28.08.2013
Outline Main Theorem (Eisenbud) Let R = S/(f ) be a hypersurface. Then MCM(R) MF S (f ) Survey of the talk: 1 Define hypersurfaces. Explain, why they fit in our setting. 2 Define. Prove, that they form a Frobenius category. 3 Prove the equivalence.
Regular local rings Definition A Noetherian local ring S is called regular if Example k[x 1,..., X n ] (X1,...,X n) k[[x 1,..., X n ]] Kdim S = gldim S <. Lemma A regular local ring S is Gorenstein. Proof. gldim S < Ext i S (k, S) = 0 for i 0 indim S <
Quotients of Gorenstein rings Let S be regular local. Let 0 f S be a non-unit. Lemma A regular local ring is a domain. Proposition Let S be a Noetherian local ring. Let f S be a non-unit, non-zero divisor. Then indim S/(f ) S/(f ) = indim S S 1. Corollary S Gorenstein S/(f ) Gorenstein.
Hypersurfaces Definition Let S be a regular local ring and 0 f S be a non-unit. Then R := S/(f ) is called a hypersurface. Main Theorem (Eisenbud) Let R = S/(f ) be a hypersurface. Then MCM(R) MF S (f )
Definition Let S be a ring, f S. A matrix factorisation of f is a pair of maps of free S-modules such that ϕ ψ F G F ϕ f 1 F F G F G ψ f 1 G ϕ
Examples Example Let S = C[[x, y]], f = y 2 x 2. Then a matrix factorisation is given by y x x y y x x y S 2 S 2 S 2 Example 1 F F f 1 F F F
Morphisms of matrix factorisations Definition A morphism between two matrix factorisations of f is given by: F G F α ϕ ψ F G F β ϕ ψ Composition of two morphisms is given componentwise.denote the category of matrix factorisations by MF S (f ). α
Additive Categories Recall A category A is called additive if A has a zero object. Every Hom-Set is an abelian group and composition is bilinear. There exists a coproduct of every two objects.
Additive Categories Proposition The category MF S (f ) is additive: A has a zero object. 0 0 0 is a zero object. Every Hom-Set is an abelian group and composition is bilinear. (α, β) + (α, β ) = (α + α, β + β ). There exists a coproduct of every two objects. ϕ S ψ S ϕ T ψ T ( S 0 S 1 S 0 ) ( T0 T 1 T 0 ) ϕ S 0 0 ϕ T ψ S 0 0 ψ T := S 0 T 0 S 1 T 1 S 0 T 0
Important Lemmas Lemma Let (ϕ, ψ) MF S (f ). Then ϕ is a monomorphism. Proof. Let ϕ(z) = 0. Then 0 = ψϕ(z) = f z. Thus z = 0.
and exact sequences Lemma ϕ ψ Let F G F be a matrix factorisation. Then ϕ ϕ ψ ϕ F G F is exact, where F = F /(ff ). Proof. ϕψ = f 1 G ϕψ = 0 Let z ker ϕ.then ϕ(z) fg = ϕψg z Im(ψ).
Proposition Up to isomorphism one can assume F = G. Proof. ϕ ψ Let F G F be a matrix factorisation. Then ϕ ϕ ψ ϕ F G F is exact. Localize at a minimal prime p, you get artinian modules. Thus l(g p ) = l(im ψ p ) + l(ker ψ p ) = l(f p ). But here rank(g p ) = l(g p )/l(r p ).
Proof (continued). Thus F = G, say via A.Then the following provides an isomorphism between matrix factorisations: F G F 1 F ϕ Aϕ A ψ ψa 1 F F F 1 F
Exact categories Recall An additive category A with a class of short exact sequences { A B C } is called exact if 1 B B 1 B 0, 0 B B B Composition of admissible monomorphisms is admissible monomorphism Composition of admissible epimorphisms is addmisible epimorphism Pushout of an admissible monomorphism and an arbitrary map yields an admissible monomorphism. Pullback of an admissible epimorphism and an arbitrary map yields an admissible epimorphism.
Exact sequences Proposition The category MF S (f ) is an exact category with exact sequences given by: S 0 T 0 U 0 S 1 T 1 U 1. This means in particular: admissible epimorphism = all epimorphisms = split epimorphisms admissible monomorphism = split monomorphisms
MF S (f ) is exact Proof. 1 T0 0 T 0 T 0 is exact. 1 T1 0 T 1 T 1 Same for the dual situation. Composition of admissible monomorphisms is admissible monomorphism follows from Composition of split monomorphisms is split monomorphism Same for dual situation.
MF S (f ) is exact (continued) Proof (continued). 1 0 S 0 S 0 S 0 µ T 0 T 0 S 0 1 0 µ 0 0 1 is the pushout of the canonical split map and an arbitrary map µ. That this holds for MF S (f ) as well follows componentwise.
Exact subcategories of abelian categories Remark (added after the talk, after remarks from R. Buchweitz, M. Kalck and M. Severitt) Any exact category can be embedded into an abelian category by an abstract abelian hull -construction. In the case of MF S (f ) there are several possibilities to embed it into: The abelian category of chain complexes, by mapping (ϕ, ψ) to (..., 0, ϕ, 0, ψ, 0,... ). The category of modules over S( 1 2 ). The category of graded modules over the Z 2 -graded algebra S[y]/(y 2 = f ), where y = 1.
