Krull, Gelfand-Kirillov and Filter Dimensions of Simple Affine - - PDF document
Krull, Gelfand-Kirillov and Filter Dimensions of Simple Affine Algebras Vladimir Bavula Talks/lectbonn 1 Plan 1. Gelfand-Kirillov Dimension, Examples. 2. Filter Dimension. 3. Analog of the Inequality of Bernstein for Simple Affine
Vladimir Bavula
∗
∗Talks/lectbonn
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Simple Affine Algebras.
and Filter Dimensions for Simple Affine Al-
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module=left module K is a field of char 0 N and R are sets of natural and real numbers
number or ∞ defined as γ(f) := inf{r ∈ R : f(i) ≤ ir for suff. large i >> 0} is called the degree (or growth) of f. For functions f, g : N → N: γ(f + g) ≤ max{γ(f), γ(g)}, γ(fg) ≤ γ(f) + γ(g), γ(f ◦ g) ≤ γ(f)γ(g), where f ◦g is the composition of the functions.
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Let A be an affine (≡ finitely generated) al- gebra with generators x1, . . . , xn. Then A is equipped with a standard finite dimensional fil- tration A = ∪i≥0 Ai, A0 = K, A1 = K +
∑ n
i=1 Kxi, Ai := Ai 1, i ≥ 2.
Let M be a finitely generated A-module and M0 be a finite dimensional generating subspace
dimensional filtration M = ∪i≥0 Mi, Mi = AiM0. Definition (Gelfand-Kirillov, 1966). The Gelfand-Kirillov dimension of the A-module M: GK(M) := γ(i → dim Mi). The Gelfand-Kirillov dimension of the algebra A: GK(A) := γ(i → dim Ai).
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GK(M) and GK(A) do not depend on the choice
nomial ring in n indeterminates. Pn = ∪i≥0 Pn,i, Pn,0 = K, Pn,1 = K+
∑ n
i=1 KXi,
Pn,i =
∑
{KXα | |α| ≤ i}, Xα := Xα1
1 · · · Xαn n ,
|α| = α1 + · · · + αn.
(n+i
n
)
= (i+n)(i+n−1) · · · (i+1)/n! = in/n!+· · · .
The n’th Weyl algebra An = K < X1, . . . , Xn, ∂1, . . . , ∂n >
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defining relations: ∂iXj − Xj∂i = δij, the Kronecker delta, XiXj − XjXi = ∂i∂j − ∂j∂i = 0, i, j = 1, . . . , n. The algebra An is a simple Noetherian infinite dimensional algebra canonically isomorphic to the ring of differential operators with polyno- mial coefficients An ≃ K[X1, . . . , Xn, ∂/∂X1, . . . , ∂/∂Xn], Xi ↔ Xi, ∂i ↔ ∂i/∂Xi, i = 1, . . . , n.
An,i =
∑
{KXα∂β, |α| + |β| ≤ i}, dim An,i =
(2n + i
2n
)
= i2n/(2n)! + · · · .
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Lemma 1. Let A be a simple affine inf. dim. algebra and let M ̸= 0 be a f.g. A-module. Then dim M = ∞, hence GK(M) ≥ 1.
algebra homomorphism A → HomK(M, M), a → (m → am), is injective, so ∞ = dim A ≤ dim HomK(M, M) and dim M = ∞. Theorem 2. (The inequality of Bernstein, 1972). For a nonzero finitely generated mod- ule M over the Weyl algebra An, GK(M) ≥ n. Let X be a smooth irreducible algebraic vari- ety of dimension n. Let D(X) be the ring of differential operators on X.
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X = Sn := {(xi) ∈ Kn+1 : x2
1 + · · · + x2 n+1 =
1}, D(Sn). The alg. D(X) is a simple affine Noetherian
Theorem 3. (McConnell-Robson). For a nonzero finitely generated D(X)-module M, GK(M) ≥ n.
iA := inf{GK(M), M is a nonzero finitely generated A − module}? To find an approximation for iA was a main motivation for introducing the filter dimen- sion.
