33 Let R be a Noetherian local ring and let E be a finitely generated torsionfree R–module having a rank e > 0.
(a) If E is free locally in depth one, but not free, then `(E) ≥ e + 1.
(b) If E is free locally in depth two and is orientable, and ν(E) ≥ e + 2, then `(E) ≥ e + 2.
Proposition 34 Let R be a Cohen–Macaulay local ring of dimension d > 0 with infinite
residue field and E a finitely generated R–module having a rank e. If R(E) is Cohen–
Macaulay then
r(E) ≤ `(E) − e ≤ d − 1.
Corollary 35 Let R be a Cohen–Macaulay ring and let E be a finitely generated torsionfree
R–module having a rank. If R(E) is Cohen–Macaulay then E is free locally in codimension
one.
Proposition 36 Let R be a Cohen–Macaulay local ring of dimension 2 with infinite residue
field and let E be a finitely generated torsionfree R–module having a rank.
(a) R(E) is Cohen–Macaulay if and only if E is free locally in codimension 1 and r(E) ≤
1.
(b) E is of linear type if and only if µ(E℘ ) ≤ dim R℘ + e − 1 for every prime ℘ of R of
positive codimension.
Proposition 37 Let R be a Cohen–Macaulay local ring of dimension d with infinite residue
field and let E be a finitely generated torsionfree orientable R–module that satisfies µ(E℘ ) ≤
dim R℘ + e − 1 for every prime ℘ of R such that 1 ≤ dim R℘ ≤ d − 1.
(a) Assume d = 3. If r(E) ≤ 1 then R(E) is Cohen–Macaulay. If E satisfies µ(E℘ ) ≤
dim R℘ + e − 1 for every prime ℘ of R of positive codimension, then E is of linear
type.
(b) Assume d ≤ 4. If r(E) ≤ 2 and depth E ≥ 2, then R(E) is Cohen–Macaulay.
(c) Assume d = 5. If R is Gorenstein, r(E) ≤ 2, and depth E ≥ 4, then R(E) is Cohen–
Macaulay.
Theorem 38 Let R be a Gorenstein local ring with infinite residue field and let E be a
finitely generated R–module with rank e which has projective dimension one and is torsionfree locally in codimension 1. Let a1 , . . . , an where n ≥ s be a set of generators of E and let
ϕ be a matrix presenting E with respect to these generators. Let e ≤ s ≤ n be an integer
for which µ(E℘ ) ≤ dim R℘ + e − 1 for every prime ℘ of R such that 1 ≤ dim R℘ ≤ s − e.
21
(a) The following are equivalent:
(i) R(E) is Cohen–Macaulay and `(E) ≤ s;
(ii) r(E) ≤ `(E) − e ≤ s − e;
(iii) r(Ep ) ≤ s − e for every prime p with dim Rp = `(Ep ) − e + 1 = s − e + 1, and
`(E) ≤ s;
(iv) Up to elementary row operations, In−s (ϕ) is generated by the maximal minors of
the last n − s rows of ϕ;
(v) Up
P to a triangular change of the given generators of E, Fs (E) = F0 (E/U ) for
U = si=1 Rai .
(b) If the equivalent conditions of (a) hold then U is a reduction of E with rU (E) = r(E).
Furthermore, E is of linear type, or else, `(E) = s and r(E) = s − e.
Proposition 39 Let R = k[Y1 , . . . , Yd ] be a polynomial ring in d ≥ 2 variables over a field
and let E be a finitely generated R–module that has projective dimension one and satisfies
µ(E℘ ) ≤ dim R℘ + e − 1 for every prime ℘ of R such that 1 ≤ dim R℘ ≤ d − 1. Assume
that E has a presentation matrix ϕ whose entries are linear forms (so that, in particular,
E is graded). Write n = ν(E), Y = Y1 , . . . , Yd , T = T1 , . . . , Tn for a new set of variables,
and B(ϕ) for the matrix whose entries are linear forms in k[T1 , . . . , Tn ] satisfying T · ϕ =
Y · B(ϕ). Then R(E) is Cohen–Macaulay and R(E) ∼
= R[T1 , . . . , Tn ]/(T · ϕ, Id (B(ϕ))).
The following was proved more generally for torsionfree modules. We include a version
for ideals only.
Proposition 40 Let R be a Gorenstein local ring of dimension 3 with infinite residue field.
Let J ⊂ R be an ideal satisfying the following conditions:
(a) µ(J℘ ) ≤ dim R℘ for every prime ℘ of R (in particular, J is 3-generated)
ϕ2
ϕ1
(b) The minimal free resolution of J is of the form 0 → R −→ R3 −→ R3 → J → 0,
where I1 (ϕ1 ) ⊂ I1 (ϕ2 )2 .
Set I = J : I1 (ϕ2 ). Then
(i) J has codimension two; in particular, I is a codimension two proper ideal
(ii) I/J ' R/I1 (ϕ2 ); in particular, I has projective dimension one, hence is a perfect ideal
(iii) J is a minimal reduction of I and r(I) = rJ (I) = 2
(iv) R(I) is Cohen–Macaulay.
We finish this part with an amusing criterion of perfection.
Proposition 41 Let R be a Gorenstein local ring such that (dim R − 3)! is invertible in R.
Let I ⊂ R be an ideal satisfying the following conditions:
(i) I is an unmixed codimension two proper ideal
22
(ii) I has projective dimension at most two
(iii) S(I/I 2 ) = grI (R) and is R/I-torsionfree.
Then I is a perfect ideal.
Question 42 Is there a counter-example if the hypothesis on the invertibility of (dim R−3)!
fails?
Such an instance would look rather odd.
3
3.1
Miscelânia
Álgebras graduadas direcionais
O tema central desta seção é uma extensão natural da teoria clássica dos produtos de Segre
(vide referências [9], [16] and [73] para uma visão de conjunto).
Definition 43 Seja A um anel comutativo e M, um semigrupo comutativo (notação aditiva). Uma A-álgebra M-graduada é uma A-álgebra comutativa que é soma direta de subgrupos aditivos
M
S=
Su ,
u∈M
tal que S0 = A e Su Sv ⊂ Su+v para todos u, v ∈ M. Em particular, cada Su é um A-módulo
- dito a componente ou a parte homogênea de grau u de S.
Esta noção se estende, de modo evidente, a G-graduações, com G grupo abeliano. Nosso uso
destas será apenas fortuito (por exemplo, na consideração de anéis de frações de álgebras
M-graduadas).
Suporemos, doravante, que os A-módulos Su (u ∈ M) são finitamente gerados. Isto não
acarreta, automaticamente, que S seja finitamente gerada como A-álgebra no sentido de existir um subconjunto finito M ⊂ M tal que S = A[⊕u∈M Su ]. A parte mais importante deste
trabalho tratará, prioritariamente, de A-álgebras graduadas finitamente geradas, passando,
desta forma, ao largo do famoso problema da Teoria dos Invariantes.
Uma situação marcante é o das M-álgebras graduadas, onde M é produto direto finito
de semigrupos. Entre estas, destacam-se as Nm -álgebras graduadas (e seu análogo, as
Zm -álgebras graduadas). Neste caso, usaremos a terminologia já consagrada na literatura:
álgebras multigraduadas, com o número de cópias de N subentendido.
Assim, uma álgebra multigraduada tem a feição
S=
M
S(u1 ,...,um ) .
(6)
(u1 ,...,um )∈N×...×N
O elemento (u1 , . . . , um ) é, então, chamado multigrau.
Nas aplicações geométricas, S(0,...,0) = k é um corpo e S é finitamente gerada. Quase
sempre suporemos também que os geradores não são quaisquer, mas têm de fato multigrau
da forma (0, . . . , 0, ui , 0, . . . , 0), para várias escolhas de ui ∈ N - em cujo caso, deverı́amos
23
dizer que S é uma álgebra multigraduada multicoordenada, mas isto carregaria demasiado
a terminologia.
Finalmente, temos a noção de uma álgebra multistandard, que é uma álgebra multigraduada gerada por elementos de multigraus (0, . . . , 0, 1, 0, . . . , 0), com i = 1, . . . , m. Neste
caso, o protótipo é o anel de polinômios B = k[X1 , . . . , Xm ] em m conjuntos de variáveis
mutuamente independentes.
Um ideal J ⊂ B é chamado multihomogêneo se é homogêneo, separadamente, em
cada um dos conjuntos de variáveis de B. Neste caso, temos J = ⊕J(u1 ,...,um ) , onde
J(u1 ,...,um ) = J ∩ B(u1 ,...,um ) . É evidente que o anel residual B/J é uma álgebra multistandard. Inversamente, toda tal álgebra é isomorfa a um anel residual B/J .
