# THE COHOMOLOGY OF SHEAVES

@inproceedings{Nicolaescu2011THECO, title={THE COHOMOLOGY OF SHEAVES}, author={Liviu I. Nicolaescu}, year={2011} }

A fast introduction to the the construction of the cohomology of sheaves pioneered by A. Grothendieck and J.L. Verdier. The approach is from the point of view of derived categories, though this concept is never mentioned. 1. Sheaves Let R commutative ring with 1. We denote by RMod the category of R-modules. For any topological space X we denote by OpenpXq the collection of open subsets. It can be organized as a category in which the morphisms are given by inclusions. A pre-sheaf of R-modules is…

## 136 Citations

Combinatorial part of the cohomology of the nearby fibre

- Mathematics
- 2022

Let f : X → S be a unipotent degeneration of projective complex manifolds of dimension n over a disc such that the reduction of the central fibre Y = f(0) is simple normal crossings, and let X∞ be…

Two cohomology theories for structured spaces

- Mathematics
- 2020

In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps…

Talk 3: Review of constructible sheaves and operations on sheaves

- Mathematics
- 2019

with the obvious restriction maps. Given x ∈ X , we can choose an open neighborhood U ⊂ X such that π−1(U) ' U , so Γ(π : Y → U) ' F . In particular, F|U (V ) = C〈F 〉 for any V ↪→ U , and F|U is a…

Coarse sheaf cohomology

- Mathematics
- 2022

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coeﬃcients produce interesting…

Irreducible symplectic complex spaces

- Mathematics
- 2012

In chapter 1 we define period mappings of Hodge-de Rahm type for certain submersive, yet not necessarily locally topologically trivial, morphisms of complex manifolds. Generalizing Griffiths's…

The Parity of Lusztig's Restriction Functor and Green's Formula

- Mathematics
- 2021

. Our investigation in the present paper is based on three important results. (1) In [13], Ringel introduced Hall algebra for representations of a quiver over ﬁnite ﬁelds and proved the elements…

Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

- Mathematics
- 2020

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a…

Perverse sheaves learning seminar: Sheaf functors and constructibility

- Mathematics
- 2018

Conventions 1.1. Varieties are (quasiprojective) complex varieties, and we will use their complex topology when dealing with constructibility of sheaves and complexes. Morphisms of varieties are…

A Sheaf-Theoretic Construction of Shape Space

- MathematicsArXiv
- 2022

. We present a sheaf-theoretic construction of shape space—the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to…

Homotopical Aspects of Mixed Hodge Theory

- Mathematics
- 2012

In the present work, we analyse the categories of mixed Hodge complexes and mixed Hodge diagrams of differential graded algebras in these two directions: we prove the existence of both a…

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