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Let A be a simple affine algebra with the filtra- tion F = {Ai}, A = ∪i≥0 Ai. Define the return function νF : N → N of A: νF(i) := inf{j ∈ N : AjaAj ∋ 1 for all 0 ̸= a ∈ Ai}, where AjaAj is the subspace of A generated by the products xay, for all x, y ∈ Aj.
fdim A := γ(νF). The filter dimension does not depend on the choice of F.
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Theorem 4. (B., 1996). Let A be a simple affine infinite dimensional algebra.
M: GK(M) ≥ GK(A) fdim(A) + max{fdim(A), 1}.
and let F = {Ai} be the filtration of A.
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Let M0 be a fin. dim. gen. subspace of the A-module M: M = ∪i≥0 Mi, Mi = AiM0, i ≥ 0. It follows from the definition of the return func- tion ν = νF of A that, for every 0 ̸= a ∈ Ai, 1 ∈ Aν(i)aAν(i). Now, M0 = 1M0 ⊆ Aν(i)aAν(i)M0 ⊆ Aν(i)aMν(i), so the linear map Ai → HomK(Mν(i), Mν(i)+i), a → (m → am), is injective, hence dim Ai ≤ dim HomK(Mν(i), Mν(i)+i) = dim Mν(i) dim Mν(i)+i. Using the elementary properties of the degree, we have GK(A) := γ(dim Ai) ≤ γ(dim Mν(i) dim Mν(i)+i)
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≤ γ(dim Mν(i)) + γ(dim Mν(i)+i) ≤ γ(dim Mi)γ(ν) +γ(dim Mi) max{γ(ν), 1 = γ(i → i)} = GK(M)(fdim A + max{fdim A, 1}), since GK(M) = γ(dim Mi) and fdim A = γ(ν).
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Theorem 5. (B., 1998). Let D(X) be the ring of differential operators on a smooth irre- ducible algebraic variety X of dimension n.The filter dimension fdim D(X) = 1.
generated D(X)-module M, GK(M) ≥ n. Proof. GK(M) ≥ GK(D(X)) fdim(D(X)) + max{fdim(D(X)), 1} = 2n 1 + max{1, 1} = 2n 2 = n.
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K.dim , the Krull dimension (in the sense of Rentschler-Gabriel, 1967) Theorem 6 (Rentschler-Gabriel, 1967) Let An be the Weyl algebra. Then K.dim An = n. Theorem 7 (McConnell-Robson) Let X be a smooth irreducible algebraic variety of dim
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titive if, given any fin. gen. S-module M, there is an integer n > 0 s. t. for every chain
M = M0 ⊃ M1 ⊃ . . . ⊃ Mm with GK(Mi/Mi+1) = GK(M), one has m ≤ n. Lemma 8. D(X) is a finitely partitive alg., and for any fin. gen. D(X)-module M, GK(M) is a natural number. Theorem 9 (B., 1998) Let A be a finitely par- titive simple affine algebra with GK(A) < ∞. Suppose that the Gelfand-Kirillov dimension of every finitely generated A-module is a natural
ated A-module M, the Krull dimension K.dim(M) ≤ GK(M)− GK(A) fdim(A) + max{fdim(A), 1}. In particular, K.dim(A) ≤ GK(A)(1− 1 fdim(A) + max{fdim(A), 1}).
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By Theorem 9, K.dim D(X) ≤ 2n(1 −
1 1+max{1,1})
= 2n(1 − 1
2) = 2n 2 = n.
K.dim D(X) ≥ n, easy. ******************************************* A generalization to AFFINE ALGEBRAS is given in
Joseph theorem for affine algebras”, Proc. Ed- inburgh Math. Soc. 42 (1999), no.2, 311– 332. (Faithful and Schur Dimensions)
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