Passamos, entrementes, ao principal conceito desta seção.
Definition 44 Seja S uma álgebra multistandard e sejam p1 , . . . , pm ≥ 0 inteiros. A
subálgebra (p1 , . . . , pm )-direcional de S é o anel
M
S(p1 u,...,pm u) .
u∈N
Observemos que ∆ = ∆(p1 , . . . , pm ) := {(p1 u, . . . , pm u) | u ∈ N} é um subsemigrupo de
e que o anel introduzido é, de fato, uma A-álgebra N-graduada, que pode ser facilmente
regraduada de modo a se tornar standard. A direção (p1 , . . . , pm ) estando subentendida,
denotaremos a álgebra direcional por S∆ . Em termos de geradores, se S = k[x1 , . . . , xm ],
então
S∆ = k[xα1 1 · · · xαmm | |αj | = pj , 1 ≤ j ≤ m].
Nm
Em particular, uma subálgebra direcional é uma álgebra tórica (no sentido lato, não
necessariamente normal).
O lema seguinte coleta alguns resultados técnicos úteis.
Lemma 45 Seja S = B/J = ⊕(u1 ,...,um ) S(u1 ,...,um ) uma álgebra multistandard e seja S∆ a
subálgebra direcional na direção (p1 , . . . , pm ). Então:
(i) S∆ ' B∆ /J, onde J = B∆ ∩ J = ⊕u∈N J(p1 u,...,pm u) .
(ii) De modo mais explı́cito, J é gerado pelos polinômios da forma M F , onde F é um gerador
de J de multigrau (d1 , . . . , dm ) e M é um multimonômio Xβ1 · · · Xβm , onde (β1 , . . . , βm ) ∈
Nm satisfaz a condição de que βj = upj − dj , ∀j, com u ≥ 1 o menor possı́vel.
A demonstração é direta, a partir das definições.
Example 46 Ponhamos X1 = {X1 , X2 }, X2 = {Y1 , Y2 } e J = (X13 Y2 − X23 Y1 ) ⊂ B =
k[X1 , X2 ; Y1 , Y2 ]. Aqui, F = X13 Y2 − X23 Y1 tem multigrau (3, 1). Seja B∆ a subálgebra
direcional na direção (2, 3), isto é:
B∆ = k[X12 Y13 , X12 Y12 Y2 , X12 Y1 Y22 , X12 Y23 , X1 X2 Y13 , X1 X2 Y12 Y2 , X1 X2 Y1 Y22 , X1 X2 Y23 ,
X22 Y13 , , X22 Y12 Y2 , X22 Y1 Y22 , X22 Y23 ].
Então, J é gerado pelos produtos Xβ1 Yβ2 F , com |β1 | = 1, |β2 | = 5. Vemos, assim, que
J é gerado por 10 binômios que são imediatamente obtenı́veis. Podemos ainda obter a
apresentação de S∆ como anel de restos de k[U], com #U = 12. O ideal de apresentação
24
será gerado pelas relações polinomiais dos 12 geradores de B∆ e pelos polinômios (todos de
grau 2, neste caso) que resultam de substituir os monômios constituintes de cada gerador
de J, que pertencem a B∆ , pelos Us correspondentes. Um cálculo em Macaulay fornece 55
geradores mı́nimos, todos de grau 2 e diz-nos que Proj S∆ é uma curva de grau 11 em P11 .
Logo, trata-se da curva racional normal em P11 !
Algumas álgebras bigraduadas
Entre as álgebras multigraduadas, distinguem-se as bigraduadas. Um dos exemplos mais
básicos é o da álgebra simétrica S(E) de um R = k[X]-módulo E gerado por elementos de
um grau fixo r ≥ 1. Mais precisamente, seja
ϕ
R(−d1 ) ⊕ · · · ⊕ R(−ds ) −→ R(−r)m −→ E −→ 0
a apresentação homogênea de E. Então S(E) ' k[X, T]/J , onde J é gerado por elementos
dos vários bigraus (dj −r, 1), 1 ≤ j ≤ s, na bigraduação standard de k[X, T]. Pelo Lema 45,
o ideal de apresentação da álgebra direcional S(E)∆ (de direção (1, 1)) no anel k[X, T]∆ =
k[Xi Tj ] é gerado pelos polinômios dos vários graus dj − r que são produtos dos geradores
de J por monômios em T de grau dj − r − 1, respectivamente.
Example 47 Se E admite apresentação linear (i.é., se ker ϕ acima é gerado por vetores
de grau r + 1 na graduação de R(−r)m ), então o ideal de apresentação de S(E)∆ é gerado
pelos próprios geradores de J . Este é o caso de muitos módulos importantes, por exemplo,
módulos de secções de certos fibrados homogêneos sobre Pn−1 [70].
Example 48 Se E é gerado em grau fixo, os elementos de R-torção de S(E) geram um
ideal primo bihomogêneo A ⊂ S(E). Segue-se que a álgebra de Rees R(E) = S(E)/A tem
estrutura natural de k-álgebra bigraduada.
O caso
P acima em que E = I ⊂ R é um ideal merece destaque especial. Neste caso,
R(I) ' s≥0 I s ts ⊂ k[X][t] é a álgebra de Rees usual de I. O anel residual gr I (R) =
P
s s+1 é outra álgebra bigraduada importante.
s≥0 I /I
consideraremos, com algum detalhe, o caso de uma álgebra de Rees, já mencionado nos
exemplos da Seção 2.1.
Dado um ideal homogêneoI ⊂ R = k[X] na graduação standard, denotaremos por It
o k-espaço vetorial das formas de grau t pertencentes a I. Consideremos a álgebra de
Rees R(I) = ⊕s≥0 I s ts , onde I é um ideal gerado por formas f1 , . . . , fr de graus (digamos)
d1 ≤ · · · ≤ dr , respectivamente. Ora, R(I) = k[X][f1 t, . . . , fr t] ⊂ k[X, t] é bigraduada pela
bigraduação induzida pela bigraduação standard de k[X, t]. Nesta, o bigrau de Xi é (1, 0),
enquanto que o de fj t é (dj , 1), de modo que a bigraduação induzida não é standard.
Dada uma direcção (p, q), se ∆ = ∆(p,q) , tem-se imediatamente S∆ = ⊕s≥0 (I sq )sp tsq .
q é gerado (não necessariamente de maneira mı́nima) pelas formas
Observemos que o ideal IP
q1
qr
de grau d1 · · · dr onde j qj = q, logo é gerado por formas de grau no máximo qdr . Isto
implica em que, se t ≥ qdr , o k-espaço vetorial (I q )t é gerado pelas formas Xα F , onde F
é gerador de I q |α| = t − gr(()F ). Daqui resulta, imediatamente, que se dr ≤ p/q, então
S∆ == k[(I q )p tq ] ' k[(I q )p ] ⊂ k[X].
25
Como caso especial, se dj = d ∀j e (p, q) = (d + 1, 1), então S∆ = k[Id+1 t] ' k[Id+1 ] ⊂
k[X]. Regraduando os fj t’s, definindo bideg(fj t) = (0, 1), R(I) vem a ser uma álgebra
bigraduada standard, a qual denotaremos por S a seguir. Ora, neste caso, ve-se que se
∆ = ∆(1,1) , então S∆ = k[Id+1 t] ' k[Id+1 ] = k[Xi fj |1 ≤ i ≤ n, 1 ≤ j ≤ r] ⊂ k[X].
Outras álgebras multigraduadas
Seja Z ⊂ Rd um submódulo graduado de Rd (na N-graduação natural
de Rd induzida
P
pela graduação standard de R = k[X]). Em outras palavras, Z = j RVj , onde Vj é um
vetor de Rd cujas coordenadas têm todas o mesmo grau rj . Outra forma equivalente é
dizer que Z é o primeiro módulo de sizigias de um R-módulo graduado gerado em grau fixo
(o caso em que Z é um ideal homogêneo de R sendo incluido como o primeiro módulo de
2
sizigias de R/I!). Pelo Exemplo
P 48, sabemos que a álgebra de Rees R(Z) é N -graduada.
Mais precisamente, R(Z) = v≥0 Sv (Z)/t(Sv (Z)), onde t(M ) ⊂ M denota o submódulo
de torção do R-módulo M . Como R-submódulo de Sv (Rd ), Sv (Z)/t(Z) é N-graduado pela
N-graduação de Sv (Rd ) induzida pela de Rd . Então, R(Z)(u,v) = Sv (Z)u /t(Sv (Z))u .
