Minors and Categorical Resolutions

Transcription

1 SINGULARITIES and COMPUTER ALGEBRA Conference in Honor of Gert-Martin Greuel s 70th Birthday Minors and Categorical Resolutions Yuriy Drozd Institute of Mathematics of the National Academy of Sciences of Ukraine Pfalz-Akademie Lambrecht June 24, 2015

2 Origin: Drozd Greuel (2000): classification of vector bundles on cyclic projective configurations / Burban Drozd (2001): derived categories of coherent sheaves on cyclic projective configurations. Burban Drozd (2003): Auslander algebra of curves with nodes and cusps and its tilting to a finite dimensional algebra of global dimension 2. Burban Drozd Gavran (2015) (this talk).

3 What is it about? Well-known and easy: coherent sheaves over P 1 representations of the Kronecker algebra. Beilinson D(Coh P n ) DΛ for a good finite dimensional algebra Λ. Many generalizations. In all cases the variety was smooth and, in some sense, rational. Moreover, smooth is indispensable if we want Λ to be regular, i.e. of finite global dimension. Burban Drozd (2003): replace O X by some non-commutative extension A such that there is a good correspondence between DO X and DA, then find Λ for A. Good correspondence non-commutative resolution in the sense of Van den Bergh or Kuznetsov Lunts. In the case of curves it has been done and is now presented.

4 So, it is a report on a joint work with I.Burban and V.Gavran. The details can be found in: arxiv: [mathag] arxiv: [mathag] Contents: (1) Minors and bilocalization (2) Case of non-commutative curves (3) König s resolution (4) Rational case: tilting (5) Examples

5 1. Minors and bilocalization A non-commutative scheme is a pair (X, A), where X is a scheme and A is an O X - algebra quasi-coherent as O X -module. If X is affine, A is just a sheafification of a O-algebra, where O is the ring of regular functions on X. We always suppose that X is quasi-compact and separated. We call (X, A) noetherian if X is noetherian and A is coherent as O X -module. We call (X, A) projective if X is projective and A is coherent. Qcoh A = the category of quasi-coherent A-modules. Coh A = the category of coherent A-modules. DA = D(Qcoh A). D pf A = the full subcategory of compact objects from DA. As X is quasi-compact and separated, usual considerations show that F D pf A if and only if it is perfect, i.e. locally quasi-isomorphic to a finite complex of locally projective coherent A-modules.

6 A minor of a non-commutative scheme (X, A) is a non-commutative scheme (X, E), where E = (End A P) op, where P is a coherent locally projective A-module. Then there are functors: where I Qcoh Q I Qcoh A I! Q = A/J P, F F F! Qcoh E J P = Im( ev : P E P A ), P = Hom A (P, A), F = Hom A (P, ), F = P E, F! = Hom E (P, ), I is the natural embedding, I = Q A, I! = Hom A (Q, ).

7 I Qcoh Q I Qcoh A I! F F F! Qcoh E Moreover, Qcoh E is the Serre quotient Qcoh A/ Im I and Im I = Ker F = P = { M Hom A (P, M) = 0 } is both localizing and colocalizing subcategory in Qcoh E. In particular, The natural morphism 1 Qcoh E FF is an isomorphism. The natural morphism FF! 1 Qcoh E is an isomorphism. The functor F is a full embedding and its essential image is (Im I) = { M Hom A (M, C) = Ext 1 A (M, C) = 0 for all C Im I }. Moreover, (Im I) consists of all A-modules M such that for every point x X there is an exact sequence P 1 P 0 M x, where P i are multiples (maybe infinite) of P x. The functor F! is a full embedding and its essential image is (Im I) = { M Hom A (C, M) = Ext 1 A (C, M) = 0 for all C Im I }. Moreover, (Im I) consists of all A-modules M such that there is an exact sequence M I 1 I 2, where I i are some injective modules from Im F!. (Note that this condition can also be verified locally.)

8 Passing to derived categories and functors, we obtain an analogous diagram LI D Q A I DA RI! We keep the notations I and F as these functors are exact, so there left and right derived functors coincide and are just componentwise application of I or F. D Q A denotes the full subcategory of complexes whose cohomologies are actually Q-modules. Again DE DA/D Q A, where D Q A is both localizing and colocalizing, and The natural morphism 1 DE F LF is an isomorphism. The natural morphism F RF! 1 DE is an isomorphism. The functor LF is a full embedding, maps D pf E to D pf A and its essential image is LF F RF! DE (Im I) = { M Hom A (M, C [n]) = 0 for all C Im I, n Z }. The functor RF! is a full embedding and its essential image is (Im I) = { M Hom A (C [n], M ) = 0 for all C Im I, n Z }. The functor LI induces an equivalence DA/ Im LF D Q A. The functor RI! induces an equivalence DA/ Im LF! D Q A. There are semi-orthogonal decompositions DA = D Q A, Im LF = Im RF!, D Q A. (Note that both images are equivalent to DE but do not coincide.)

