Day convolution and the Hodge filtration on THH by Saul Glasman V-- Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of z CD Doctor of Philosophy in Mathematics QLL = <~ - at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 @Massachusetts Institute of Technology 2015. All rights reserved. Parts of this thesis were first published in Mathematical Research Letters in 2015, published by International Press. A uthor .. ...................... Department of Mathematics April 29, 2015 Signature redacted" Certified by.... e ......... ........I.................... Clark Barwick Cecil and Ida B. Green Career Development Associate Professor Thesis Supervisor Accepted by. Signature redacted.................... William Minicozzi Chairma Depart ent Committee on Graduate Theses Day convolution and the Hodge filtration on THH by Saul Glasman Submitted to the Department of Mathematics on April 29, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract This thesis is divided into two chapters. In the first, given symmetric monoidal oc- categories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary prop- erties. Thesis Supervisor: Clark Barwick Title: Cecil and Ida B. Green Career Development Associate Professor 3 4 Acknowledgments This thesis, as well as the author's mathematical career, owes an inestimably enor- mous debt to nearly five years spent in the company of Clark Barwick's deep insight and inexhaustibly patient guidance. Clark proposed both of the projects contained in this thesis and taught me everything I needed to see them to completion. I'd like to thank MIT for providing a hospitable working environment and a the MIT-Harvard topology bloc for being an absolute powerhouse. There's nowhere like Boston to be a topology student. I'd also like to thank Jacob Lurie for writing the Two Indispensibles, Higher Topos Theory and Higher Algebra, and thus promoting quasicategory theory to something that anyone sufficiently motivated could use to do real work. 5 6 Contents 1 Introduction 9 2 Day convolution for oo-categories 13 2.1 The Day convolution symmetric monoidal oo-category ......... 13 2.2 The symmetric monoidal Yoneda embedding .... .......... 24 3 A spectrum-level Hodge filtration on topological Hochschild homol- ogy 31 3.1 Preliminaries on twisted arrow categories, ends and coends ....... 31 3.2 Tensor products of commutative ring spectra with spaces ...... 38 3.3 The Hodge filtration .... ........ ........ ...... 42 3.4 M ultiplicative structure ..... ........ ....... ..... 47 3.5 The layers of the filtration and Adams operations ... ....... 51 A Commutative algebras and their modules 57 7 8 Chapter 1 Introduction In this thesis, we seek to begin the development of a theory of functorial, spectrum- level Hodge filtrations on topological Hochschild homology (THH) of commutative ring spectra and its cousins, such as TR and TC. What is a Hodge filtration? The original Hodge filtration is a filtration on the de Rham complex of a commutative ring R Q*(R) -( --. -+( R*(R)) " 4 - (Q*(R)) = R where the complex (Q*(R)) " is the "stupid truncation" of Q*(R), which is not a homotopy invariant of * (R) but depends on R itself: it has the same terms as Q* (R) in degrees at most n, and is zero in degrees greater than n. This filtration gives rise to a spectral sequence known as the Hodge to de Rham spectral sequence. In a celebrated sequence of papers beginning with [71, Deligne proved that for a C-algebra R, the spectral sequence degenerates at E2 , giving rise to a mixed Hodge structure on the de Rham cohomology of R. In the special case where R is regular, the Hochschild-Kostant-Rosenberg theorem [121 gives an isomorphism of graded abelian groups HInH,(R) Loda ~ * (R) where HH* is Hochschild homology. In [14], Loday interpreted the Hodge filtration as 9 the -y-filtration associated to a A-ring structure that exists on the Hochschild homology of an arbitrary commutative ring R, thus generalizing the Hodge filtration in an algebraic direction. Rationally, this filtration is canonically split, and it coincides with the decomposition by eigenspaces of the Adams operations that exist on any rational A-ring. Later, Pirashvili [24], working rationally, used functor homology to give a general- ization of the rational Hodge decomposition to what he called "higher-order Hochschild homology". In the present work, we take Pirashvili's approach and run with it in a homotopy-theoretic direction, using a homotopy coend formula for topological Hochschild homology and its "higher-order" variants to give spectrum-level filtrations with properties analogous to those of Loday and Pirashvili's original examples. We work with