Por outro
P lado, oα anel de polinômios B = R[t1 , . . . , td ] = R[t] é bigraduado pondo
B(u,v) =
|α|=v Ru t . A álgebra de Rees R(Z) de Z é bigraduamente isomorfa à RP
subálgebra S do anel de polinômios B gerada pelos polinômios da forma dl=1 fj,l tl , onde
Vj = (fj,1 , . . . , fj,d ) é um gerador de Z. Notemos que um tal polinômio
tem bigrau (rj , 1),
P
onde rj = gr(()fj,1 ) = . . . = gr(()fj,d ). Mais explicitamente, S = (u,v) S(u,v) , onde
S(u,v) =
X
X
X
Ru−Pj rj αj (
f1,l tl )α1 · · · (
fm,l tl )αm .
|α|=v
l
l
P
Em particular, S(u,v) = (0) para u < j rj αj .
A álgebra bidirecional de direção (p, q) de S é facilmente expressável a partir da forma
desta, mas a aparência não é especialmente atraente. No caso especial em que rj = d ∀ j,
tem-se S∆(p,q) = (0) se p/q ≤ d (isto é, não existe álgebra direcional se o coeficiente angular
da direção é menor ou igual ao grau dos geradores).
Particularizando o módulo Z a um ideal homogêneo I ⊂ R, ve-se que
X
S∆(p,q) =
(I qu )pu tqu .
u
Particularizando
P mais, se Z = I ⊂ R é um ideal homogêneo gerado em grau fixo d,
então S∆(d+1,1) = u (I u )(d+1)u tu .
Remark 49 Esta última álgebra direcional foi o tema central de [65], onde foi chamada
de álgebra diagonal. A razão para esta terminologia advem do fato de que, nesta situação,
S pode ser regraduada pondo S(u,v) = (I v )u+dv tv . Nesta nova N2 -graduação, os geradores
de S têm bigrau (1, 0) ou (0, 1), de modo que a álgebra (1, 1)-direcional (isomorfa, sem
preservar graus, à álgebra (d + 1, 1) direcional anterior) é não trivial .
Normalidade de álgebras direcionais
Existe uma técnica completamente geral, descoberta no século passado pelos especialistas em Teoria dos Invariantes, para detectar quando certas propriedades de um anel de
26
polinômios se transferem a um subanel de invariantes pela ação de um grupo: trata-se do
operador de Reynolds. Modernamente, a existência de um operador de Reynolds numa
extensão de anéis R ⊂ S traduz-se como um homomorfismo de R-módulos S → R cuja
restrição a R é a identidade. É claro que a existência de um tal homomorfismo de aumentação significa precisamente que R é somando direto de S como R-módulo, o que torna
particularmente agradável a transferência de certas propriedades de um a outro anel.
No presente contexto de álgebras direcionais, temos também um homomorfismo de
Reynolds.
Theorem 50 Seja S = B/J = ⊕(u1 ,...,um ) S(u1 ,...,um ) uma álgebra multistandard e seja S∆
a subálgebra direcional na direção (p1 , . . . , pm ). Então a extensão S∆ ⊂ S admite um
homomorfismo de Reynolds.
Proof. Pondo P = (p1 , . . . , pm )N ⊂ Nm , temos uma decomposição de k-espaços vetoriais
M
S = S∆ ⊕ (
S(q1 ,...,qm ) ).
(q1 ,...,qm )∈ Nm \P
Por outro lado, como P é subsemigrupo de Nm , segue facilmente que o segundo somando
é, de fato, um S∆ -submódulo de S.
Remark 51 A demonstração acima é um caso particular do princı́pio geral: se S é uma
álgebra graduada por um semigrupo G e se H ⊂ G é um subsemigrupo, então o subanel
SH := ⊕h∈H Sh de S é um somando direto de S como SH -módulo.
Corollary 52 Se a álgebra multistandard S é normal, então toda subálgebra direcional de
S é normal.
3.2
Ω-álgebras
The guiding principle is to associate several algebras to the so-called diagonal ideal in a
tensor product A ⊗k A, for a given k-algebra essentially of finite type.
3.2.1
A álgebra das estrelas tangentes
In this section we introduce some other algebras, whcih are very much related to intersection
theory in algebraic geometry.
The following notation will prevail thoughout.
k
R
A
D
Ω(A/k)
SA/k
TA/k
a base field
a ring of fractions of a polynomial ring k[X] = k[X1 , . . . , Xn ]
a k-algebra of the form R/I, I ⊂ R an ideal
the kernel of the multiplication map A ⊗k A → A, generated by the
residues of the elements Xi ⊗ 1 − 1 ⊗ Xi (diagonal ideal )
the module of Kähler k-differentials of A
the symmetric algebra of the A-module Ω(A/k)
the associated graded ring gr D (A ⊗k A)
27
The scheme defined by the algebra SA/k is called the Zariski cone of (the variety defined
by) A, while the one defined by TA/k , following the suggestion of Johnson’s, will be named
the tangent star cone of (the variety defined by) A.
The algebras SA/k and TA/k will be called the Zariski tangent algebra and the tangent
star algebra, respectively, according to the terminology that has been introduced in [66].
The well-known identification Ω(A/k) ' D/D2 yields a surjective homomorphism
SA/k −→ TA/k .
Various ways of measuring the kernel of this map were introduced in [66], based on
earlier development of the theory of algebras of linear type.
Definition 53 (i) A has the expected star dimension if
dim SA/k = dim TA/k .
(ii) A is set theoretically starlike linear if
(SA/k )red = (TA/k )red .
(iii) A is starlike linear if
SA/k = TA/k .
Clearly, (iii) ⇒ (ii) ⇒ (i). We will see that none of these implications is reversible.
Although these definitions look rather inconspicuous, they become natural as soon as one
works out their meaning in concrete cases. Thus, for example, (i) above at least implies that
A is generically reduced, hence reduced if it is also unmixed (in particular, a 0-dimensional A
having the expected star dimension must be a product of fields). If, moreover, A ' k[X]/I,
with I a perfect ideal of codimension two, then (i) can be simply be restated by saying
that A is reduced and, locally in codimension one, (isomorphic to) a hypersurface ring – in
higher codimension, the condition required by (i) is automatically satisfied in this case.
Example 54 The following examples may be worth keeping in mind for their behaviour
pattern. In all of them, k stands for a field.
1. A = k[X, Y, Z]/(XY, XZ, Y Z)
This is a Cohen–Macaulay reduced ring of dimension one.
The presentation ideal J of TA/k over the polynomial ring A[T, U, V ] contains the element
T U V which is a non-zero-divisor on the subideal J(1) generated in degree one (by the linear
relations coming from the transposed Jacobian matrix of the generators XY, XZ, Y Z).
Therefore, dim SA/k > dim TA/k , so A has not the expected star dimension.
However, by the same token as Exercise 4, it is easy to check that the “cylinder” A[T ] =
k[X, Y, Z, T ]/(XY, XZ, Y Z) on A = k[X, Y, Z]/(XY, XZ, Y Z) does have the expected star
dimension. Geometrically, this may look rather intriguing but, from the algebraic viewpoint,
it is rather explicable.
2. A = k[X, Y ]/(X 2 , XY )
Again this is a ring of dimension one, but not reduced (clearly, not Cohen-Macaulay
since it has even an embedded prime).
28
The presentation ideal J of TA/k over the polynomial ring A[T, U ] is generated by J(1)
plus the extra relations T 3 , T 2 U – a result that can be obtained by means of a computation
in the program Macaulay. Since T is a zero-divisor on J(1), it is clear that J and J(1)
have the same codimension (which is 2). Therefore, A has the expected star dimension.
However, A is not set theoretically starlike linear since SA/k and TA/k do not share the same
minimal primes, e.g., (X, Y ) is a minimal prime of J(1) which does not even contain J.
The reduced structure (i.e., the underlying set theoretic geometry) is easily found: in
4-space X, Y, T, U , (SA/k )red is the union of two concurrent planes while (TA/k )red is one
single plane.
3. A = k[X, Y ]/(F (X, Y )), with F (X, Y ) a square-free polynomial.
Geometrically, the situation is as good possible since one has a reduced plane curve.
At any rate, it is easy to verify that A has the expected star dimension. Also, geometers
will undoubtedly find no hardship in realizing that SA/k and TA/k have the same reduced
structure (i.e., that A is set theoretically starlike linear). However, most everybody will be
stymied if asked whether the two cones are the same. The fact that this is indeed the case
was originally proved in [30] as a special case of a hypersurface in n-space. In this work, it
will come out as a special case of a more general class of varieties, namely, locally complete
intersections in n-space (see Section 4).
Problem 55 How does one find, without computer resources, the ideal-theoretic defining
equations of TA/k in case A is a one-dimensional ring of the form k[X1 , . . . , Xn ]/I?
