Krull, Gelfand-Kirillov and Filter dimension

Vladimir Bavula

In the first lecture a connection will be established between Krull, Gelfand-Kirillov and Filter dimension of simple affine algebras, and an analog of Bernstein’s inequality will be proved for the nonzero finitely generated modules over simple affine algebras. These results will be applied to compute the Krull dimension of the ring of differential operators on a smooth affine variety. Detailed proofs will be given.

In the second lecture we give a review of (recent) results on the Krull dimension of skew Laurent extensions and generalized Weyl algebras with left Noetherian coefficients, based on papers of P. Gabriel, K.R. Goodearl, D. Jordan, T.J. Hodges, T.H. Lenagan, J.C. McConnell, J.J. Pettit, R. Rentschler, J.C. Robson, P. Smith, F. Van Oystaeyen, R.B. Warfield and V. Bavula.

These lectures can be considered as an addition to and continuation of the lectures given by Tom Lenagan.

1 Krull, Gelfand-Kirillov and Filter dimensions of simple affine algebras

Let K be a field, a module means a left one, ⊗ = ⊗_{K}. Let A_{n} be the n’th Weyl algebra, i. e. the K−algebra with 2n generators X_{1}, …, X_{n}, ∂_{1}, …, ∂_{n} and relations

∂_{i}X_{j} − X_{j}∂_{i} = δ_{ij}, the Kronecker delta, X_{i}X_{j} − X_{j}X_{i} = ∂_{i}∂_{j} − ∂_{j}∂_{i} = 0.

If char K = 0, then A_{n} is a simple Noetherian algebra.

Let K.dim, GK and fil.dim be the (left) Krull (in the sense of Rentschler and Gabriel, [RG]), Gelfand-Kirillov and Filter dimension respectively.

Theorem 1.1 ( the Bernstein’s inequality, [Be]). Let A_{n} be the n’th Weyl algebra over a field of characteristic zero. Then GK(M) ≥ n for every nonzero finitely generated A_{n}-module M.

For each function f from N_{0} to R′, where R′ = {r ∈ R, r ≥ 1} we can associate a number γ(f) ∈ R, a degree of f,

γ(f) = inf{r ∈ R : f(n) ≤ n^{r} for sufficiently large n >> 0}.

Let A be an affine algebra with generators a_{1}, …, a_{s}. Then A is equipped with a standard finite dimensional filtration F: A = ∪_{i≥0} A_{i}, where A_{0} = K, A_{1} = K+∑ _{i=1}^{s} Ka_{i}, A_{i} := A_{1}^{i}, i ≥ 2.

If the algebra A is simple, define the return function ν_{F} : N_{0} → N_{0} ∪ {∞} of A associated with F in the following way

ν_{F} (i) = inf{j ∈ N_{0} ∪ {∞} : A_{j}aA_{j} ∋ 1 for all 0 ≠ a ∈ A_{i}},

where A_{j}aA_{j} is the subspace of A generated by the products xay, for all x, y ∈ A_{j}. Definition. ([Bav1]). The filter dimension of A is defined as fil.dim A = γ(ν_{F}).

Theorem 1.2 ([Bav1]). (An analog of the Bernstein’s inequality for the simple affine algebras).

Let A be a simple affine algebra; then

GK(A) ≤ GK(M)(fil.dim A + max{fil.dim A, 1})

for every nonzero finitely generated A−module M. Moreover, if A is infinite dimensional, then

fil.dim A ≥ 1/2.

We say that an algebra S is (left) finitely partitive ([Jos 1] or [MR], 8.3.17) if, given any finitely generated S-module M, there is an integer n > 0 such that for every chain

M = M_{0} ⊃ M_{1} ⊃ … ⊃ M_{m}

with GK(M_{i}/M_{i+1}) = GK(M), one has m ≤ n.

Theorem 1.3 ([Bav2]). Let A be a finitely partitive simple affine algebra with GK(A) < ∞. Then, for any nonzero finitely generated A-module M, the Krull dimension

K.dim(M) ≤ GK(M) − GK(A)/(fil.dim A + max{fil.dim A, 1}).

In particular case,

K.dim(A) ≤ GK(A)(1 − 1/(fil.dim A + max{fil.dim A, 1})).

Let B be an integral regular domain, affine over a field of characteristic zero and D(B) be its ring of differential operators. It is well-known that D(B) is a simple affine Noetherian algebra ([MR], ch. 15).

Theorem 1.4 ([Bav2]). The filter dimension

fil.dim D(B) = 1.

As an application we compute the Krull dimension of D(B).

Theorem 1.5 ([MR], ch. 15).

K.dimD(B) = GK(D(B))/2 = K.dim(B).

Theorem 1.6 ([MR], 15.4.3). Let M be a nonzero finitely generated D(B)-module. Then

GK(M) ≥ GK(D(B))/2 = K.dim(B).

2 Krull dimension of skew Laurent extensions and generalized Weyl algebras

Let R be a ring, σ ∈ Aut(R) and a ∈ Z(R). Denote by S (resp. T) a skew Laurent extension S = R[X, X^{−1}; σ] (resp. a generalized Weyl algebra T = R(σ, a)). Note that if a = 1 we have S = T.

