All Questions
7 questions
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
6
votes
1
answer
478
views
Unbounded acyclic resolutions
Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
- 28.7k
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
1
vote
1
answer
634
views
Inception of modern view of Sheaf Cohomology in Mathematical Literature
From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
- 37
0
votes
0
answers
308
views
Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
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