Structured spaces: categories, sheaves, bundles, schemes and cohomology theories

Cohomology of GKM-sheaves

Let $T$ be a compact torus and $X$ be a a finite $T$-CW complex (e.g. a compact $T$-manifold). In earlier work, the second author introduced a functor which assigns to $X$ a so called GKM-sheaf $\mathcal{F}_X$ whose ring of global sections $H^0(\mathcal{F}_X)$ is isomorphic to the equivariant cohomology $H^*_T(X)$ whenever $X$ is equivariantly formal (meaning that $H^*_T(X)$ is a free module over $H^*(BT))$. In the current paper we prove more generally that $H^0(\mathcal{F}_X) \cong H^*_T(X)$ if and only if $H_T^*(X)$ is reflexive, and find a geometric interpretation of the higher cohomology $…

Splitting vector bundles outside the stable range and A^1-homotopy sheaves of punctured affine spaces

We discuss the relationship between the ${\mathbb A}^1$-homotopy sheaves of ${\mathbb A}^n \setminus 0$ and the problem of splitting off a trivial rank $1$ summand from a rank $n$-vector bundle. We begin by computing $π_3^{{\mathbb A}^1}({\mathbb A}^3 \setminus 0)$, and providing a host of related computations of "non-stable" ${\mathbb A}^1$-homotopy sheaves. We then use our computation to deduce that a rank $3$ vector bundle on a smooth affine $4$-fold over an algebraically closed field having characteristic unequal to $2$ splits off a trivial rank $1$ summand if and only if its third Chern c…

Blown-up intersection cochains and Deligne's sheaves

In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to prove that the sheaf versions of the functors of blown-up intersection cochains are realizations of Deligne's sheaves. This proves that Deligne's sheaves can be incarnated at the level of complexes of sheaves by soft sheaves of perverse differential graded algebras. We also study Poincaré and Verdier dualities of blown-up intersections sheaves with the use o…

Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.

On the Bott-Chern characteristic classes for coherent sheaves

Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham cohomology. The main result of this paper shows that such characteristic classes are independent of the Hermitian metric and only dependent on the homotopy class of the cohesive module in the homotopy category. This extends the Bott-Chern cohomology to the bounded derived catego…

A Small Model for the Cohomology of Some Principal Bundles

Let G be a compact, connected and simply connected Lie group, and ΩG the space of the loops in G based at the identity. This note shows a way to compute the cohomology of the total space of a principal ΩG-bundle over a manifold M, from the cohomology of G, the differential forms on M and the characteristic classes of the bundle. The equivariant situation is also treated.

超细层与相互作用分解

Extra-fine sheaves and interaction decompositions

本文引入拓扑空间上的超细层概念，将其应用于配备Alexandroff拓扑的偏序集，推导相互作用分解的求和-相交条件，计算整体截面维数并推广线性边际问题计数公式，还给出相关上同调比较定理。

Betti数的均匀逼近

Uniform approximation of Betti numbers

本文将Lefschetz关于仿射概形上同调逼近的原理实现为均匀形式，利用Jouanolou技巧、Deligne一般基变换公式的动形版本及反常层理论，提出了一种独立于上同调界定域上拟射影概形Betti数的新方法。

Pontryagin对偶与profinite模层

Pontryagin duality and sheaves of profinite modules

本文研究profinite群的Pontryagin对偶理论，填补了在离散模系数上同调研究中缺失的对偶构造这一理论空白。

Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve

We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

A commutative P^1-spectrum representing motivic cohomology over Dedekind domains

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated étale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application…