Apresentação da álgebra tangente de Zariski. The preceding presentation of Ω(A/k)
as an A-module yields a corresponding presentation of the Zariski tangent algebra as
an A-algebra, namely, if T = T1 , . . . , Tn are presentation variables such that Ti 7→ dXi
(mod ∂(I/I 2 )) then
SA/k ' A[T]/(TΘ).
It may be worthwhile mentioning yet another related algebra, to wit, the symmetric algebra
SA⊗k A (D) of the ideal D ⊂ A ⊗k A
In order to deal with a presentation of the latter, we identify A ⊗k A with the ring
k[X, U]/(I(X) + I(U)), where I(X) = I and I(U) is obtained from I by the substitution X 7→ U. Let T = T1 , . . . , Tn denote new variables (“ presentation variables”) over
k[X, U]/(I(X) + I(U)) and define a surjective homomorphism
ρ : k[X, U, T]/(I(X) + I(U)) → SA⊗k A (D)
by the assignment Ti 7→ Xi − Ui (mod I(X) + I(U)), where Xi − Ui is taken in degree 1.
It is well known that the presentation ideal ker ρ is generated by T-linear polynomials
with coefficients in k[X, U]/(I(X) + I(U)). Now, for each of these generators whose coefficients actually belong to D, we consider an arbitrary lifting to k[X, U, T] and denote by
D(X − U) ⊂ k[X, U, T] the ideal generated by these liftings. We informally call the latter
the syzygetic generators of the presentation ideal ker ρ.
We now introduce fresh generators, to be called the Taylor generators of ker ρ. Namely,
for each generator fj (X) of I(X), consider the Taylor expansion of fj (X) at the point U
and collect the polynomial coefficients of the linear terms Xi − Ui in an arbitrary fashion to
29
P
get an expression of the form
gij (Xi − Ui ) – modulo D(X − U), it will be immaterial the
way one collects them, due to the Koszul relations.
P Let T (I) denote the ideal of k[X, U, T]
generated by the corresponding T-linear forms
gij Ti .
Our result claims that, together the two kinds not only generate but are also natural
generators.
Proposition 56 Let char k = 0. Then, in the above notation, one has:
1. There is a presentation
SA⊗k A (D) ' k[X, U, T]/(D(X − U) + T (I) + I(X) + I(U)).
2. The above presentation of SA⊗k A (D) modulo D yields the earlier presentation of the
Zariski tangent algebra by means of the transposed Jacobian matrix.
Proof. (1) Quite generally, for any ring B and any ideal b, from a presentation
0 −→ Z −→ B n −→ b −→ 0,
one obtains a presentation of the conormal module
0 −→ Z/Z ∩ bB n −→ B n /bB n −→ b/b2 −→ 0.
This says that Z = Z ∩ bB n + L, where L is generated by lifted generators. Applying to
the present situation, with B = A ⊗k A and b = D, it will suffice to show that the Taylor
expansions at U of the generators f1 , . . . , fm of I, read modulo the ideal I(X) + I(U), are
liftings of the differentials
n
n
X
X
∂f1
∂fm
dXi , . . . ,
dXi
∂Xi
∂Xi
i=1
i=1
read modulo I = I(X).
Now, for that, in the Taylor expansion
fj (X) − fj (U) =
n
X
∂fm
i=1
∂Xi
(U)(Xi − Ui ) +
1
2
X
1≤i≤k≤n
∂ 2 fj
(U)(Xi − Ui )(Xk − Uk ) + . . .
∂Xi ∂Xj
one collects terms as follows:
fj (X) − fj (U) = (
X ∂fj
∂fj
+
(Xk − Uk ) + . . .)(X1 − U1 )
∂X1
∂X1 ∂Xk
k≥2
X ∂fj
∂fj
+ (
+
(Xk − Uk ) + . . .)(X2 − U2 )
∂X2
∂X2 ∂Xk
k≥3
+ ....
It is clear that, modulo D, these yield the above differentials.
(2) The second part follows immediately from the preceding proof.
Although much less ubiquitous than the tangent algebras studied in these notes, the
symmetric algebra SA⊗k A (D) is nonetheless important. We next mention connections to
central theories.
30
• (Relation to deformation theory) Under suitable conditions, the presentation ideal of
SA⊗k A (D) is actually generated by the Koszul relations on the generators of D and the
Taylor relations T (I). This requires that the ideal D be syzygetic in the terminology
of [69] or, in the language of deformation functors [40], that T2 (A|k, A) = 0.
• (Relation to embedding dimension of affine varieties) Let A stand for a finitely generated algebra over an infinite field k. In[72, Theorem 1] the following estimate is
obtained
edim A ≤ dim SA⊗k A (D).
(This is not stated as such in loc. cit, but it turns out to be the same statement at least
if grade D > 0 due to Proposition 2). Here edim A stands for the affine embedding
dimension of A, i.e., the least n ≥ 0 such that there is a surjective k-homomorphism
k[X1 , . . . , Xn ] −→ A.
The proof given in [72] hinges on a rather involved geometric argument, so one would
naturally wonder if there is a simpler algebraic proof.
We note en passant that if A is a finite type equidimensional k-algebra having the
expected star dimension (see below) then the above estimate reduces to edim A ≤ 2 dim A+1
(cf. Proposition 59 and, specially, Remark 60), which is a classical result in the case of a
smooth A. Thus, it seems even more natural to ask for a direct algebraic proof of this
inequality for algebras A having the expected star dimension. It would suffice to show that,
for such algebras, the following upper bound always works:
edim A ≤
sup {edim Am } + 1.
m∈MaxA
Actually, one is tempted to formulate the
Conjecture 57 Let k be an infinite (maybe perfect) field and let A be a finite type equidimensional k-algebra. Then edim A ≤ 2 dim A.
Álgebras tangentes de Zariski com a dimensão esperada. Let k be a field and let A
be a k–algebra essentially of finite type; write A = W −1 B, B finitely generated k–algebra,
B ⊂ A, W a multiplicative set in B.
As before, write
0 → D −→ A ⊗ A = A ⊗k A −→ A → 0;
then D/D2 = Ω(A/k).
A typical prime ideal of A will be denoted by ℘. Identifying Spec(A) with V (D) ⊂
Spec(A ⊗k A), will allow for the slight confusion of denoting the inverse image of ℘ in
A ⊗k A also by ℘.
Lemma 58 [66, Lemma 2.2] Let k stand for a field and let B ⊂ A be k-algebras such that
B is of finite type over k and A is a ring of fractions of B. Let ℘ ∈ Spec (A) = V (D) ⊂
Spec (A ⊗k A); then
dim(A ⊗k A)℘ = 2 dim A℘ + trdegk A℘ /℘A℘ .
If A is locally equidimensional, then (A ⊗k A)℘ is equidimensional and quasi–unmixed.
31
We refer to [66, loc. cit.] for a proof of this lemma which uses but standard machinery.
The next proposition also appears in [66].
Proposition 59 Assume that k is a perfect field, A is a k-algebra essentially of finite type,
locally equidimensional and equicodimensional. Then the following are equivalent:
(a) dim SA/k = dim TA/k (i.e., A has the expected star dimension).
(b) edim(A℘ ) ≤ 2 dim A℘ , for all ℘ ∈ Spec(A).
(c) µ(D℘ ) ≤ dim(A ⊗k A)℘ for every ℘ ∈ V (D).
(d) Let B = k[X]/(f1 , . . . , fm ) ⊂ A such that A is a ring of fractions of B. Then
ht It (Θ) ≥ ht (f1 , . . . , fm ) − t + 1,
for 1 ≤ t ≤ ht (f1 , . . . , fm ), where Θ denotes the transposed Jacobian matrix of
f1 , . . . , fm modulo the ideal (f1 , . . . , fm ).
Proof. (a) ⇔ (c). (c) is the usual condition that follows from the linear type property - it
is exactly equivalent to the preliminary dimension equality, which is the content of (a).
(b) ⇔ (c) It is well known (cf., e.g., [15]) that
µ(D℘ ) = µ(Ω(A/k)℘ ) = edim(A℘ ) + dim B/℘.
The equivalence follows now from Lemma 58.
(c) ⇔ (d): (Assuming that A is generically a complete intersection) This follows from the
usual back-and-forth process between local number of fenerators and height of Fitting ideals
of a module.
Remark 60 We note that the equivalence (a) ⇔ (b) holds with no assumption on the field
k. Moreover, if A is k-affine (or local) equidimensional then all hypotheses are fulfilled and
we can freely use this equivalence. More particularly, if A is a graded affine k-domain (or a
local domain) then condition (b) is equivalent to dim SA/k = 2 dim A.
The principle of Cartan–Eilenberg–Serre
We now consider the technique of reduction to the diagonal. We note that it will be
mainly used in the opposite direction to that used in the classical sense!
