General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications On support varieties over complete intersections David A. Jorgensen 1 (new stuff at the end is joint with Petter Bergh 2 ) 1 University of Texas at Arlington 2 NTNU, Norway Maurice Auslander Distinguished Lectures and International Conference, 2013 Woods Hole, MA
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications The General Idea of Support Varieties M � V ( M ) Associate to an R -module M and algebraic set in some affine (or projective) space whose properties reflect homological characteristics of M .
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications The General Idea of Support Varieties M � V ( M ) Associate to an R -module M and algebraic set in some affine (or projective) space whose properties reflect homological characteristics of M . Throughout, R ring, k = ¯ k , M , N f.g. R -modules.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications The Typical Situation Let A = ⊕ i ≥ 0 A i be a commutative graded ring with A i = 0 for i odd. Suppose for every M there is a homomorphism of graded algebras η M : A → Ext ∗ R ( M , M ) such that for every N and ξ ∈ Ext ∗ R ( M , N ) we have ξ · η M ( a ) = η N ( a ) · ξ for every a ∈ A
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications The Typical Situation Let A = ⊕ i ≥ 0 A i be a commutative graded ring with A i = 0 for i odd. Suppose for every M there is a homomorphism of graded algebras η M : A → Ext ∗ R ( M , M ) such that for every N and ξ ∈ Ext ∗ R ( M , N ) we have ξ · η M ( a ) = η N ( a ) · ξ for every a ∈ A Then A is called a ring of central cohomology operators .
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Support and Varieties The cohomological support of ( M , N ) is Supp A ( M , N ) = { p ∈ Spec A | Ext ∗ R ( M , N ) p � = 0 }
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Support and Varieties The cohomological support of ( M , N ) is Supp A ( M , N ) = { p ∈ Spec A | Ext ∗ R ( M , N ) p � = 0 } When A finitely generated over A 2 with A 0 = k , then the support variety of ( M , N ) is V A ( M , N ) = ( Supp A ( M , N ) ∩ MaxSpec A ) ∪ { A ≥ 1 } and V A ( M ) = V A ( M , k ) .
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications This construction fits all known classical cases where support varieties are defined:
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications This construction fits all known classical cases where support varieties are defined: Group algebras kG for finite groups; A is then even part of the cohomology ring.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications This construction fits all known classical cases where support varieties are defined: Group algebras kG for finite groups; A is then even part of the cohomology ring. Finite dimensional algebras; A is the even part of the Hochschild cohomology ring.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications This construction fits all known classical cases where support varieties are defined: Group algebras kG for finite groups; A is then even part of the cohomology ring. Finite dimensional algebras; A is the even part of the Hochschild cohomology ring. Complete intersections; A is a subring of the cohomology ring, generated by central elements of degree 2.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Special case: complete Intersections Now assume that Q is a local (meaning also Noetherian) ring with maximal ideal n and residue field k , R = Q / ( f ) where f = f 1 , . . . , f c is a regular sequence in n 2 .
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Special case: complete Intersections Now assume that Q is a local (meaning also Noetherian) ring with maximal ideal n and residue field k , R = Q / ( f ) where f = f 1 , . . . , f c is a regular sequence in n 2 . In this case we have A = R [ χ 1 , . . . , χ c ] as the ring of cohomology operators , defined from the Eisenbud operators 1980. (deg χ i = 2, 1 ≤ i ≤ c ) Example For Q = k [[ x , y ]] , R = Q / ( x 2 , y 2 ) , and M = k , the Eisenbud operators are defined by ...
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications A theorem of Gulliksen 1974 tells us when Ext ∗ R ( M , N ) is a finitely generated graded module over R [ χ 1 , . . . , χ c ] Theorem If Ext ∗ Q ( M , N ) is finitely generated over R, then Ext ∗ R ( M , N ) is a finitely generated graded module over R [ χ 1 , . . . , χ c ] .
