All Questions
Tagged with cohomology homological-algebra
134
questions
3
votes
2
answers
145
views
Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
- 595
2
votes
0
answers
92
views
Koszul cohomology associated with a regular sequence
Let $A$ be a local Noetherian ring and $M$ be an $A$-module. Let $\mathfrak{a}$ be an ideal of $A$ generated by a regular $M$-sequence $s_1,\cdots,s_r$. Let $K_\bullet(s_1,\cdots,s_r;M)$ be the Koszul ...
- 419
1
vote
1
answer
399
views
Why should we study the total complex?
Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
- 165
2
votes
1
answer
265
views
Exact sequence for relative cohomology + normal crossing divisors
Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.
Is it true that there is an exact sequence
$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
- 211
9
votes
0
answers
285
views
How acyclic models led to idea of model categories
The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category.
Could ...
- 5,780
4
votes
4
answers
616
views
Canonical product in sheaf cohomology
EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.3k
5
votes
1
answer
1k
views
Basic question on the de Rham theorem
There is a modern nice proof of the de Rham theorem based on sheaf theory.
The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
- 21.3k
3
votes
0
answers
232
views
Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?
A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
- 1,745
5
votes
1
answer
273
views
Axioms of derivators
I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute ...
- 884
5
votes
2
answers
452
views
How to define cohomology of algebraic structures?
I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain
\begin{align*}
\cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
- 775
5
votes
1
answer
172
views
Restriction vs. multiplication by $n$ in Tate cohomology
$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and ...
- 13.8k
7
votes
1
answer
553
views
Long exact sequences for parametrized cohomology
I'm reading Michael Shulman's articles on cohomology in HoTT here and here, as well as Floris van Doorn's thesis here.
Given $E: Z \to \mathsf{Spectrum}$ a family of spectra over a homotopy type $Z$, ...
- 5,953
3
votes
1
answer
423
views
Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?
Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout (i.e. amalgamated free product) $G_1 \sqcup_H G_2$. Suppose that we have ...
- 5,953
6
votes
0
answers
462
views
“Cohomological equation” in dynamical systems
Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$
Then, being able to find a suitable $h$ ...
- 1,494
4
votes
1
answer
212
views
Poincaré dual of the Alexander dual of the fundamental class of a knot is given by a Seifert surface
Let $K\subset S^3$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening with self linking number $0$. I will denote $F(B^2\times K)$ by $(B^2\times K)$ for simplicity. ...