In considering ordinary Koszul complexes, we will henceforth focus on a set of generators
of ϕ(F ) ⊂ R rather than on ϕ itself. Accordingly, given a sequence b of elements of a ring B
and a B-module E, the Koszul complex (resp. the Koszul homology) of b with coefficients
in E will be indicated by K• (b; E) (resp. H• (b; E)).
Theorem 61 [54, Ch. V, B] Let k be a field, let R stand for a ring of fractions of the polynomial ring k[X1 , . . . , Xn ] and let M and N be R-modules. Then there is an isomorphism
of graded R-modules
H• (∆; M ⊗k N ) ' Tor R
• (M, N ),
where ∆ denotes the sequence {Xi ⊗ 1 − 1 ⊗ Xi | 1 ≤ i ≤ n} ⊂ R ⊗k R.
32
Proof. One may clearly assume that R = k[X1 , . . . , Xn ]. Quite generally, since ∆ is a
regular R ⊗k R-sequence such that R ⊗k R/(∆) ' R, one has
R⊗ R
k (M
H• (∆; M ⊗k N ) ' Tor •
⊗k N , R).
The rest follows from a well known result in [8, IX, 2.8] to the effect that
R⊗ R
k (M
Tor •
⊗k N , R) ' Tor R
• (M, N ).
Remark 62 We will be mainly interested in gathering information on the diagonal ideal D
from the homology modules Tor R
• (M, N ). The main cases will have M = N = A or N = A,
M = ωA . In such situations one has a reasonable grasp of the modules Tor R
i (M, N ).
We now go back to the notation that has been set up at the beginning of Subsection 3.2.1.
By definition, the diagonal ideal D = ker A ⊗k A → A is generated by the residues of the
elements of ∆. The next result could be rightly called main theorem as it draws practically
on all the material developed so far.
Theorem 63 Let k be a field, let A be a locally Cohen–Macaulay k-algebra, essentially of
finite type over k and consider any fixed presentation A ' R/I, with R a ring of fractions
of the polynomial ring k[X] = k[X1 , . . . , Xn ]. Let ∆ = {Xi ⊗ 1 − 1 ⊗ Xi | 1 ≤ i ≤ n} ⊂
k[X] ⊗k k[X] and let further E be a finitely generated A-module. Assume that the following
conditions hold:
(i) depth Tor R
i (A, E)m ≥ dim Am − ht IM + i, for every integer i ≥ 0 and every maximal
ideal M ⊇ I, where m = M/I.
(ii) µ(Ω(A/k)℘ ) ≤ dim A℘ + n − ht IP , for every prime ideal P ⊇ I, where ℘ = P/I.
Then:
(a) The approximation complex M• (∆; E ⊗k A) is acyclic and
H0 (M• (∆; E ⊗k A)) ' SA/k ⊗A E
' TA/k ⊗A E ' gr D (E ⊗k A).
(b) If A is Cohen-Macaulay then SA/k ⊗A E ' TA/k ⊗A E is a Cohen–Macaulay SA/k module.
(c) If Supp(E) = Spec(A) then (SA/k )red ' (TA/k )red (in other words, A is set theoretically starlike linear).
Proof. It is an application of Theorem 18, with C := A ⊗k A, J := (Delta)/(I ⊗ 1 +
1 ⊗ I) = D and E := E (considered as an A ⊗k A-module via the map A ⊗k A → A).
The remaining identifications come from Theorem 61, from D/D2 = Ω(A/k) and from
dim A℘ + n − ht IP = 2 dim A℘ + trdegk A℘ /℘A℘ = dim(A ⊗k A)℘ by Lemma 58.
33
The hypothesis in Theorem 18(b) is satisfied in the present contex since
H1 (∆, E ⊗k A) ' Tor R
1 (A, E) ' I ⊗R E ' (I ⊗R A) ⊗A E
' Tor R
1 (A, A) ⊗A E ' H1 (∆; A ⊗k A) ⊗A E
Remark 64 Theorem 63 admits a formulation with k replaced by a noetherian ring and
A being flat over it [66]. This has some interest because of deformation theory [31].
On the other end of the spectrum, if k is a perfect field and A is equicodimensional –
a property Nagata used to include in the definition of a Cohen–Macaulay noetherian ring
– then by Proposition 59, condition (ii) of Theorem 63 is but the condition that A has the
expected star dimension.
Corollary 65 Let k be a field and let A be a k-algebra essentially of finite type over k
satisfying the following conditions:
(i) Tor R
i (A, A) is Cohen–Macaulay for every i ≥ 0.
(ii) µ(Ω(A/k)℘ ) ≤ dim A℘ + n − ht IP , for every prime ideal P ⊇ I, where ℘ = P/I.
(iii) A is (locally) Gorenstein.
Then A is starlike linear and SA/k = TA/k is a Gorenstein ring.
Proof. The starlike linear part follows from Theorem 63 with E = A and the Gorenstein
part is proved in [20, Theorem 9.1].
There are other consequences and variants of Theorem 63 which are of interest. However,
most of them actually fall off Theorem 18, so we will not expand on them referring rather
to [20].
3.2.2
Selecta
This section will briefly survey reasonably broad classes of ideals that can be approached
from the point of view of the preceding section. At the end we will discuss a question posed
by van Gastel and give counterexamples to it in the spirit of some of the chosen classes
herein.
Álgebras de estrelas de ideais licci. Following current terminology, we say that a
noetherian local algebra A is licci over a field k if it admits a presentation A ' R/I with
R a localization at a prime ideal of a polynomial ring over k and I, an ideal belonging to
the linkage class of a complete intersection.
Theorem 66 Let k be a perfect field and let A be a k-algebra essentially of finite type over
k satisfying the following conditions:
(i) A is licci locally at every one of its maximal ideals;
(ii) edim A℘ ≤ 2 dim A℘ for every ℘ ∈ Spec (A).
Then:
(a) SA/k ⊗A ωAm ' TA/k ⊗A ωAm for every maximal ideal m of A and this module is a
Cohen–Macaulay (SA/k )m -module.
34
(b) A is set theoretically starlike linear.
(c) Moreover, if A is Gorenstein then D ⊂ A ⊗k A is normally torsion free if and only if
edim A℘ ≤ 2 dim A℘ − 1 for every non-minimal ℘ ∈ Spec (A).
(d) If A is Gorenstein then it is starlike linear and SA/k ' TA/k is a Gorenstein ring.
Proof. By [7, 6.2.11], Tor R
i (Am , ωAm ) is a maximal Cohen–Macaulay Am -module for every
i ≥ 0 and every maximal ideal m of A. The assertions then follow from Theorem 63,
Remark 64, Proposition 65 and [20, Theorem 9.1]
Remark 67 (Locally complete intersections) A special case of the preceding theorem is
that of a locally complete intersection A with the expected star dimension. Here, of course,
the modules Tor R
i (A, A) are locally free (i.e., projective) over A.
Estudo de caso: ideais perfeitos de codimensão 2. Perfect ideals of codimension
two form the first class of licci ideals for which the assumption on having the expected
star dimension actually implies starlike linearity (not just set theoretically). Here is a more
precise version of this result.
Proposition 68 Let k be a perfect field and let A be a k-algebra essentially of finite satisfying the following conditions:
(i) For every maximal ideal m of A, Am ' R/I where R is a polynomial ring over k localized
at a prime ideal and I is a perfect ideal of codimension at most 2.
(ii) For every prime ideal ℘ ∈ Spec (A), edim A℘ ≤ 2 dim A℘ .
Then
(a) SA/k = TA/k is a Cohen–Macaulay ring;
(b) D is normally torsionfree if and only if edim A℘ ≤ 2 dim A℘ −1 for every non-minimal
prime ℘ ∈ Spec (A).
Proof. By [20, Theorem 9.1] and Remark 64, it suffices to show the depth condition on
the modules Tor R
i (Am , Am ), with m a maximal ideal of A. We may then assume that A is
local and that ht I = 2. Therefore, we have only to deal with Tor R
i (A, A) (i = 1, 2). By
the result of [1], H1 (I) is a Cohen–Macaulay module. Thus, on one hand, from the exact
sequence
0 −→ H1 (I) −→ Am −→ I/I 2 −→ 0
2
one gets depth Tor R
1 (A, A) = depth I/I ≥ dim A − 1.
On the other hand, the symmetric power S2 (ωA ) is also a Cohen–Macaulay module [75,
2.1(b)]. This implies that
Tor R
2 (A, A) ' Hom (ωA , A) ' Hom (S2 (ωA ), ωA )
is Cohen–Macaulay as well.
Remark 69 Condition (ii) in the preceding Proposition can in the present circumstances
be restated to the effect that A is a reduced ring and a hypersurface locally in codimension
one. Likewise, the above condition in order that TA/k be torsionfree over A is equivalent to
requiring that A be normal and a hypersurface locally in codimension two.
