We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in ...

In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H D (ΘS(− logD)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H D (ΘS(−...

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL.2;C/. The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twist...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.

Let $(A, I)$ be a bounded prism, and $X$ smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ relative to $A$. prove that if is proper $\Spf(A/I)$ dimension $n$, then cohomology crystal perfect complex $A$-modules with tor-amplitude in degrees $[0,2n]$. also establish Poincar\'e duality for reduced crystals, i.e. ...

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...

Steenrod operations were defined by Voedvodsky in motivic cohomology in order to prove the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings [5]. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. W...

Overview 1 1. Mymotivation: K-theory of schemes 2 2. First steps in homological algebra 3 3. The long exact sequence 6 4. Derived functors 8 5. Cohomology of sheaves 10 6. Cohomology of a Noetherian Affine Scheme 12 7. Čech cohomology of sheaves 12 8. The Cohomology of Projective Space 14 9. Sheaf cohomology on P̃2 16 10. Pushing around sheaves, especially by the Frobenius 17 11. A first look at...

Given a compact complex manifold $X$ and integrable Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$, we introduce double structure on $A^{\bullet,\bullet}(X)$ naturally determined by $\phi$ study its Bott-Chern cohomology. In particular, establish deformation theory for cohomology use it to compute the deformed Iwasawa holomorphically parallelizable Nakamura manifold. The $\partial\bar{...