author={Liviu I. Nicolaescu},
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… 
Combinatorial part of the cohomology of the nearby fibre
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
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
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
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
. 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
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
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
. 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
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


Resolutions of unbounded complexes
Various types of resolutions of unbounded complexes of sheaves are constructed, with properties analogous to injectivity, flatness, flabbiness, etc. They are used to remove some boundedness
Algebraic D-modules
Presented here are recent developments in the algebraic theory of D-modules. The book contains an exposition of the basic notions and operations of D-modules, of special features of coherent,
Manin: Methods of Homological Algebra, 2nd Edition,Springer-Verlag
  • 2003