Projective objects in MF S (f ) Lemma With respect to the defined exact structure (1, f ) is projective. Proof. We can assume that S 0 = S 1 =: S. Have to show every admissible epimorphism to (1, f ) splits. S ϕ ψ S α µ 1 S α ϕµ S S f
Injective objects in MF S (f ) Lemma With respect to the defined exact structure (1, f ) is injective. Proof. Have to show every admissible monomorphism from (1, f ) splits. β S S νϕ 1 S f ϕ β S S Note: Every admissible monomorphism splits. ν ψ
Frobenius categories Recall An exact category is called Frobenius category if projective and injective objects coincide and the category has enough projectives and injectives. Proposition MF S (f ) is a Frobenius category with prinjective objects add(1, f ) (f, 1).
MF S (f ) is a Frobenius category Proof. We have already seen that (1, f ) (f, 1) is prinjective. The following diagram shows enough prinjectives: ϕ ψ ϕ 1 ( 1, ϕ) S 0 S 0 S 0 S 0 1 ψ 1 0 0 f (ψ,1) S 0 S 0 S 0 S 0 ϕ 1 f 0 0 1 ( 1, ϕ) S 0 S 0 S 0 S 0 ψ ϕ
The Main Theorem again Theorem Let R = S/(f ) be a hypersurface. Then we have: MF S (f )/ add(1, f ) CM(R) and MF S (f ) CM(R).
The proof Proof I: MF S (f )/ add(1, f ) CM(R) Let X : MF S (f )/ add(1, f ) (ϕ, ψ) CM(R) Coker ϕ This is well-defined: ϕ 0 F G Coker ϕ 0 ff = ϕψf ϕg, hence f Coker(1, f ) = 0. Coker ϕ = 0. }{{} an R-module The key lemma before shows: Coker ϕ has a periodic free R-resolution.
The proof (continued) Recall R Cohen-Macaulay of dimension d. Then for any n d: Proof I (continued). Ω n (M) = 0 or Ω n (M) is Cohen-Macaulay. Coker ϕ = Ω 2n Coker ϕ is Cohen-Macaulay. ϕ F G Coker ϕ 0 α β ϕ F G Coker ϕ 0 defines the functor on morphisms.
A small amount of commutative algebra Proposition Let R = S/(f ) be a hypersurface. M CM(R) prdim S M = 1. Proof. prdim S M = Kdim(S) depth(m) Auslander-Buchsbaum formula = Kdim(S) Kdim(R) M CM(R) = 1 Krull s principal ideal theorem
The other direction of the proof Proof II: CM(R) MF S (f )/ add(1, f ) Let M CM(R), then prdim S M = 1: fm = 0 fs 1 ϕs 0. x S 1!y S 0 : fx = ϕy Set ψ : S 1 S 0 : x y. 0 F ϕ G M 0. (ϕ, ψ) gives a matrix factorisation. Define Y(M) := (ϕ, ψ). fx = ϕy, fx = ϕy f (x + x ) = ϕ(y + y ) ψ(x + x ) = y + y s S fxs = ϕ(ys) ψ(xs) = ys.
Proof II (continued). M, M CM(R). 0 F G M 0 ϕ ϕ 0 F G M 0 Minimal projective resolutions are unique up to isomorphism. Gives a functor CM(R) MF S (f )/ add(1, f ). Equivalence is now obvious. X (f, 1) = Coker(f ) = R, hence MF S (f ) CM(R).
Interchanging ϕ and ψ Corollary M = X (ϕ, ψ) ΩM = X (ψ, ϕ). Proof. ψ ϕ R 1 R 0 R 0 0 X (ψ, ϕ) R X (ϕ, ψ) 0
Frobenius categories and homotopy Proposition The equivalence before can also be written as: H 0 (MF S (f )) CM(R). Proof. null-homotopic factors through prinjective ϕ ψ S 0 S 0 S 0 α h β k α T 0 T 0 T 0 ϕ ψ with β = ϕ h + kψ and α = ψ k + hϕ.
Proof of homotopy factoring Proof (continued) β = ϕ h + kψ and α = ψ k + hϕ. ϕ S 0 S 0 S 0 1 ϕ f 0 0 1 S 0 S 0 S 0 S 0 S 0 S 0 ψ ψ 1 1 0 0 f 1 ϕ (ψ k,h) (k,ϕ h) (ψ k,h) ϕ ψ T 0 T 0 T 0
Proof of factoring homotopy Proof (continued) α = α 2 α 1 + α 2 α 1 and β = β 2β 1 + β 2 β 1 α 1 α 1 ϕ S 0 S 0 S 0 β 1 f 0 β 1 0 1 U 0 U 0 U 0 U 0 U 0 U 0 ψ α 1 1 0 α 1 0 f (α 2,α 2 ) (β 2,β 2 ) (α 2,α 2 ) ϕ ψ T 0 T 0 T 0
The end Proof. α = α 2 α 1 + α 2α 1 β = β 2 β 1 + β 2β 1 ( ) ( ) ( ) ( ) f α1 β1 ϕ β1 α1 ψ = β 1 ϕ β 1 f = α 1 ψ α 1 (ϕ α 2, ϕ α 2) = (β 2 f, β 2) (ψ β 2, ψ β 2) = (α 2, α 2f ) ϕ S 0 S 0 S 0 ψ α α 2 β 1 β β 2 α 1 α T 0 T 0 T 0 ϕ ψ