35
Example 70 Let X be an m × m − 1 matrix of indeterminants over a perfect field k, let
R := k[X] and let I ⊂ R stand for the ideal generated by the maximal minors of the matrix
X. Setting A = R/I, the assumptions of Proposition 68 are fulfilled, so A is starlike linear,
TA/k is a Cohen–Macaulay domain.
Ideais de Cohen–Macaulay de codimensão 3. The exact conditions we ought to tailor
in order to have starlike linearity for codimension three ideals are far from clear. In the
next section we will see examples of such ideals that are of the expected star dimension and
nevertheless not even set theoretically starlike linear.
The next proposition gives one result in the positive direction. The main condition,
homological in nature, may not look so natural but it takes place in many situations.
Proposition 71 Let k be a perfect field of characteristic 6= 2 and let A be a k-algebra
essentially of finite type satisfying the following conditions:
(i) For every maximal ideal m of A, Am ' R/I where R is a polynomial ring over k localized
at a prime ideal and I is a perfetct ideal of codimension at most 3 such that the first Koszul
homology module H1 (I; R) is Cohen–Macaulay.
(ii) For every prime ideal ℘ ∈ Spec (A), edim A℘ ≤ 2 dim A℘ .
Then
(a) SA/k = TA/k is a Cohen–Macaulay ring.
(b) D is normally torsionfree if and only if edim A℘ ≤ 2 dim A℘ −1 for every non-minimal
℘ ∈ Spec (A).
Proof. Again it suffices to show the following depth inequalities
depth Tor R
i (A, A) ≥ dim A − (3 − i), 1 ≤ i ≤ 3
where A is assumed to be local and A ' R/I with ht I = 3.
The cases i = 1, 3 are taken care of exactly as in the proof of Proposition 68. As for
2
i = 2, by [69] one has Tor R
2 (A, A) ' ∧ I since I is syzygetic and 1/2 ∈ R. On the other
hand, Weyman’s resolution [80] implies that pd ∧2 I ≤ 4. Therefore
depth ∧2 I ≥ dim R − 4 = dim A − 1.
Example 72 The primeval example of a perfect ideal I ⊂ R such that H1 (I; R) is Cohen–
Macaulay is one that has deviation d(I) := µ(I) − ht I at most two. Thus, codimension
three perfect ideals generated by at most 5 elements satisfy condition (i) of Proposition 71.
For them, A = R/I is starlike linear provided it has the expected dimension. It will follow
that any specialization thereoff having the expected star dimension will be set theoretically
starlike linear.
The assumption on H1 (I; R) being Cohen–Macaulay is not necessary as the following
instance puts forward.
36
Example 73 Consider the ideal I ⊂ k[X1 , . . . , X6 ] generated by the following polynomials:
f1 = X2 X4 + X3 X6 ,
f2 = X 3 X 5 + X 1 X 6 ,
f4 = X2 X3 + X2 X4 + X2 X6 + X62
f3 = X 1 X 2 − X 2 X 5 + X 3 X 5 − X 5 X 6 ,
f5 = X32 + X3 X4 + X3 X6 − X4 X6 ,
f6 = X1 X3 + X1 X4 + X4 X5 + X1 X6 .
This ideal I is perfect of codimension three (and deviation three) and has analytic spread
`( X)(I) = 5 (in particular, I is not of linear type).
Actually, H1 (I; R) is not Cohen-Macaulay as I is not syzygetic: a quadratic relation
lying outside (X)R[T] being
T1 T2 + T2 T4 − T2 T5 + T3 T5 + T1 T6 − T4 T6 ,
with Ti 7→ fi .
The corresponding projective variety cut by quadrics in P5 has degree 4 and is reduced
and irreducible as is shown by means of Noether normalization using the method given in
[76]. Alternatively, one can show under a calculation with Macaulay that the ring R/I is
(R2 ), hence R/I is a normal domain (since it is Cohen–Macaulay) and the corresponding
projective variety is non-singular.
A second calculation in Macaulay grants that the Zariski tangent algebra and the tangent
star algebra are isomorphic. Sufficient evidence from a wasteful computation in the same
program indicates that the algebra is indeed Cohen–Macaulay (possibly, a normal domain
too).
This example is curious in a number of other ways as well. Let us look at the ideal I
more closely.
First, its transposed jacobian matrix has rank 5 over R = k[X] with cokernel L such
that
Φ
Θ
0 −→ R −→ R6 −→ R6 −→ L −→ 0,
where I = (Φt ). Another computation in Macaulay yields that ht I5 (Θ) = 2 when char
k 6= 2, thus showing that L is isomorphic to an ideal of k[X] in this case. This ideal can be
taken as the ideal generated by the cubic polynomials obtained by cancelling the common
factor among the six 5 × 5 minors of a 5 × 6 submatrix of rank 5 of Θ (in particular, it has
codimension two).
Of course, we have Ω(R/I) ' L/IL, a locally free module in the punctured spectrum of
R/I.
Further, L is self–dual in the sense of jacobian duality, i.e., it and its jacobian dual [62]
are isomorphic.
Let us remark that if char k = 2 then the cokernel L has torsion and, moreover, (Φt ) =
(X).
Estudo de caso: ideais determinantais. Determinantal rings are representative of a
more general pattern which is notable in the theory of star algebras. The coordinate rings
of the Veronese and Segre varieties, to be considered later, are examples of special interest.
Typically, the results for this class are more or less definitive, at least in the generic
case. We will content ourselves in quoting the following theorem, whose parts are collected
from [66].
37
Theorem 74 Let R be a regular domain and let (X) be an r × s(r ≤ s) matrix of indeterminants over R. Let t be an integer such that 1 ≤ t ≤ r. Then:
(a) R[X]/It (X) has the expected star dimension if and only if

or
 t=1
t=r
or

t = r − 1, s ≤ r + 1
(b) If R is a localization of a polynomial ring over a field then R[X]/It (X) is set theoretically starlike linear if and only if
t = 1 or
t=r
The proof of (a) is by induction, using the well-known inversion–localization trick for
matrices of indeterminants. The argument for (b) is a consequence of Proposition 82 for
one direction, but a lot more involved for the other direction.
3.2.3
O problema de van Gastel
The rings considered in this section will be again finite type algebras over a field, i.e., of
the form A := R/I, where R = k[X] (k perfect) and I ⊂ (X) an arbitrary ideal.
The following question was raised by L. van Gastel [13].
Question 75 Let A be a reduced affine ring having the expected star dimension. Is A set
theoretically starlike linear?
The preceding sections dealt with this and similar questins from their positive side. Here
we we show that the answer to this question is negative in general.
The question can be so rephrased as to ask whether the two tangent algebras have
the same minimal primes. We observe initially that the hypothesis on the expected star
dimension already implies generic reducedness of A.
Let now J(1) and J stand, respectively, for the presentation ideals of SA/k and TA/k on
the polynomial ring k[X, T]. We know (see, e.g., [26]) that a necessary condition for the
set theoretic equality of SA/k and TA/k is the analytic independence of the generators of D
at the maximal ideal
(X, U)/I(X) + I(U)) ⊂ k[X, U]/I(X) + I(U)) ' A ⊗k A.
This latter condition translates to J ⊂ (X).
All the counterexamples to follow will trespass this condition by exhibiting directly a
(global) relation of analytic independence. In particular, the following weaker version of
van Gastel’s is momentarily open:
Question 76 Assume A = k[X]/I is reduced and has the expected star dimension. If the
generators of D are analytically independent at (X, U)/I(X) + I(U)), is it true that SA/k
and TA/k have the same reduced structure?
38
If one is willing to sacrifice unmixedness, then a “smallest” example that answers the
question negatively is very likely to be the following ideal of codimension two.
Example 77 Let X = X1 , X2 , X3 , X4 and let I be generated by
X1 X2 ,
X2 X 3 ,
X3 X 1 ,
X1 X 4 .
It is easily seen directly that dim SA/k = dim TA/k (= 4). Let T = T1 , T2 , T3 , T4 be the
presentation variables. Then T1 T2 T3 ∈ J.
In order to construct a negative instance which is moreover prime, one can upgrade the
example to a Cohen-Macaulay one. In particular, codimension three must be the case.
Example 78 (char k 6= 2) Let X = X1 , X2 , X3 , X4 , X5 , X6 and let I be generated by
X1 X2 ,
X 2 X3 ,
X 3 X1 ,
X 1 X4 ,
X 2 X5 ,
X 3 X6 .
I is Cohen-Macaulay of codimension three and deviation (= µ(I) − ht I) three (moreover,
it is of linear type).
