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 Algebras. 4. Inequality between Krull, Gelfand-Kirillov and Filter Dimensions for Simple Affine Al- gebras. Applications to D -modules. 2

1. Gelfand-Kirillov Dimension module=left module K is a field of char 0 N and R are sets of natural and real numbers Definition . For a function f : N → N , the real number or ∞ defined as γ ( f ) := inf { r ∈ R : f ( i ) ≤ i r 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. 3

Let A be an affine ( ≡ finitely generated ) al- gebra with generators x 1 , . . . , x n . Then A is equipped with a standard finite dimensional fil- tration A = ∪ i ≥ 0 A i , A 0 = K, ∑ n i =1 Kx i , A i := A i A 1 = K + 1 , i ≥ 2 . Let M be a finitely generated A -module and M 0 be a finite dimensional generating subspace of M , M = AM 0 . The module M has a finite dimensional filtration M = ∪ i ≥ 0 M i , M i = A i M 0 . Definition (Gelfand-Kirillov, 1966) . The Gelfand-Kirillov dimension of the A -module M : GK( M ) := γ ( i → dim M i ) . The Gelfand-Kirillov dimension of the algebra A : GK( A ) := γ ( i → dim A i ) . 4

GK( M ) and GK( A ) do not depend on the choice of the filtrations. Example . Let P n = K [ X 1 , . . . , X n ] be the poly- nomial ring in n indeterminates. ∑ n P n = ∪ i ≥ 0 P n,i , P n, 0 = K, P n, 1 = K + i =1 KX i , { KX α | | α | ≤ i } , X α := X α 1 1 · · · X α n ∑ P n,i = n , | α | = α 1 + · · · + α n . ( n + i ) • dim P n,i = n = ( i + n )( i + n − 1) · · · ( i +1) /n ! = i n /n !+ · · · . • GK( P n ) = n . The n ’th Weyl algebra A n = K < X 1 , . . . , X n , ∂ 1 , . . . , ∂ n > 5

defining 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 , i, j = 1 , . . . , n. The algebra A n is a simple Noetherian infinite dimensional algebra canonically isomorphic to the ring of differential operators with polyno- mial coefficients A n ≃ K [ X 1 , . . . , X n , ∂/∂X 1 , . . . , ∂/∂X n ] , X i ↔ X i , ∂ i ↔ ∂ i /∂X i , i = 1 , . . . , n. • { X α ∂ β } is a K -basis of A n . • A filtration: A n = ∪ i ≥ 0 A n,i , { KX α ∂ β , | α | + | β | ≤ i } , ∑ A n,i = ( 2 n + i = i 2 n / (2 n )! + · · · . ) dim A n,i = 2 n • GK( A n ) = 2 n . 6

2. Filter Dimension 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. Proof . The alg. A is simple, so the nonzero algebra homomorphism A → Hom K ( M, M ) , a �→ ( m �→ am ) , is injective, so ∞ = dim A ≤ dim Hom K ( M, M ) and dim M = ∞ . Theorem 2 . ( The inequality of Bernstein, 1972 ). For a nonzero finitely generated mod- ule M over the Weyl algebra A n , 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 . 7

Example . X = K n , D ( K n ) = A n ; X = S n := { ( x i ) ∈ K n +1 : x 2 1 + · · · + x 2 n +1 = 1 } , D ( S n ). The alg. D ( X ) is a simple affine Noetherian inf. dim. algebra with GK( D ( X )) = 2 n . Theorem 3 . ( McConnell-Robson ). For a nonzero finitely generated D ( X )- module M , GK( M ) ≥ n. Question . How to find (estimate) the number i A := inf { GK( M ) , M is a nonzero finitely generated A − module } ? To find an approximation for i A was a main motivation for introducing the filter dimen- sion. 8

Let A be a simple affine algebra with the filtra- tion F = { A i } , A = ∪ i ≥ 0 A i . Define the return function ν F : N → N of A : ν F ( i ) := inf { j ∈ N : 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 . (B., 1996). The filter dimension of A : fdim A := γ ( ν F ) . The filter dimension does not depend on the choice of F . 9

