3
votes
2answers
146 views
Closed monoidal structure on the derived category of sheaves
Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the …
2
votes
2answers
193 views
Internal hom of sheaves
Consider a topos, i.e. the category $Shv$ of sheaves on a Grothendieck site $T$ with values in abelian groups. The category $Shv$ is symmetric monoidal with $\otimes$, the tensor p …
12
votes
2answers
289 views
Is there a “categorical” description of Grothendieck’s algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the followi …
3
votes
1answer
170 views
Simplifying the definition of a geometric context using sieves?
On Pages 1-3 of Cours 2 of Toën's Master Course on Stacks, he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over an …
1
vote
2answers
163 views
Cohomology with compact support for coherent sheaves on a scheme
Is there a notion (for schemes or just locally ringed spaces) of cohomology with compact support? I guess there is for algebraic schemes over $\mathbf{C}$, but what about schemes i …
5
votes
1answer
203 views
Is there a description of sheaf cohomology in algebraic-topological terms?
Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology?
In more detail: Any sheaf …
7
votes
3answers
215 views
Sheaves as full reflective subcategories
Hello everyone.
My question is concerned with the following statement.
"Having a grothendieck topology on a category C is equivalent to having a full reflective subcategory Sh(C) …
0
votes
1answer
108 views
O_X module with support Z \subset X vs O_S module?
Given a $O_X$ module $\cal F$ whose support is a closed subscheme $Z \subset X$. Under what conditions can we say that $ \cal F$ is an $O_S$ module ( how far off is $\cal F$ an $O_ …
8
votes
2answers
189 views
Cohomology of a sheaf of functions locally constant along a foliation
Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known ab …
26
votes
8answers
795 views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space X we can define the sheaf cohomology of X in
I. the Grothendieck style (as the right derived functor of the global sections functor Gamma(X,-))
or
II. …
16
votes
4answers
481 views
What is the right version of “partitions of unity implies vanishing sheaf cohomology”
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ …
10
votes
3answers
337 views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover ${X_i}$ in two different way …
15
votes
2answers
215 views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare?
Let's take the concrete special case of free loops spaces. I think this is the most studied …
4
votes
3answers
321 views
Relative version of sheaf cohomology?
Is there a relative version of sheaf cohomology?
EDIT: I rather mean the cohomology of pairs.
3
votes
1answer
145 views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category …
