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Contributions to the Stable Derived Categories of Gorenstein Rings

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Contributions to the Stable Derived Categories of Gorenstein Rings Contributions to the Stable Derived Categories of Gorenstein Rings by Vincent G´elinas A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2018 by Vincent G´elinas Abstract Contributions to the Stable Derived Categories of Gorenstein Rings Vincent G´elinas Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2018 The stable derived category Dsg(R) of a Gorenstein ring R is defined as the Verdier quotient of the bounded derived category Db(mod R) by the thick subcategory of perfect complexes, that is, those with finite projective resolutions, and was introduced by Ragnar-Olaf Buchweitz as a homological measure of the singularities of R. This thesis contributes to its study, centered around representation-theoretic, homological and Koszul duality aspects. In Part I, we first complete (over C) the classification of homogeneous complete intersection isolated Z b singularities R for which the graded stable derived category Dsg(R) (respectively, D (coh X) for X = proj R) contains a tilting object. This is done by proving the existence of a full strong exceptional collection of vector bundles on a 2n-dimensional smooth complete intersection of two quadrics X = 2n+2 V (Q1;Q2) ⊆ P , building on work of Kuznetsov. We then use recent results of Buchweitz-Iyama- Z Yamaura to classify the indecomposable objects in Dsg(RY ) and the Betti tables of their complete 1 2 resolutions, over RY the homogeneous coordinate rings of 4 points on P and 4 points on P in general position. In Part II, for R a Koszul Gorenstein algebra, we study a natural pair of full subcategories whose lin Z intersection H (R) ⊆ Dsg(R) consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if R is absolutely Koszul in the sense of b lin ∼ Z Herzog-Iyengar, in which case there is an equivalence of triangulated categories D (H (R)) = Dsg(R). We then relate the heart to modules over the Koszul dual algebra R!. As first application, we extend the Bernstein-Gel'fand-Gel'fand correspondence beyond the case of exterior and symmetric algebras, or more generally complete intersections of quadrics and homogeneous Clifford algebras, to any pair of Koszul dual algebras (R; R!) with R absolutely Koszul Gorenstein. In particular the correspondence holds for the coordinate ring of elliptic normal curves of degree ≥ 4 and for the anticanonical model of del Pezzo surfaces of degree ≥ 4. We then relate our results to conjectures of Bondal and Minamoto on the graded coherence of Artin-Schelter regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimension ≥ 4. ii To the memory of Ragnar-Olaf Buchweitz. To my Mary. iii Acknowledgements This entire thesis is owed to Ragnar-Olaf Buchweitz, who has always provided me with the most stellar example of a mathematician, and of a human being. That I have made it all the way here is largely the result of having been surrounded by absolutely outstanding people, to whom thanks are due but for which words hardly seem enough. To Ben, who along with Ragnar has taught me all of the mathematics that I know. To Ozg¨ur,who¨ is always high in spirit, for all the good times, for being the best conference roommate, and for being a solid friend all this time. To Louis-Philippe, for being amazingly kind, thoughtful, and also for smooth driving. I owe an immense debt to my supervisors Colin and Joel, who have been extremely generous with their time, advice and mentorship. The same goes for my committee members John Scherk and Paul Selick, who have helped me on many occasions and given me solid advice during my time here. To Jemima, for all the constant help and for taking care of all of us. To the inhabitants of the math house, for some of the best food I've had in grad school. Finally to Mary McDonald, who pushes me and challenges me, who makes our lives comfortable and interesting, who constantly gets me to try new things, who shares her interests with me, and who has always been the best partner for exploring the road ahead. iv Contents 1 Introduction 1 1.1 Overview of the thesis . .3 1.2 Preliminaries . 10 1.3 Background: Maximal Cohen-Macaulay modules over Gorenstein rings . 13 I MCM Modules and Representation Theory of Algebras 25 2 Graded Gorenstein rings with tilting MCM modules 26 2.1 Some history: Tilting theory, exceptional singularities and low dimension . 26 2.2 Hodge theory obstruction in higher dimension . 35 2.3 Cones over smooth projective complete intersections . 42 3 Classifications of MCM modules and Betti tables over tame curve singularities 54 3.1 The tilting modules . 