3. Analog of the Inequality of Bernstein for Simple Affine Algebras Theorem 4 . (B., 1996). Let A be a simple affine infinite dimensional algebra. 1. fdim A ≥ 1 / 2. 2. For every nonzero finitely generated A − module M : GK( A ) GK( M ) ≥ fdim( A ) + max { fdim( A ) , 1 } . Proof . 2. Let A = K < x 1 , . . . , x n > = ∪ i ≥ 0 A i and let F = { A i } be the filtration of A . 10

Let M 0 be a fin. dim. gen. subspace of the A -module M : M = ∪ i ≥ 0 M i , M i = A i M 0 , i ≥ 0 . It follows from the definition of the return func- tion ν = ν F of A that, for every 0 ̸ = a ∈ A i , 1 ∈ A ν ( i ) aA ν ( i ) . Now, M 0 = 1 M 0 ⊆ A ν ( i ) aA ν ( i ) M 0 ⊆ A ν ( i ) aM ν ( i ) , so the linear map A i → Hom K ( M ν ( i ) , M ν ( i )+ i ) , a �→ ( m �→ am ) , is injective, hence dim A i ≤ dim Hom K ( M ν ( i ) , M ν ( i )+ i ) = dim M ν ( i ) dim M ν ( i )+ i . Using the elementary properties of the degree, we have GK( A ) := γ (dim A i ) ≤ γ (dim M ν ( i ) dim M ν ( i )+ i ) 11

≤ γ (dim M ν ( i ) ) + γ (dim M ν ( i )+ i ) ≤ γ (dim M i ) γ ( ν ) + γ (dim M i ) max { γ ( ν ) , 1 = γ ( i → i ) } = GK( M )(fdim A + max { fdim A, 1 } ) , since GK( M ) = γ (dim M i ) and fdim A = γ ( ν ) . 12

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 . • ( McConnell-Robson ). For a nonzero finitely generated D ( X )- module M , GK( M ) ≥ n. Proof . GK( D ( X )) GK( M ) ≥ fdim( D ( X )) + max { fdim( D ( X )) , 1 } 1 + max { 1 , 1 } = 2 n 2 n = 2 = n. 13

4. Inequality between Krull, Gelfand-Kirillov and Filter Dimensions for Simple Affine Algebras. Applications to D -modules K.dim , the Krull dimension (in the sense of Rentschler-Gabriel , 1967) Theorem 6 ( Rentschler-Gabriel , 1967) Let A n be the Weyl algebra . Then K . dim A n = n . Theorem 7 ( McConnell-Robson ) Let X be a smooth irreducible algebraic variety of dim n . Then K . dim D ( X ) = n . 14

Definition . An algebra S is called finitely par- titive if, given any fin. gen. S -module M , there is an integer n > 0 s. t. for every chain of submodules M = M 0 ⊃ M 1 ⊃ . . . ⊃ M m with GK( M i /M i +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 number. Then, for any nonzero finitely gener- ated A -module M , the Krull dimension GK( A ) K . dim( M ) ≤ GK( M ) − fdim( A ) + max { fdim( A ) , 1 } . In particular, 1 K . dim( A ) ≤ GK( A )(1 − fdim( A ) + max { fdim( A ) , 1 } ) . 15

• ( McConnell-Robson ). K . dim D ( X ) = n . Proof . GK D ( X ) = 2 n and fdim D ( X ) = 1. By Theorem 9, 1 K . dim D ( X ) ≤ 2 n (1 − 1+max { 1 , 1 } ) = 2 n (1 − 1 2 ) = 2 n 2 = n. K . dim D ( X ) ≥ n , easy. ******************************************* A generalization to AFFINE ALGEBRAS is given in V. Bavula and T. Lenagan, ”A Bernstein-Gabber- Joseph theorem for affine algebras”, Proc. Ed- inburgh Math. Soc. 42 (1999), no.2, 311– 332. ( Faithful and Schur Dimensions) 16