Auslander-Yoneda algebras and derived equivalences. Changchang Xi ( ~) ccxi/

Transcription

1 International Conference on Operads and Universal Algebra, Tianjin, China, July 5-9, Auslander- and derived Changchang Xi ( ~) xicc@bnu.edu.cn ccxi/

2 Abstract In this talk, we shall present a method of constructing derived between. In particular, we show that, for a self-injective algebra A and an A-module X, there is a derived equivalence between the of A X and A Ω(X) for any admissible subset Φ of N 0, where Ω is the Heller loop operator of A. This is a joint with Hu and Koenig.

3 Schedule of the talk Auslander- and corollaries

4 Definition of admissible sets N Φ = {0,1,2,3, }, the set of all natural numbers. N Definition Φ is called admissible if (1) 0 Φ and (2) for p,q,r Φ with p + q + r Φ, we have p + q Φ q + r Φ. Remark. Definition can be done for monoids.

5 Examples of admissible sets Examples: {0,i,j}, {0,1,3,6,9}, {0,1,3,5,7,,2n + 1}, Φ subset contains 0, = Φ m := {x m x Φ} adm. for m 3, {0,1,2,4} is NOT admissible.

6 Notations and definitions A: f. dim. algebra over a field k, A-mod: category of all f.g. left A-modules, X,Y A-mod Ext i A (X,Y): the i-th cohomology of X and Y. Definition algebra of a module X: M Ext A(X) := ExtA(X,X), i i=0 Multiplication: Concatenation of long exact sequences. This is an N-graded algebra.

7 Definition of Φ-AY Φ N, define E Φ A L (X) := ExtA i (X,X), i Φ Multiplication: x ExtA i (X,X), y Extj A (X,X), { xy if i + j Φ x y = 0 otherwise.

8 General model: Φ: subset of a monoid G, A = L A i : G-graded k-algebra, that is, i G A(Φ):= L A g, g Φ A i : k-space, A g A h A gh,g,h G, Multipl.: a g A g, a h A h, { a g a h if gh Φ, a g a h = 0 otherwise.

9 Definition of Φ-AY- Question: Is EA Φ (X) associative with 1? No, for arbitrary Φ. Yes, if Φ is admissible.

10 Definition of Φ-AY- Definition X A-mod, Φ: adm. subset of N, EA Φ (X) defined above is called Φ-AY-algebra of X. Examples: Φ = {0}, Endomorphism. Φ = N, ; Hochschild cohomology. Φ = 2N, Even. Φ = {0,i}, Trivial extension of End(X) by Ext i A (X,X).

11 Remark Warning: In general, EA Φ (X) is neither subalgebra nor quotient algebra of the algebra ExtA (X)! Φ = {0,1,3,9}, A = k[x]/(x 2 ), X = k

12 Category of complexes of A-modules A: f. dim. algebra. Complex X = (X i,d i ) i Z : a sequence in A-mod X 1 d 1 X 0 d 0 X 1 d 1 d i d i+1 = 0 for all i. Morphism f = (f i ) i Z : X Y : X 1 dx 1 X 0 d 0 X 1 f 1 f 0 f 1 Y 1 dy 1 Y 0 Each square commutes. d 0 Y Y 1

13 C(A): category of complexes with morphisms of complexes. f : X Y is quasi-iso. if f i induces isomorphism: H i (X ) H i (Y ) for all i, where H i (X ) := Ker(d i )/Im(d i 1 ), the i-th cohomology of X, f is null-homotopic if s i : X i Y i 1 such that, for all i, f i = d i Xs i+1 + s i d i 1.

14 Homotopy category Homotopy category K (A): Objects = objects of C(A), Morphism set: Hom K (A) (X,Y ) := Hom C(A)(X,Y ) {f g, } where, for f,g : X Y, we write f g if f g is null-homotopic.

15 category category D(A) of A: Localization of K (A) at all quasi-isomorphisms. That is, Objects of D(A) = Objects of K (A), but adding additional morphisms such that every quasi-iso. is invertible. D(A) is triangulated category.

16 X is bounded: almost all X i = 0. C b (A): category of bounded cpxs, K b (A): homotopy category of bounded cpxs, D b (A): derived category of bounded cpxs.

17 Definition A, B: f. dim. A and B is called derived equivalent if D b (A) and D b (B) are equivalent as triangulated categories. We write: A der B

18 1960 : Verdier + Grothendieck. After: Heller, Happel, Rickard, Keller,, Many branches: Algebraic geometry, representation theory, mathematical physics, Invariants: number of simple modules, Hochschild co- and homology, cyclic Hochshild co- and homology, finiteness of global dimension,

19 X,Y A-mod, Ext i A (X,Y) Hom D b (A)(X,Y[i]) Hom D b (A)(X[ i],y) where [1] is the shift functor of complexes. A-mod is embedded in D b (A) fully and faithfully.

20 The notion of almost (D, Φ)-split sequences. Φ: subset of N D: additive full subcat. of A-mod M: A-module add(m): additive subcat. gen. by M in A-mod

21 Almost (D, Φ)-split sequences Definition An exact sequence 0 X f M g Y 0 in A-mod is called almost (D, Φ)-split if (1) M D, (2) for any object D in D and i Φ, the induced maps Hom D b (A)(f,D [i]) and Hom D b (A)(D [ i],g) are surjective.

22 Almost (D, Φ)-split sequences M,D D, i Φ D [ i] g f 0 g X M Y 0 f D [i]

23 Example: P: projective-injective A-module, = any exact sequence 0 X P Y 0 is almost (add(p), Φ)-split sequence.

24 Theorem Φ: adm. set, M: A-module, 0 X M 1 Y 0: alm. (add(m),φ)-split seq. s.t. Ext i A (M,X) = 0 = Exti A (Y,M) for 0 i Φ = E Φ A der (X M) E Φ A (M Y).

25 Idea of the proof: Λ := E Φ A (X M), Γ := EΦ A (M Y) Construct a tilting complex over Λ, Prove the endomorphism algebra of the tilting complex is iso. to Γ.

26 Corollary. Φ: arbitrary adm. subset of N, A: Quasi-Frobenius algebra, X: A-module, = E Φ A (A X) der E Φ A (A Ωi A (X)).

27 Further comments The main result can be formulated for a triangulated category. In this case, Φ can be taken as an adm. subset of Z, The shift functor is replaced by an auto-equivalence functor, Exact sequence is generalized to triangle, Φ-AY algebra E Φ (X) is replaced by its quotient algebra.

28 Thank you!