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Relative Singularity Categories Relative singularity categories Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn vorgelegt von Martin Kalck aus Hamburg Bonn 2013 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn 1. Gutachter: Prof. Dr. Igor Burban 2. Gutachter: Prof. Dr. Jan Schr¨oer Tag der Promotion: 29.05.2013 Erscheinungsjahr: 2013 Summary In this thesis, we study a new class of triangulated categories associated with sin- gularities of algebraic varieties. For a Gorenstein ring A, Buchweitz introduced the b triangulated category Dsg(A) = D (mod − A)= Perf(A) nowadays called the triangu- lated category of singularities. In 2006 Orlov introduced a graded version of these categories relating them with derived categories of coherent sheaves on projective varieties. This construction has already found various applications, for example in the Homological Mirror Symmetry. The first result of this thesis is a description of Dsg(A), for the class of Artinian Gorenstein algebras called gentle. The main part of this thesis is devoted to the study of the following generalization of Dsg(A). Let X be a quasi-projective Gorenstein scheme with isolated singularities, F a coherent sheaf on X such that the sheaf of algebras A = EndX (OX ⊕ F) has finite global dimension. Then we have the following embeddings of triangulated categories b b D Coh(X) ⊇ Perf(X) ⊆ D Coh(X) : Van den Bergh suggested to regard the ringed space X = (X; A) as a non-commutative resolution of singularities of X. We introduce the relative singularity category b ∆X (X) = D Coh(X) = Perf(X) as a measure for the difference between X and X. The main results of this thesis are the following (i) We prove the following localization property of ∆X (X): b M M D (Abx − mod) ∆ ( ) ∼= ∆ A := : X X Obx bx x2Sing(X) x2Sing(X) Perf(Obx) Thus the study of ∆X (X) reduces to the affine case X = Spec(O). (ii) We prove Hom-finiteness and idempotent completeness of ∆O(A) and determine its Grothendieck group. (iii) For the nodal singularity O = k x; y =(xy) and its Auslander resolution A = EndO(O ⊕ m), we classify all indecomposableJ K objects of ∆O(A). (iv) We study relations between ∆O(A) and Dsg(A). For a simple hypersurface sin- gularity O and its Auslander resolution A, we show that these categories determine each other. (v) The developed technique leads to the following `purely commutative' statement: =∼ ∼ M SCM(R) −!Dsg(X) = MCM(Obx); x2Sing(X) where R is a rational surface singularity, SCM(R) is the Frobenius category of special Cohen{Macaulay modules, and X is the rational double point resolution of Spec(R). 4 Contents 1. Introduction 6 1.1. Global relative singularity categories 7 1.2. Local relative singularity categories 9 1.3. Frobenius categories and Gorenstein rings for rational surface singularities 14 1.4. Singularity categories of gentle algebras 17 1.5. Contents and Structure 18 Acknowledgement 19 2. Frobenius categories 20 2.1. Definitions, basic properties and examples 20 2.2. Idempotent completion 26 2.3. Auslander & Solberg's modification of exact structures 27 2.4. A Morita type Theorem for Frobenius categories 29 2.5. The Buchweitz{Happel{Keller{Rickard{Vossieck equivalence 33 2.6. Singularity Categories of Iwanaga{Gorenstein rings 37 2.7. A tale of two idempotents 39 2.8. Alternative approach to the `stable' Morita type Theorem 41 2.9. Frobenius pairs and Schlichting's negative K-theory 42 2.10. Idempotent completeness of quotient categories 45 3. DG algebras and their derived categories 48 3.1. Notations 48 3.2. Definitions 49 3.3. The Nakayama functor 52 3.4. Non-positive dg algebras: t-structures, co-t-structures and Hom-finiteness 53 3.5. Minimal relations 57 3.6. Koszul duality 57 3.7. Recollements 60 4. Global relative singularity categories 68 4.1. Definition 68 4.2. The localization property 69 5. Local relative singularity categories 78 5.1. An elementary description of the nodal block 78 5.2. The fractional Calabi{Yau property 92 5.3. Independence of the Frobenius model 96 5.4. Classical vs. relative singularity categories 100 5.5. DG-Auslander algebras for ADE{singularities 107 5.6. Relationship to Bridgeland's moduli space of stability conditions 110 6. Special Cohen{Macaulay modules over rational surface singularities 111 6.1. Rational surface singularities 111 6.2. The McKay{Correspondence 113 6.3. Special Cohen{Macaulay modules 116 5 6.4. A derived equivalence 118 6.5. SCM(R) is a Frobenius category 123 6.6. Main result 127 6.7. Examples 129 6.8. Concluding remarks 132 7. Singularity categories of gentle algebras 135 7.1. Main result 135 7.2. Examples 137 7.3. Proof 139 References 143 6 1. Introduction This thesis studies triangulated and exact categories arising in singularity theory and representation theory. Starting with Mukai's seminal work [124] on Fourier-type equivalences between the derived categories of an abelian