Abstract of \Extending Grothendieck topologies to diagram categories and Serre functors on diagram schemes" by Henning Arn´or Ulfarsson,´ Ph.D., Brown University, May 2009. We study Serre functors and related constructions. A Serre functor on a triangulated category was defined by Bondal and Kapranov to be an auto-equivalence inducing certain natural dualities on homorphism sets in the category. In the special case of the bounded derived category of complexes of coherent sheaves on a smooth scheme Y the Serre functor is given by twisting by the dualizing sheaf and shifting by the dimension of the scheme. In the work of Lunts it is shown that a Serre functor exists if the scheme Y is replaced by a diagram scheme, which is a collection of schemes connected by morphisms; or more precisely a functor X : D ! Schemes where D is some category, often called the shape of the diagram. We will give a description of the Serre functor for certain diagram schemes. Related to the description of the Serre functor for diagram schemes is the study of the category DiagSchemes of diagram schemes and how it inherits properties from the category of schemes. We will consider inheritance of Grothendieck topologies and construct a general method for diagrammatizing any topology in such a way that desirable properties are not lost in the process. We will then consider how to carry prestacks and stacks over along with a Grothendieck topology. A part of this work is joint with Jonathan Wise, at Stanford University. Extending Grothendieck topologies to diagram categories and Serre functors on diagram schemes by Henning Arn´or Ulfarsson´ M. Sc., Brown University, 2006 B. Sc., University of Iceland, 2004 Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island May 2009 c Copyright 2009 by Henning Arn´or Ulfarsson´ This dissertation by Henning Arn´or Ulfarsson´ is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Dan Abramovich, Director Recommended to the Graduate Council Date Dan Abramovich, Reader Date Thomas Goodwillie, Reader Date Stephen Lichtenbaum, Reader Approved by the Graduate Council Date Sheila Bonde Dean of the Graduate School iii Vita Henning Ulfarsson´ was born in 1981 in Reykjav´ık, Iceland. He received his bachelor's degree in mathematics from The University of Iceland in 2004 and his master's degree in mathematics from Brown University in 2006. Henning is married to Emil´ıaL´oaHalld´orsd´ottirand together they have three daughters. iv Acknowledgements First I thank my wife, Emil´ıa, and my daughters, Salka, Kolka and Hekla, who have given me endless support and joy in my pursuit of becoming a mathematician. I also want to thank my parents who have always encouraged me to follow my dreams. I am grateful also to my advisor, Dan Abramovich, without whose gentle nudging I might never have finished this thesis. I also want to thank the many friends I've had through graduate school. In particular, I'm glad to count Conni & Joseph, Panagiotis, Katherine & Jonathan, Michelle & David, Steffen, Noah, Qile and Kevin among the friends I've had since I started graduate school. I am also grateful to everyone in Brown's math department, especially Natalie, Doreen, and Audrey, for all their help. v Contents 1 Introduction 1 1.1 A short note on the literature . 2 1.2 The structure of this thesis . 2 I Diagram Categories 4 2 Configuring categories 5 2.1 Constructing the configured category ♦C ......................... 5 2.2 2-categorical aspects of ♦C ................................. 9 3 Configuring the base of a fibered category 12 3.1 Connections with poset categories . 16 4 Configuring sites and mixing topologies 22 4.1 Constructing the necessary fibered products . 22 4.2 The configured topologies . 26 4.3 Subcanonicity of the configured topology . 31 5 Configuring stacks 33 II Serre Functors 36 6 Configurations 37 6.1 Poset schemes and configuration schemes . 37 6.2 The order topology on a poset . 39 6.3 Poset categories and configuration categories . 40 6.4 A note on Serre duality . 41 6.5 Building adjunctions . 42 vi 7 Constructing the Serre functor for the inclusion of a divisor 45 7.1 Describing the Serre functor in terms of the individual Serre functors . 45 7.2 Describing the Serre functor in terms of the dualizing sheaves . 47 7.3 Conjectural description of the trace map . 51 A Review of some preliminaries 54 A.1 Some category theory . 54 A.2 Posets . 59 B Auxiliary material 61 B.1 A simpler version of ♦C .................................. 