Related papers: Godement resolution and operad sheaf homotopy theo…
Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in Arkhipov and {\O}rsted (2018a, 2018b), we define a dg-model for the derived category of…
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the \'etale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three…
For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…
We continue our study of the monoidal category $D(G)$. At the level of cohomology we transfer the duality functor to the derived category of Hecke dg-modules. In the process we develop a more general and streamlined approach to the…
We build resolutions for general twisted tensor products of algebras. These bimodule and module resolutions unify many constructions in the literature and are suitable for computing Hochschild (co)homology and more generally Ext and Tor for…
In this short note we show that the homotopy category of smooth compactifications of smooth algebraic varieties is equivalent to the homotopy category of smooth varieties over a field of characteristic zero. As an application we show that…
We develop a theory of minimal models for algebras over an operad defined over a commutative ring, not necessarily a field, extending and supplementing the work of Sagave in the associative case.
We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The…
We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coefficients in the graded exterior sheaf of the natural sheaf. This builds on the results of our previous paper, where this homology was computed…
We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for…
From the `cofree' cooperad $T'(A[-1])$ on a collection $A$ together with a differential, we construct an $L_\infty$-algebra structure on the total space $\bigoplus_nA(n)$ that descends to coinvariants. We use this construction to define an…
We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models…
We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible $G$-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a…
We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skew-commutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for…
Action of the graded Grothendieck-Teichmueller (GT) group on a resolution of the operad of Gerstenhaber algebras (GA) is defined. It is shown that the induced Lie algebra action is homotopically non-trivial (i.e. the induced map from the…
We define and study binary operations for homotopy groups with coefficients. We give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of…
We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of…
We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally…
For $k$ a field of positive characteristic and $X$ a smooth variety over $k$, we compute the Hochschild cohomology of Grothendieck's differential operators on $X$. The answer involves the derived inverse limit of the Frobenius acting on the…
Boij-S\"oderberg theory focuses on the properties and duality relationship between two types of numerical invariants. One side involves the Betti table of a graded free resolution over the polynomial ring. The other side involves the…