9 In some cases we can obtain more information Theorem 1. Suppose that the ideal J P = Im( ev : P E P A ) is flat as right A-module. Then D Q A DQ. Moreover, in this case gl.dim A max { gl.dim Q + lp.dim A J P + 2, gl.dim E }, where lp.dim A M = max { x X pr.dim Ax M x }. The last inequality is useful when we deal with the so called quasi-hereditary rings and schemes.

10 Example. Let F be a coherent A-module, A F = End A (A F) op. A F can be considered as an algebra of matrices ( ) A F A F = F, B where B = (End A F) op. If P = ( A F ) considered as AF -module, then A (End AF P) op, so A is a minor of A F and we have the situation of six functors diagram as above. Note that in this case Q B/J, where J = Im( F A F B ). We call a noetherian non-commutative scheme A strictly Gorenstein if X is equidimensional, A is Cohen Macaulay as O X -module and inj.dim A A = dim X. Theorem 2. In the situation of the Example above, let A be strictly Gorenstein and F be maximal Cohen Macaulay (over O X ). Then the restrictions of the derived functors LF and RF! (both from DA to DA F ) onto D pf A coincide. Thus this restriction is both left and right adjoint to F : DA F DA.

11 We use these results to obtain categorical resolutions for the derived categories of quasicoherent modules over non-commutative curves, i.e. such noetherian non-commutative schemes (X, A) that X is a curve (an excellent equidimensional scheme of dimension 1) and A is torsion free and coherent as O X -module. We also suppose that this non-commutative curve is reduced, i.e. A has no nilpotent ideals. Let R be the A-ideal such that R x = A x if A x is hereditary (i.e. gl.dim A x = 1) and R x = rad A x otherwise. We denote by A the endomorphism algebra of R considered as right A-module. It is known that A = A if and only if A is hereditary. As X is excellent and A is reduced, any ascending chain of over-rings of A stabilizes, so there is a chain A = A 1 A 2 A 3... A n A n+1 = Ã, where A i+1 = A i and à is hereditary. Note that if we deal with a usual curve, i.e. A = O X, à is a normalization of O X. We call n the level of A.

12 Set K = End A (A 1 A 2... A n+1 ) op. It is identified with the algebra of matrices (a ij ) with a ij A ij = Hom A (A i, A j ). Note that A ij = A j if i < j. Thus K is of the form A 1 A 2 A 3... A n A n+1 A 21 A 2 A 3... A n A n+1 K = A 31 A 32 A 3... A n A n A n+1,1 A n+1,2 A n+1,3... A n+1,n A n+1 We call the non-commutative curve (X, K) the König s resolution of the non-commutative curve (X, A). It was first considered (in a bit another form) by König for orders over discrete valuation rings. We denote by P i the locally projective K-module formed by the i-th column of this matrix presentation of K, in particular P = P 1 and P = P n+1. Then A i = (End K P i ) op, so all these rings are minors of K and we have the corresponding six functors diagrams. Especially we are interested in the diagram D Q K LĨ Ĩ RĨ! DK where Q = K/J P. (One easily sees that J P K-module.) L F F R F! DÃ is locally projective both as right and as left

13 Denote by e i the diagonal idempotents of K ε k = n+1 i=k+1 I k = Kε k K, Q k = A/I k. Then I n = J P and Q n = Q. Set also Ĩ k = I k /J P. Note that all Q k are supported on sing X and are artinian algebras. So they can be identified with the artinian algebras of their global sections. e k, We prove the following main result about the König s resolution. Theorem 3. Q is an artinian quasi-hereditary algebra with a heredity chain of ideals I = { Ĩk 1 k n }. Hence gl.dim Q 2n 2 and gl.dim A 2n. Thus Qcoh K is an abelian category of finite global dimension and the category Qcoh A is its bilocalization. So the corresponding functor F : DK DA can be considered as a categorical resolution of the singular derived category DA. If the non-commutative curve (X, A) is strictly Gorenstein, this resolution is weakly crepant in the sense of Kuznetsov, i.e. the restrictions of the left and right adjoint functors of F onto D pf A coincide.