61 B.2 Some observations on the topologies on ♦C ....................... 62 C Homological algebra of general shapes 66 C.1 Introduction . 66 C.2 Replacing the source Z ................................... 66 C.3 Generalizing the category of complexes . 67 Bibliography 77 vii Chapter 1 Introduction This project started with my advisor's question about what the Serre functor [3] looks like for diagram schemes, i.e., diagrams X : S ! Sch in the category of schemes1. Any smooth scheme Y has a dualizing sheaf !Y which gives us Serre duality in the bounded derived category of complexes of coherent sheaves on Y, Db(Y ); i.e., for any two complexes E; F in Db(Y) we have a natural isomorphism L _ RHom(E; F ) ' RHom(F; E ⊗ !Y [n]) ; b b L where n is the dimension of Y . We define a functor S : D (CohY ) ! D (CohY ), S(E) = E ⊗ !Y [n], and call it the Serre functor. The Serre functor is a powerful tool for working with coherent sheaves and Lunts proved that a Serre functor exists when the scheme Y is replaced by certain types of diagram schemes in [13], but did not given a concrete description of it. A simple example of a diagram scheme is the inclusion of a divisor into a scheme, D ! Y ; this is given by the functor X : {•1 !•2g ! Sch, defined by X(•1) = D, X(•2) = Y and X(•1 !•2) the inclusion of D into Y . I was able to give a description of the Serre functor for this type of diagram scheme with Jonathan Wise and we have a forthcoming preprint on the proof [16]. This makes up Part II of this thesis. Part I contains an investigation of how the category of diagram schemes inherits properties from the category of schemes. In [13] where Lunts proved the existence of the Serre functor he did not construct a category around diagram schemes, so I defined morphisms between diagram schemes and constructed the category DiagSch2. Lunts defined and used poset categories3 to describe sheaves on diagram schemes and I wanted to give a more natural description, in terms of fibered categories. When I had taken care of the above I asked if it were possible to carry Grothendieck topologies and stacks with you when you \diagrammatize", i.e., if you start with a site (C; T ) and you form the diagram category DiagC, is there an extension of T to DiagC, call it DiagT , such that 1. DiagT restricted to C is equivalent to T . 1Here X is a functor and S is any category, sometimes called the shape of the diagram. 2It turned out that this category had been used in [11, 12], see the note on the literature below, as well as many other places. 3A poset category is a collection of categories \glued together along a poset". 1 2 2. If T is subcanonical4 then so is DiagT . 3. If F ! C is a stack in T , then DiagF ! DiagC is a stack in DiagT . I found two candidate topologies for DiagT , one which satisfies only (1), and another which satisfies (1), (2) and (3). The latter one is constructed in such a way that \both the objects and the arrows in the diagrams are covered", without going into too much detail. 1.1 A short note on the literature Diagrams of schemes have been studied before, such as by Illusie in [11], [12], where the a category of sheaves over a fixed configuration scheme is studied as a topos. Here we will be interested in looking at configuration sheaves over many configuration schemes at once and in some sense this should be a topos glued from the individual topoi. We will not delve too deeply into this connection here. See also Remark 2.4. 1.2 The structure of this thesis Part I Preliminary material on posets, fibered categories, Grothendieck topologies and stacks can be found in Appendix A. In Section 2 we consider a generalization of Lunts's configuration schemes by 1. replacing the category of schemes with an arbitrary category C; and 2. replacing the fixed poset S with a category of categories S called the category of shapes. We define the category of diagrams, denoted ♦C, whose objects are functors X : S ! C, where S is an object of S. We give examples of diagram categories and look at some of their basic properties. In Section 3 we replace the category of coherent sheaves on Sch by an arbitrary fibered category F ! C and show in Theorem (3.1) that ♦F ! ♦C is also fibered. To get a better description of the fiber ♦F(X) for a diagram X : S ! C we define the category of liftings X in Definition 3.2. Lunts uses poset categories to describe these same fibers when S is a (finite) poset and in Theorem (3.13) we show how this is related to our liftings X by using the slightly more general pseudo-poset categories.