Again A has the expected star dimension, as is readily checked from the heights of the
various Fitting ideals, using Proposition 59.
The presentation ideal J of the tangent star algebra TA/k has an extra generator as
before, namely, T1 T2 T3 corresponding to the (odd) cycle structure in the graph defined by
the generators of I.
Actually, a computation in Macaulay shows that (provided char k 6= 2) then
J = (J(1), T1 T2 T3 ) and J(1) = M ∩ J,
where M = (X)k[X, T].
Remark 79 We note en passant that the tangent star algebra TA/k is still Cohen-Macaulay.
This holds even if char(k) = 2, though in this case one has J = (J(1), X3 T1 T2 , T1 T2 T3 ) hence
the finer equality J(1) = M ∩ J is no longer valid.
Out of Example 78, using the deformation-linkage argument in the last section of [62],
one can produce many prime ideals preserving similar data.
Example 80 Here is an example linking the preceding one with the regular sequence
X1 X2 + X1 X4 ,
X2 X 3 + X 2 X 5 ,
X3 X 6 .
After introducing generic variables Z1 , Z2 , Z3 and linking again, one finds the following
(non-homogeneous) prime ideal in a polynomial ring in 9 variables.
X3 X4 Z1 + X4 X5 Z1 + X3 X6 Z1 + X4 X6 Z1 + X5 X6 Z1 − X1 X4 Z2 − X1 X6 Z2 + X1 X4 ,
X2 X3 Z1 + X2 X5 Z1 + X2 X6 Z1 − X3 X6 Z1 − X1 X2 Z2 + X1 X6 Z2 + X1 X2 Z3
+X1 X4 Z3 + X1 X2 ,
2
X3 Z1 + X3 X5 Z1 + X3 X6 Z1 − X1 X3 Z2 + X1 X3 Z3 + X1 X5 Z3 + X1 X3 ,
X4 X5 Z2 + X2 X6 Z2 + X4 X6 Z2 + X5 X6 Z2 − X22 Z3 − X2 X4 Z3 + X2 X5 ,
X3 X4 Z2 + X3 X6 Z2 + X22 Z3 + X2 X4 Z3 + X2 X3 ,
X3 X4 Z3 + X4 X5 Z3 + X2 X6 Z3 + X3 X6 Z3 + X4 X6 Z3 + X5 X6 Z3 + X3 X6 .
39
The symmetric algebra SA/k still has the expected dimension (which is now 12). On
the other hand, A is not set theoretically starlike linear by as the present link is also a
deformation of the original ideal.
Remark 81 For the purpose of getting smaller counterexamples, one may specialize down
the Z-variables. Again, it would not be too difficult to apply same primality test as in [76]
to show that a suitable specialization is still prime.
The preceding examples seem somewhat fragmentary. Next we present two counterexamples to van Gastel’s question which are important from the viewpoint of deformation
theory and free resolutions.
1. O cone projetante da superfı́cie de Veronese. This surface is certainly ubiquitous
in classical Algebraic Geometry. The approach taken here is rather simple.
As is well known, the projecting cone of the surface is ideal-theoretically defined by the
2 × 2 minors of the symmetric matrix


X1 X2 X3
 X2 X4 X5  .
X3 X5 X6
Let R := k[X] = k[X1 , . . . , X6 ] and A := R/I, where I is the ideal generated by these
minors. Also well known is that A is a normal Cohen–Macaulay isolated singularity of
of dimension 3. Let T = T1 , . . . , T6 be presentation variables for SA/k (or TA/k ) over A.
Finally, let D(T) stand for the determinant of the above matrix evaluated on the variables
T.
A classical argument will show that D(T) belongs to the presentation ideal of TA/k over
A. This yields the required relation of analytic dependence.
More generally, one has the following intriguing phenomenon:
Proposition 82 Let k be a field, let R = k[X] = [X1 , . . . , Xn ] and let I ⊂ (X) be any ideal
of R. If there exists an r × r matrix M, whose entries are linear forms in R and whose
determinant is non-zero, such that moreover I[(r+1)/2] (M) ⊂ I, then the ring (R/I)(X) is
not set theoretically starlike linear.
Proof. The canonical generators {xi ⊗ 1 − 1 ⊗ xi | 1 ≤ i ≤ n} of D (here xi denotes the
residue of Xi in B = (R/I)(X) ) are minimal generators. Therefore, as remarked above, they
would have to be analytically independent at the maximal ideal (xi ⊗ 1, 1 ⊗ xi | 1 ≤ i ≤ n)
were B to be set theoretically starlike linear. Now, expanding the determinant of the
r × r matrix M ⊗ 1 − 1 ⊗ M by Laplace, using multilinearity and the assumption that the
[(r + 1)/2] × [(r + 1)/2] minors of M belong to the given ideal I, one finds
det(M ⊗ 1 − 1 ⊗ M) ⊂ I[(r+1)/2] ⊗ R + R ⊗ I[(r+1)/2] ⊂ I ⊗ R + R ⊗ I,
which yields a relation of analytic dependence.
Going back to the example, it remains to verify that A has the expected star dimension.
But this is clear by Proposition 59, (b), since A is an isolated singularity (alternatively, by
a computation in Macaulay using (d) of the same proposition).
40
Remark 83 (1) The ideal I above is interesting from the purely algebraic viewpoint. First,
the preceding argument shows a posteriori that its first Koszul homology module H1 (I; R)
is not Cohen–Macaulay (cf. Proposition 71). Nonetheless, it is known [35] that it is an ideal
of linear type, i.e., its blowing-up algebra coincides with the residual scheme (symmetric
algebra of I). This seemingly makes the counterexample very tight.
(2) The star algebra TA/k is a Cohen–Macaulay domain (computation in Macaulay).
(3) As in Example 78, here too the finer equality J(1) = M ∩J hold true provided char k 6= 2.
2. O cone projetante da variedade de Segre Σ2,2 . It is well known that this cone
is ideal-theoretically defined by the 2 × 2 minors of a generic 3 × 3 matrix. The argument
is then ipsis-literis the same as the one for the Veronese, a relation of analytic dependence
being given once more by the determinant evaluated at T. Thus, this yields an additional
counterexample to van Gastel’s question.
We close this section with a remark about the characteristic of the ground field k.
The difference between characteristic 2 and other characteristics turns out to be deeper
than it appears on the surface, at least for ideals generated by quadratic polynomials.
Practically all the explicit examples discussed so far have in common the following
feature. Let I ⊂ R = k[X] be a homogeneous ideal generated by quadrics and let E stand
for the R-module defined as the cokernel of the transposed jacobian matrix of a set of
generators of I. Assume that E has rank zero as R-module. Then the maximal minors
of the jacobian matrix Θ, viewed in the T-variables, belong to the presentation ideal J of
TA/k (and, clearly, not to J(1)).
In the case of the veronesean and the Segre threefold, the module E indeed has rank
zero if char 2 6= 0 (but positive rank otherwise!) and, moreover, the maximal minors of Θ
are indeed powers of generators of J not lying in J(1).
This yields an efficient, as yet not completely understood, method of verifying the deficiency of the corresponding secant variety.
3.2.4
A álgebra de Gauss de um módulo
Definition 84 Let R be a positively graded ring with graded maximal ideal m and let M
be V
a finitely generated graded R-module having (generic) rank r. The special fiber algebra
R( r M ) ⊗R R/m is called the (abstract) Gauss algebra of M and denoted GM (R).
Basic lemmata are as follows.
Lemma 85 If R is a standard domain over a field and M is finitely generated in the same
degree then GM (R) is a domain.
Proof. This of course holds, more generally, for any module with a rank generated in
the same degree (not just the wegde of such a module). The argument goes as follows.
Let k be the base field. Under the assumed conditions, k = R/m and the special fiber is
graded-isomorphic to a k-subalgebra of the Rees algebra and the latter is a domain.
Next we show that, essentially, any abstract Gauss algebra is the abstract Gauss algebra
of a module of projective dimension one.
41
Lemma 86 Let R be a standard ring over a field k and let M be finitely generated in
the same degree. Then there is a module M 0 , finitely generated in the same degree and of
projective dimension one such that the respective Gauss algebras GM 0 (R) and GM (R) are
isomorphic as graded k-algebras.