14 Using several steps of the minor construction, we prove the following result about the structure of the category DK. Let Ā i = A i /A i+1,i. We also identify it with the semisimple artinian algebra of its global sections. Theorem 4. There are semi-orthogonal decompositions of DK where Recall that semi-orthogonal means that DK = S 1, S 2,..., S n, T = T, S n,..., S 2, S 1, S i S i DĀ i, T T DÃ. Ext m K (C i, C j ) = Extm K (C j, C i ) = 0 for C i S i, C i S i, all m and i > j, as well as Ext m K (F, C i ) = Extm K (C i, F ) = 0 for F T, F T and all m.

15 Suppose now that a non-commutative curve (X, A) is rational and projective over an algebraically closed field k, i.e. all irreducible components of X are rational projective curves over k and A is a central O X -algebra. Then the hereditary algebra à is actually a kind of Geigle Lenzing weighted projective line (introduced here in Lambrecht in 1985), so it has a tilting module T, i.e. a coherent Ã-module which generates Dà and is such that Ext m (T, T ) = 0 for m 0. Therefore Dà DR, where R = à (Endà T ) op is a canonical algebra of Ringel, that is a k-algebra given by the quiver α 11 α 12 α 1r α 21 α 22 α 2r α k1 1, αk2 1,2.... αk r 1,r α k1 1 α k2 2 with relations α j = α 1 + λ j α 2 for 3 j r, where α j = α kj j... α 2j α 1j, all λ i are non-zero and different. Certainly, if r = 2, it is the path algebra of a quiver of type Ãk 1 +k 2. In particular, if r = 2, k 1 = k 2 = 1, it is the Kronecker algebra. α k rr

16 Let k = Q k e k and k = Hom Āk (e k Q k, Ā k ), the dual to e k Q k, be the standard and costandard Q-modules with respect to the heredity chain of ideals I. We also set T = FT and T = HT. Theorem 5. If a non-commutative curve (X, A) is rational and projective over an algebraically closed field k, then ˆT = Q[ 1] T is a tilting complex in DK, i.e. it is a compact generator and Hom DK ( ˆT, ˆT [m]) = 0 for all m 0. Therefore, DK DΛ, where Λ = (End DK T ) op is a finite dimensional quasi-hereditary k-algebra of the form ( ) Q E Λ =, 0 R where Q = Γ(X, Q), R = (Endà T ) op is a canonical algebra and E = Ext 1 K (Q, T ). Theorem 6. We have two semi-orthogonal set of generators of DÃ, namely D pf K = 1, 2,..., n, T n+1 = T, n,..., 2, 1 n+1.

17 If we deal with a usual curve with only simple plane singularities in the sense of Arnold, we can explicitly calculate the quasi-hereditary algebra Q = Γ(X, Q). Namely, every singular point of type A m gives a component of Q described by the quiver 1 α 1 β 1 2 α 2 β 2 3 α n 1 (n 1) n β n 1 with relations Here n = [ ] m+1 2 is the level. β k α k = α k+1 β k+1 if 1 k < n 1, β n 1 α n 1 = 0.

18 A singular point of type D m gives a component of Q described by the quiver with relations α 1 1 β 1 β 2 α 2 β 2 3 α 3 β 3 4 α n 1 (n 1) n β n 1 Here n = [ ] m 2. β k α k = α k+1 β k+1 if 1 k < n 1, β n 1 α n 1 = 0, β α 1 = 0, β 2 β = 0. Finally, the component of Q corresponding to a singularity of type E 6 is the same as for D 4, while that corresponding to a singularity of type E 7 or E 8 is the same as for D 5.

19 If X is rational, one can calculate the whole algebra Λ such that DK DΛ. For instance, let X has 2 components X 1, X 2 and 3 singular points x 1 X 1 of type E 6, x 2 X 1 X 2 of type D 7 and x 3 X 2 of type A 5. Then the quiver of Λ is α ξ 1 η ξ 2 η γ 11 γ 12 γ 13 γ 23 γ 21 γ 22 γ 32 γ β 1 β 1 α 21 β 2 β α 31 β α 22 β (E 6 ) (D 7 ) α 32 β (A 5 )

20 HOPE WE ALL WILL MEET HERE FOR THE 80th ANNIVERSARY OF GERT-MARTIN