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Results tagged with cohomology
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user 6074
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
17
votes
Accepted
Yoga of six functors for group representations?
In fact, I think the correct answer is that all of this works, except that it is $\pi_\ast$ that gives group cohomology, and $\pi_!$ that gives group homology. …
- 21.3k
15
votes
Accepted
Conjectures of Peter Scholze about q-de Rham complex: examples
First, I should say that in my paper with Bhatt on prismatic cohomology, much of the content of these conjectures has been proved, although the connection is not made very clear, and we only work after … This subspace does not deform to $q$-de Rham cohomology, so one gets no canonical $q$-difference equation. …
- 21.3k
27
votes
What is homology anyway?
The issue is that in the types of $6$-functor formalisms that are usually studied say in étale cohomology, there just is no left adjoint to pullback, and cohomology is not the dual of anything. … , it satisfies a projection formula (which cohomology doesn't in general), and cohomology is always the dual of homology, but not the other way around. …
- 21.3k
15
votes
1
answer
1k
views
Artin vanishing for Stein manifolds and restriction maps
With complex coefficients, a simple argument for this is to compute the cohomology in terms of the cohomology of the de Rham complex. Their theorem gives a more precise Morse-theoretic statement. … With $\mathbb C$-coefficients, it follows from the comparison with de Rham cohomology (at least when cohomology groups are finite-dimensional, which I'm happy to assume). …
- 21.3k
17
votes
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
There is a Weil cohomology theory for varieties over $\overline{\mathbb F}_p$ with values in $\mathrm{Isoc}_{\mathbb R}$. … Conjecturally, a Weil cohomology theory should even exist with values in Kottwitz' category for $F=\mathbb Q$. …
- 21.3k
18
votes
Accepted
Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Roughly speaking, a prerequisite for a stacky approach to some cohomology theory is that this cohomology theory satisfies a categorical Künneth formula: $D(X)\otimes_{D(\ast)} D(Y)\cong D(X\times Y)$. … Edit: And for rigid cohomology, I explained a construction of such a stack in my course, I hope I will one day update the notes to include it. …
- 21.3k