Related papers: A Riemann-Roch Theorem for dg Algebras

For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect…

Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the familiar pairing on the…

We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type…

Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg…

We set up a homological algebra for N-complexes, which are graded modules together with a degree -1 endomorphism d satisfying d^N=0. We define Tor- and Ext-groups for N-complexes and we compute them in terms of their classical counterparts…

This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment…

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to…

We show how to compute a certain group of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group of…

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The…

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a…

We introduce a notion of Hochschild Lefschetz class for a good coherent D-module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann-Roch formula for a D-module…

The Hirzebuch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension are explicitly given. They have cohomological, homological, Hochschild cohomological and Hochschild…

A DG algebras $A$ over a field $k$ with $H(A)$ connected and $H_{<0}(A)=0$ has a unique up to isomorphism DG module $K$ with $H(K)\cong k$. It is proved that if $H(A)$ is degreewise finite, then $RHom_A(?,K): D^{df}_{+}(A)^{op} \equiv…

For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and…

We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^*(A,A)$. We also study a $G$-algebra structure associated to this deformation theory.

On a (pseudo-)Riemannian manifold (MM,g), some fields of endomorphisms i.e. sections of End(TMM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra,…

In a previous paper the authors constructed a class of quasi-Hopf algebras $D^{\omega}(G, A)$ associated to a finite group $G$, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological…

We construct virtual fundamental classes for dg-manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend-Fantechi [BF]…