quasicategories throughout, and we will liberally reference Lurie's blockbuster volumes [15] and [17]. The structure of the paper is as follows. Chapter 3 is devoted to the construction and properties of the Hodge filtration. In order to investigate the multiplicative properties of this filtration, however, we must construct an oc-categorical analog of the Day convolution. This goal, which we accomplish in Section 2.1, is the subject of the more technical Chapter 2, and is of independent interest. We'll recall some classical facts about this construction. Let (C, Oc) and (D, ®-) be two symmetric monoidal categories such that D admits all colimits. In [6], Day equips the functor category Fun(C, D) with a "convolution" symmetric monoidal structure: If F, G : C -+ D are functors, then their convolution product F ODay G is defined as the left Kan extension of 0D o (F x G) : C x C -> D along Oc : C x C - C. According to [6, Example 3.2.2], the commutative monoids for the convolution product are exactly the lax monoidal functors from C to D. We show in Proposition 2.1.10 that the analogous proposition holds for oc-categories - that is, E, algebras for the Day convolution product are lax symmetric monoidal functors, which is to say oc-operad maps in the sense of [17, Definition 2.1.2.7] - thus answering a question of Blumberg. In Section 2.2, we construct the Yoneda embedding for a symmetric monoidal 10 category C as a symmetric monoidal functor into the presheaf category Fun(CP, Top) equipped with the Day convolution product, and deduce that our construction agrees in this special case with the Day convolution constructed by Lurie in [17, Corollary 6.3.1.121. The study of THH is far from the only application of this version of the Day convolution. An important special case is the tensor product of Mackey functors; see, for example, [23]. In a recent paper [31, Barwick develops a theory of higher- categorical Mackey functors in order to study equivariant K-theory. In forthcoming work of the author with Barwick, Dotto, Nardin and Shah [4], an equvariant theory of the Day convolution is developed, building on the material in this thesis. The stage is now set for Chapter 3. In Section 3.1, we collect some useful facts about ends and coends in the oo-categorical context. In Section 3.2, we discuss tensor products of commutative ring spectra with spaces. In Section 3.3, we define our central object of study: for each simplicial set X, we define a Hodge-like filtration on the functor CRing -+ CRing: R - R 0 X where CRing is the category of commutative ring spectra. Keeping X free, we explore the geometric structure of our filtration. In Section 3.5, we specialize to X = S1 , so that R OX - THH(R). Here we identify the graded pieces of the filtration and show that they are eigenspectra for the Adams operations V, as they must be if we are to sensibly call our filtration a Hodge filtration. In addition, we believe that our filtration coincides with the TAQ- filtration investigated by Minasian [221 and McCarthy-Minasian [20]. We will take 11 up this questions in a future paper. Finally, in Appendix A, we give an account of modules over commutative algebras in quasicategories that fills a minor gap in the literature. In a forthcoming paper [10], we will describe how to lift the Hodge filtration on THH to a filtration by cyclotomic spectra, and thus obtain a filtration on topological cyclic homology TC as well as its antecedents TR and TF. We suspect this filtration to be related to a filtration on TF, with deep connections to p-adic geometry, whose existence was conjectured by Scholze [26], and we plan to investigate this link. We also hope to show that the cyclotomic trace from K-theory to TC makes the weight filtration on K-theory (various versions of which are explicated beautifully in [11]) compatible with the Hodge filtration on TC, ideally by showing that the trace is a map of spectral A-rings (whatever these are) and by framing the filtrations on each side as 7-filtrations. In between, we plan to give explicit computations of the filtrations we define in interesting cases. 12 Chapter 2 Day convolution for oc-categories 2.1 The Day convolution symmetric monoidal 00- category Throughout this thesis, F will denote the category of finite sets, or by abuse of no- tation, the nerve of that category. Likewise, F, will denote the category of finite pointed sets and pointed maps or its nerve. Let C -+ F, and D* -+ F, be sym- metric monoidal oo-categories (see [17, Definition 2.0.0.71). To sidestep potential set-theoretic issues, we'll fix a strongly inaccessible uncountable cardinal A and as- sume that both C® and DO are A-small.