57 3.2 Graded MCM modules over the cone of 4 points on P2 in general position . 63 3.2.1 Four Subspace Problem . 65 3.2.2 Indecomposable graded MCM modules . 69 3.2.3 Betti tables of indecomposables . 76 3.3 Graded MCM modules over the cone of 4 points on P1 .................... 77 3.3.1 Weighted projective lines of genus one and braid group actions . 83 3.3.2 Matrix factorisations corresponding to simple torsion sheaves . 85 3.3.3 Betti tables from cohomology tables . 89 3.3.4 Cohomology tables of indecomposable coherent sheaves . 94 3.3.5 Betti tables of indecomposables . 101 II Absolutely Koszul Gorenstein algebras 104 4 Fano algebras, higher preprojective algebras and Artin-Schelter regular algebras 105 4.1 Finite dimensional Fano algebras and noncommutative projective geometry . 105 4.2 Fano algebras from Koszul Frobenius algebras . 113 5 Absolutely Koszul algebras and t-structures of Koszul type 120 5.1 Linearity defect and Theorem A . 121 v 5.2 The Artin-Zhang-Polishchuk noncommutative section ring and Theorem B . 144 5.3 The virtual dimension of a Koszul Gorenstein algebra . 151 6 Applications of absolute Koszulity 154 6.1 Application: BGG correspondence beyond complete intersections . 154 6.2 Application: The Coherence Conjectures of Minamoto and Bondal . 164 6.3 Discussion and conjectures . 166 A Appendix 168 A.1 Representation theory of quivers and finite dimensional algebras . 168 A.2 Derived Morita theory and Tilting theory . 175 A.3 Semiorthogonal decompositions and Orlov's Theorem . 177 vi Chapter 1 Introduction This is currently in draft form. This thesis is concerned with representation-theoretic and homological aspects of the stable derived categories of Gorenstein rings, mostly centered around questions of derived Morita theory, Koszul duality and tame classification problems. Stable derived categories were originally introduced by Ragnar-Olaf Buchweitz and later independently by D. Orlov under the name of singularity categories. The stable derived category of a Noetherian ring R is defined as the Verdier quotient b perf Dsg(R) = D (mod R)=D (mod R) of the bounded derived category of R by the thick subcategory of perfect complexes, meaning complexes with finite projective resolutions. When R is commutative, the Auslander-Buchsbaum-Serre character- isation of regular rings as the commutative rings of finite global dimension shows that the singularity category Dsg(R) vanishes precisely for R regular, and thus in general provides a measure of the singu- larities of spec R. A two-sided Noetherian, not necessarily commutative ring R is Gorenstein if both injective dimensions idim (RR) < 1 and idim (RR) < 1 are finite, in which case those dimensions are equal and define the Gorenstein dimension of R. When R is Gorenstein, a foundational theorem of Buchweitz identifies ∼ Dsg(R) = MCM(R) with the stable module category of maximal Cohen-Macaulay (MCM) R-modules, whose study interpolates singularity theory, commutative algebra and representation theory. When R is Z graded, we write Dsg(R) for the version involving graded modules. These triangulated categories have turned out to be ubiquitous in the last 30 years, and routinely appear in algebra and geometry. We list some notable examples: i. The homotopy category of matrix factorizations of f 2 S for S regular is equivalent to the stable derived category of the hypersurface singularity S=f, that is MF(S; f) =∼ MCM(S=f) by a theorem of Eisenbud. ii. The stable derived category of a self-injective algebra Λ is equivalent to its stable module category mod Λ. This applies in particular to the group algebra Λ = kG of a finite group G over a field of characteristic p > 0 dividing the order of G. 1 Chapter 1. Introduction 2 iii. The stable derived category of the group ring SG for S regular and G finite is equivalent to the stable category of lattices over SG. iv. Stable derived categories arise on the opposite side of Koszul duality equivalences with the bounded derived categories Db(coh X) of coherent sheaves on projective varieties X, as in the Bernstein- Gel'fand-Gel'fand (BGG) correspondence Z ∗ ∼ b Dsg(^V ) = D (coh P(V )) and its generalisations to complete intersections of quadrics by Buchweitz and Kapranov. v. Let X ⊆ Pn be an arithmetically Gorenstein projective variety with homogeneous coordinate ring RX , meaning that the affine cone spec RX over X has Gorenstein singularities. Then there is an Z b adjoint pair Dsg(RX ) D (coh X) between the graded stable derived category of RX and the derived category of coherent sheaves on X, which, by a theorem of Orlov, is always an equivalence when X is Calabi-Yau, and in general embeds one category inside the other according to the sign ∼ of the twist on the canonical module !RX = RX (a) of RX .
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