variety and its dual variety, derived categories of coherent sheaves have been recognised as interesting invariants of the underlying projective scheme. In particular, the relations to the Minimal Model Program and Kontsevich's Homological Mirror Symmetry Conjecture [110] are active fields of current research. On the contrary, two affine schemes with equivalent bounded derived categories are already isomorphic as schemes, by Rickard's `derived Morita theory' for rings [142]. This statement remains true if the bounded derived category is replaced by the subcategory of perfect complexes, i.e. complexes which are quasi-isomorphic to bounded complexes of finitely generated projective modules. So neither the bounded derived category Db(mod − R) nor the subcategory of perfect complexes Perf(R) ⊆ Db(mod−R) are interesting categorical invariants of a commutative Noetherian ring R. In 1987, Buchweitz [34] suggested to study the quotient category Db(mod − R) D (R) := : (1.1) sg Perf(R) As indicated below, this category is a useful invariant of R. It is known as the singularity category of R. If R is a regular ring, then every bounded complex admits a finite projective resolution and therefore we have the equality Perf(R) = Db(mod − R). In particular, the singularity category of a regular ring vanishes. Moreover, the category of perfect complexes Perf(R) can be considered as the smooth b part of D (mod−R) and Dsg(R) as a measure for the complexity of the singularities of Spec(R). If R is Gorenstein, i.e. of finite injective dimension as a module over itself, then Buchweitz proved the following equivalence of triangulated categories MCM(R) −! Dsg(R); (1.2) where the left hand side denotes the stable category of maximal Cohen{Macaulay modules. We list some properties of a singularity, which are `detected' by the sin- gularity category. (a) Orlov [132] has shown that two analytically isomorphic singularities have equivalent singularity categories (up to taking direct summands). (b) Let k be an algebraically closed field. A commutative complete Gorenstein k-algebra (R; m) satisfying k ∼= R=m has an isolated singularity if and only if Dsg(R) is a Hom-finite category, by work of Auslander [12]. (c) Let S = C z0; : : : ; zd and f 2 (z0; : : : ; zd) n f0g. Then Kn¨orrer[108] proved the followingJ equivalenceK of triangulated categories 2 2 Dsg S=(f) −! Dsg S x; y =(f + x + y ) : (1.3) J K (d) Kn¨orrer[108] and Buchweitz, Greuel & Schreyer [35] showed that a complete hypersurface ring R = S=(f) defines a simple singularity, i.e. a singularity 7 which deforms only into finitely many other singularities, if and only if the singularity category has only finitely many indecomposable objects. Recently, Orlov [131] introduced a global version of singularity categories for any Noetherian scheme X and related it to Kontsevich's Homological Mirror Symmetry Conjecture. Moreover, if X has isolated singularities, then the global singularity category is determined by the local singularity categories from above (up to taking direct summands) [132]. This work consists of four parts, which deal with (relative) singularity categories in singularity theory and representation theory. • Firstly, we study the relative singularity category associated with a non- commutative resolution of a Noetherian scheme X. This construction is inspired by the constructions of Buchweitz and Orlov and we show that its description reduces to the local situation if X has isolated singularities. • In the second part, we investigate these local relative singularity categories. In particular, we give an explicit description of these categories for Aus- 2 2 lander resolutions of A1-hypersurface singularities x0 + ::: + xd in all Krull dimensions. Moreover, we study the relations between the relative and the classical singularity categories using general techniques: for example, dg- algebras and recollements. Our main result shows that the classical and relative singularity categories mutually determine each other in the case of ADE-singularities. • The third part is inspired by the techniques developed in the second part. However, the result is `purely commutative'. Namely, we give an explicit description of Iyama & Wemyss' stable category of special Cohen{Macaulay modules over rational surface singularities in terms of the well-known singu- larity categories of rational double points. • Finally, we describe the singularity categories of finite dimensional gentle algebras. They are equivalent to finite products of m-cluster categories of type A1. 1.1. Global relative singularity categories. Let X be a Noetherian scheme. The singularity category of X is the triangulated quotient category Dsg(X) := Db(Coh(X))= Perf(X), where Perf(X) denotes the full subcategory of complexes which are locally quasi-isomorphic to bounded complexes of locally free sheaves of finite rank.
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