Theorem 2.1 ([Ho]). Let the commutative Noetherian ring R have finite Krull dimension; then

K(R[X, X^{−1}; σ]) = sup{K(R), ht q + 1 | q is a σ − semistable prime of R}.

Here ht p is the height of an ideal p and p is σ-semistable, if σ^{n}(p) = p for some integer n. If there is no such n, the ideal is called σ-unstable.

Theorem 2.2 ([BVO]). Let R be a commutative Noetherian ring with K(R) < ∞ and T = R(σ, a) be a generalized Weyl algebra. Then

K(T) = sup{K(R), ht p + 1, ht q + 1 | p is a σ − unstable prime ideal of R for which

there exist infinitely many integers i with a ∈ σ^{i}(p);

q is a σ − semistable prime ideal of R}.

Let K be a field and Usl(2) be the K-enveloping algebra of the Lie algebra

sl(2) =< X, Y, H | [H, X] = X, [H, Y] = −Y, [X, Y] = 2H >,

The Casimir element C = Y X + H(H + 1) is central in Usl(2)).

Usl(2) ≃ K[H, C](σ, a = C − H(H + 1)), X → X, Y → Y, H → H,

σ given by H → H − 1, C → C. Applying Theorem 2.2 we obtain:

Corollary 2.3 ([Sm] for K = C; see also [MR], 8.6.15) K(Usl(2)) is either two when char K = 0, or three when char K = p > 0.

Definition, [GL]. A critical R-module M is called S-clean if M is S ⊗_{R} M is a critical S-module.

Definition, [GL]. Let A be a simple R-module and let M be an arbitrary R-module. Then the height h(A : M) of A in M is defined to be the supremum of those non-negative integers n for which there exists a sequence A = A_{0}, A_{1}, …, A_{n} of S-clean R-modules such that A_{i} is isomorphic to a minor subfactor of A_{i+1}, for i = 0, …, n − 1, while A_{n} is isomorphic to a subfactor (not necessarily minor) of M.

Theorem 2.4 ([GL]). Let R be a left noetherian ring with finite Krull dimension. Then

K(S) = K(R)

unless there exists a simple R-module A such that

K(S ⊗_{R} A) = 1 and h(A : R) = K(R),

in which case

K(S) = K(R) + 1.

Definition, ([BL]). Let A be a simple R-module and let M be an arbitrary R-module.

1. If A is a-torsionfree, then h(A : M) is defined to be the supremum of those non-negative integers n for which there exists a sequence A = A_{0}, A_{1}, …, A_{n} of T -clean R-modules such that A_{i} is isomorphic to a minor subfactor of A_{i+1}, for i = 0, …, n − 1, while A_{n} is isomorphic to a subfactor (not necessarily minor) of M.

2. If A is a-torsion, then h_{+}(A : M), (h_{−}(A : M)), is defined to be the supremum of those non-negative integers n for which there exists a sequence A = A_{0}, A_{1}, …, A_{n} of R-modules such that each of the R-modules A = A_{0}, A_{1}, …, A_{i}, for some i ≥ 0, is (T, +)-clean, ((T, −)-clean), while each of the R-modules A_{i+1}, …, A_{n} is T -clean; and A_{j} is isomorphic to a minor subfactor of A_{j+1}, for j = 0, … n − 1, while A_{n} is isomorphic to a subfactor (not necessarily minor) of M.

Theorem 2.5 ([BL]). Let R be a left noetherian ring with finite Krull dimension. Then

K(T) = K(R)

unless there exists either

(i) a simple a-torsionfree R-module A such that h(A : R) = K(R) and ^{σ^{i}}A ≃ A for some i; or (ii) a simple (T, +)-clean R-module A such that h_{+}(A : R) = K(R) and the set St_{+}(A) is infinite; or (iii) a simple (T, −)-clean R-module A such that h_{−}(A : R) = K(R) and the set St_{−}(A) is infinite;

in which case

K(T) = K(R) + 1,

where St_{±}(A) = {±i ≥ 0 | σ^{∓i}(a)A = 0}.

References

[Bav1] V. V. Bavula, Filter dimension of algebras and modules, a simplisity criterion of generalized Weyl algebras, Comm. Algebra, 24 (1996), no. 6, 1971-1992.

[Bav2] V. V. Bavula, Krull, Gelfand-Kirillov and filter dimensions of simple affine algebras, J. Algebra, 206 (1998), no. 1, 33–39.

[BVO] V. V. Bavula and F. Van Oystaeyen, Krull dimension of generalized Weyl algebras and iterated polynomial rings. Commutative coefficients, Preprint, Univ. of Antwerp, (1997); J. Algebra, (to appear).

[BL] V. V. Bavula and T. H. Lenagan, Krull dimension of generalized Weyl algebras with non-commutative coefficients, Preprint, Univ. of Edinburgh, (1998); Tr. AMS, (to appear).

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[MR] J. C. McConnell and J. C. Robson, Non-commutative Noetherian rings, Wiley, 1987.

[RG] R. Rentschler and P. Gabriel, Sur la dimension des anneaux est ensembles ordonnés, C. R. Acad. Sci. Paris, Sér. A, 265 (1967), 712–715.

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