Proof. We actually claim that there is such an M 0 as follows: there is a homogeneous
R-module surjection M 0 M and there is a functorially induced map GM 0 (R) GM (R)
which is an isomorphism of k-algebras. Indeed, let F be a finitely generated graded free
module generated by homogeneous elements of the same degree mapping surjectively onto
M by a degree-preserving map. Let U denote the kernel of this surjection. There exists a
homogeneous free R-submodule U 0 of U which coincides with U locally at the associated
0
0
primes of R. Letting M 0 be the quotient F/U 0 , the natural surjection M
Vr = 0F/U ∼
F/U V
M surjects
= M is obtained by factoring out a torsion R-module, so similarly
r
onto
M
with
an
R-torsion
kernel,
and
the
fucntorially
induced
map
of
Rees
algebras
Vr 0
Vr
R( M ) → R( M ) is an isomorphism. Therefore, the induced map GM 0 (R) → GM (R)
is an isomorphism.
The reason for the terminology is that, in a particular case, the algebra is the homogeneous coordinate ring of the Plücker embedding of the Gauss image of a closed subvariety
of projective space. To see this, let R be a standard graded ring over an algebraically closed
field k - hence the homogeneous coordinate ring of a projective subvariety X ⊂ Pnk . Let M
be a finitely generated graded R-module in the same degree and consider its Gauss algebra
GM (R). By Lemma 86, we may assume that M is the cokernel of an injective map ϕ of
free graded R-modules U ,→ F whose ranks are c and c + r, respectively (recall r is the
rank of M ). Consider the set of determinants ∆j1 ...jc of ϕ of order c and the k-subalgebra
A = k[∆j1 ...jc | 1 ≤ j1 < · · · < jc ] ⊂ R. Here ∆j1 ...jc denotes the minor of ϕ indexed by the
rows j1 , . . . , jc . Since the matrix of the map is homogeneous of rank c, these determinants
are all of the same degree and do not all vanish, hence the above ring extension defines a
c+r
V
rational map γM : X 99K P( c (F/mF )) = P( c ) .
We then claim that GM (R) ' A as graded k-algebras, thus showing that GM (R) is
indeed the homogeneous coordinate ring of the image of the rational map γM . Since A is
graded-isomorphic to the special fiber R(I) ⊗R k, where I is the ideal of R generated
V by
the minors ∆j1 ...jc (cf. Lemma 85), it suffices to show an isomorphism R(I) = R( r M ).
ϕ
But the map U ,→ F induces a map
r
^
c
c
^
(∧c ϕ)∗ ^
F ' ( F )∗ −→ ( U )∗ ' R
V
whose image is the ideal I of V
R, where ∗ denotes R-dual.
This map factors through r M ,
V
r
and thus
M I. Because r M has rank one, the kernel is torsion,
Vr induces a surjection
so R( M ) ' R(I).
Applying this procedure to the case where M is the module Ωk (R) of k-differentials of R
recovers the classical Gauss image of X in its Plücker embedding (here we of course assume
that k is perfect and that R is reduced and equidimensional in order that Ωk (R) have rank
r = dim R).
That is not to say that Ωk (R) is always of projective dimension one (the case when R is
a complete intersection). But one can take for ϕ a sufficiently general (n + 1) × c submatrix
42
of the Jacobian matrix (on R) of a set of homogeneous generators of the homogeneous
defining ideal of X = Proj(R), where c is the embedding codimension of R. In fact, taking
a subideal generated by a regular sequence of length c which coincides with the defining
ideal locally at the minimal primes of R will do: we then take ϕ to be the Jacobian matrix
(on R) of the elements of this regular sequence.
We set G(R) = GΩk (R) (R). There are many interesting questions concerning this algebra. We mention some of the known results.
Theorem 87 ([55, Theorem 2.1]) Let R be a standard graded domain over a field k, of
dimension d. Assume that k is algebraically closed in the field of fractions of R. If R is not
a polynomial ring, then dim G(R) ≥ ht Fd (R), where Fi (R) denotes the ith Fitting ideal of
Ωk (R).
The analogue of the above for a local ring (R, m) essentially of finite type over a field k
is not known. Note the difficulty: dim R is no longer the rank of Ωk (R), the latter being
rather equal to dim R + tr.deg.k (R/m).
For an open question concerning a general module M , see the section of problems at
the end.
Theorem 88 ([3, Main Lemma 2.3]) Let T be a polynomial ring over a field k and let
R ⊂ T be a k-subalgebra generated by forms g = g0 , . . . , gn of the same degree such that
dim R = dim T . Then there is an isomorphism of graded k-algebras G(R) ' k[∆(g)], where
∆(g) is the set of the d × d minors of the Jacobian matrix of g.
The isomorphism in this theorem is not obvious (much less a set inclusion). It may be
called the “complementarity isomorphism” of a parametric k-algebra R.
3.2.5
A álgebra tangencial
In this portion we wish to show that the special fiber of the module of k-differentials of a
standard graded k-algebra R has a nice interpretation as the homogeneous coordinate ring
of the so-called tabgencial variety to the projective variety Proj(R).
For this, we need some general preliminaries.
Definition 89 Let B be a noetherian ring, F a free B-module. Let M ⊂ F be a submodule
and J ⊂ B an ideal. One defines the saturation of M with respect to J to be the submodule
(M : J ∞ ) := ∪t≥1 (M : J t ) ⊂ F.
Here, (M : J t ) = {f ∈ F | f J t ⊂ M }.
Let us see how saturation yields torsion.
Lemma 90 Let A be a noetherian ring and let B be a noetherian A-algebra. Let n ⊂ A be
an ideal containing a non-zero-divisor such that B is A-torsionfree locally on Spec (A)\V (n)
(i.e., Bp is Ap -torsionfree for every p ∈ Spec (A), p 6⊃ n). Then the A-torsion submodule of
B is the ideal (0) : n∞ .
43
We apply to the context of certain Rees algebras:
Proposition 91 Let R be a reduced affine k-algebra (k perfect, R not necessarily graded).
Then R(Ωk (R)) = S(Ωk (R))/((0) : J∞
R ) where JR ⊂ R is the jacobian ideal of R.
The following geometric notion is very useful in the theory of projections.
Definition 92 Let V ⊂ Pn be a projective reduced algebraic set. The tangential variety
to V (for the given embedding) is the Zariski closure of the set ∪p Tp,V ⊂ Pn , where p runs
through the smooth points of V (N.B. Here Tp,V denotes the embedded projective tangent
space to V at p).
If the embedding of V is clear from the context, we denote the tangential variety to V
by T (V ). It is known (and relatively easy to check) that T (V ) is indeed an algebraic set.
The next proposition shows this with slightly more precision.
Proposition 93 Let R be a reduced standard graded k-algebra (k algebraically closed). Let
V = Proj(R). Then T (V ) = Proj(R(Ωk (R)) ⊗R k).
4
Alguns problemas
Problem. Let R be a Noetherian local domain and M a finitely generated R-module of
rank
Vr r which is not a direct sum of a free module and a torsion module. Does the inequality
`( M ) ≥ htFr (M ) hold, where Fr (M ) is the r-th Fitting ideal of M ?
The problem is open even for the module Ωk (R) of k-differentials of a local domain
essentially of finite type over an algebraically closed field of characteristic zero!
Problem. What other Gauss algebras GM (R) of modules M play a significant role in
algebraic geometry, besides the case where M = Ωk (R)?
We know presently of no other instance!
Problem. What is the parametric expression of G(R) if R is a graded k-subalgebra of
a polynomial ring over k of smaller dimension than the ambient? Can one systematically
reduce to the case where the dimension is the same?
That seems to require some technicality, but is not out of reach.
Problem. Are there reasonably general conditions under which the equality dim G(R) =
dim R holds, besides that of R being isolated singularity (i.e., Proj(R) smooth)?
We mean algebraic conditions which are elegantly stated in terms of the algebraic data.
Problem. Are there sharp inequalities (becoming sometimes equalities) for the degree of
the image of a rational map whose base locus has arbitrary dimension?
An answer to this question would seemingly require altogether a different or souped-up
approach regarding the one in Theorem 19.
44
Problem. Find a formula for the degree of the dual hypersurface to a (non-deficient)
arithmetically Cohen–Macaulay projective variety.
Kleiman ([33]) has found such a formula for a complete intersection with (geometric)
isolated singularities in terms of certain Chern numbers. Here we mean a formula in more
down-to-earth algebraic invariants even in the case considered by Kleiman. It would seem
that the appropriate object to look at is some version of the Jacobian module attached to
the Jacobian matrix of the defining equations.
Problem. Extend the results on the Rees algebra of a module to any ring dimension
and, in the case the ring is regular (say), to higher projective dimensions. Are there any
structural results in higher projective dimension?
This question seems fairly intractable even in projective dimension two.
Problem. Relate a Bourbaki ideal of the module of Kähler differentials to the tangential
and Gauss algebras.
This is very vague but the explicit constructions deserve a better look.
45
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Departamento de Matemática, CCEN, Universidade Federal de Pernambuco,
Cidade Universitária, 50740-540 Recife, PE, Brazil
E-mail: [email protected]
50

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