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Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$ whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold. [ABSTRACT FROM AUTHOR]

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if $(Z/p)^r$ acts freely on $(S^1)^k$, then $r \leq k$. [ABSTRACT FROM AUTHOR]

The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic. [ABSTRACT FROM AUTHOR]

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras. [ABSTRACT FROM AUTHOR]

We compute the Hochschild homology of Leavitt path algebras over a field k. As an application, we show that L2 and L2 ⊗ L2 have different Hochschild homologies, and so they are not Morita equivalent; in particular, they are not isomorphic. Similarly, L∞ and L∞ ⊗ L∞ are distinguished by their Hochschild homologies, and so they are not Morita equivalent either. By contrast, we show that K-theory cannot distinguish these algebras; we have K*(L2) = K*(L2 ⊗L2) = 0 and K*(L∞) = X*(L∞ ⊗ L∞) = K*(k). [ABSTRACT FROM AUTHOR]

The article considers another construction of the p-adic period map using the de Rham cohomology of Illusie and de Jong's alterations in a bid to understand the Riemann-Hilbert correspondence in this setting. Information on the derived de Rham algebra is provided, as well as the integer rings. It cites the projective system of commutative dg algebras. Also included are information on a reinterpretation of Colmez's theorem and derived de Rham algebra in logarithmic setting.