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Fact: the action of A on Ext ∗ R ( M , k ) factors through the algebra ¯ A = A ⊗ R k = k [ χ 1 , . . . , χ c ] , so we have the support variety V ¯ A ( M ) . In other words A ( M ) = { ( b 1 , . . . , b c ) ∈ k c | φ ( b 1 , . . . , b c ) = 0 for all V ¯ A Ext ∗ φ ∈ Ann ¯ R ( M , k ) } a closed set (cone) in k c when Ext ∗ R ( M , k ) is f.g. — e.g. Q is a regular local ring.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Fact: the action of A on Ext ∗ R ( M , k ) factors through the algebra ¯ A = A ⊗ R k = k [ χ 1 , . . . , χ c ] , so we have the support variety V ¯ A ( M ) . In other words A ( M ) = { ( b 1 , . . . , b c ) ∈ k c | φ ( b 1 , . . . , b c ) = 0 for all V ¯ A Ext ∗ φ ∈ Ann ¯ R ( M , k ) } a closed set (cone) in k c when Ext ∗ R ( M , k ) is f.g. — e.g. Q is a regular local ring. Recall: if M is finitely generated and graded over k [ χ 1 , . . . , χ c ] , then b i = dim k M i grows polynomially.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Support varieties give a nice classification of R -modules: M ∼ N iff V ¯ A ( M ) = V ¯ A ( N )
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Support varieties give a nice classification of R -modules: M ∼ N iff V ¯ A ( M ) = V ¯ A ( N ) A courser classification is given by the complexity , i.e., the dimension of V ¯ A ( M ) : M ∼ N iff dim V ¯ A ( M ) = dim V ¯ A ( N )
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Support varieties give a nice classification of R -modules: M ∼ N iff V ¯ A ( M ) = V ¯ A ( N ) A courser classification is given by the complexity , i.e., the dimension of V ¯ A ( M ) : M ∼ N iff dim V ¯ A ( M ) = dim V ¯ A ( N ) One has V ¯ A ( M , N ) = V ¯ A ( M ) ∩ V ¯ A ( N ) For 0 → M 1 → M 2 → M 3 → 0 one has V ¯ A ( M r ) ⊆ V ¯ A ( M s ) ∪ V ¯ A ( M t ) for { r , s , t } = { 1 , 2 , 3 } . V ¯ A ( M ) = V ¯ A (Ω M )
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Notes: V ¯ A ( M ) was originally defined only for single module by Avramov in 1989.
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Notes: V ¯ A ( M ) was originally defined only for single module by Avramov in 1989. Tor R i ( M , N ) = 0 for i ≫ 0 = ⇒ V ( M ) ∩ V ( N ) = { 0 } — J 1997 (1995).
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Notes: V ¯ A ( M ) was originally defined only for single module by Avramov in 1989. Tor R i ( M , N ) = 0 for i ≫ 0 = ⇒ V ( M ) ∩ V ( N ) = { 0 } — J 1997 (1995). Fargo 1995 � Avramov-Buchweitz 2000 (1998)
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Notes: V ¯ A ( M ) was originally defined only for single module by Avramov in 1989. Tor R i ( M , N ) = 0 for i ≫ 0 = ⇒ V ( M ) ∩ V ( N ) = { 0 } — J 1997 (1995). Fargo 1995 � Avramov-Buchweitz 2000 (1998) The realizability question: Which cones in k c are support varieties?
General Idea of Support Varieties Complete Intersections A Simpler, More Versatile Approach Applications Notes: V ¯ A ( M ) was originally defined only for single module by Avramov in 1989. Tor R i ( M , N ) = 0 for i ≫ 0 = ⇒ V ( M ) ∩ V ( N ) = { 0 } — J 1997 (1995). Fargo 1995 � Avramov-Buchweitz 2000 (1998) The realizability question: Which cones in k c are support varieties? Answer: all . Solved by Avramov and Jorgensen in 2000.
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