We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toën's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.

the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S, we introduce a dg-category F (S, A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F (S, A) is independent of the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces.In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F (S, A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.

We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi-Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi-Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between categories of matrix factorisations. We give an elementary proof of a formula for the Chern character of the convolution generalising the Hirzebruch-Riemann-Roch formula of Polishchuk and Vaintrob. IntroductionA linear factorisation of an element W in a ring R (all our rings are commutative) is a Z/2-graded R-module X together with an odd R-linear map d : X −→ X, called the differential, which squares to W · 1 X . The pair (X, d) is called a matrix factorisation if each X i is free over R, and a finite rank matrix factorisation if each X i is free of finite rank. We write HMF(R, W ) for the homotopy category of matrix factorisations and hmf(R, W ) for the full subcategory of finite rank matrix factorisations. These objects were introduced by Eisenbud [Eis80] as a way of representing a special class of modules over local rings of hypersurface singularities, and they have played an important role in singularity theory and more recently in homological mirror symmetry and string theory. Because of this latter connection W is now often referred to as the potential.Suppose that ϕ : S −→ R is a ring morphism and that W = ϕ(V ). Viewing X as an S-module via ϕ we obtain a Z/2-graded S-module ϕ * (X) with a differential squaring to V ·1 ϕ * (X) , and we call this linear factorisation of V over S the pushforward along ϕ. In this paper we study the pushforward of finite rank matrix factorisations along ring morphisms. Even when X is finite rank the pushforward is typically not a finitely generated S-module, but we can ask: when is ϕ * (X) homotopy equivalent to a finite rank matrix factorisation, and can such a finite model for the pushforward be described concretely? We refer to this as the construction of finite pushforwards.Before stating our results we want to mention some interesting examples of pushforwards. Our first motivation was to understand the composition of integral functors between categories of matrix factorisations; let us begin with a special case, which involves the Hirzebruch-Riemann-Roch theorem for matrix factorisations.Hirzebruch-Riemann-Roch. Let k be a field of characteristic zero, W ∈ R = k x 1 , . . . , x n a polynomial and X, Y two finite rank matrix factorisations of W over R. The R-module Hom R (X, Y ) has a natural Z/2-grading and a differential making it into a matrix factorisation of zero and if the zero locus of W has an isolated singularity at the origin then Hom R (X, Y ) has finite-dimensional cohomology over k. One then defines the Euler characteristicThe Hirzebruch-Riemann-Roch theorem for matrix factorisations, recently proven by Polishchuk and Vaintrob [PV10], expresses this Euler characteristic in terms of the Chern characters of X and Y using the general Riemann-Roch theorem for dg-algebras of Shklyarov [Shk07]. Our point of view is that pushing Hom...