Questions tagged [cohomology]
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, ...
1,483 questions
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On the sum of Kloosterman sum of degree 4
I have a puzzle which needs some helps from the experts here.
Let the Kloosterman sum of degree 4 be as follows:$$S_4(1,n;c)=\hskip 0.5em \sideset{_{}^{}}{^{\ast}_{}}\sum\limits _{x,y,z \bmod c} e \...
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Reference on homology and cohomology at the undergraduate level, towards $\infty$-categories
I am planning a one-semester course in algebraic topology for final-year undergraduate mathematics students. The intended focus is on homology and cohomology, and it would be the first encounter of ...
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Restriction of a positive-frequency scalar Penrose class to a Hopf torus in a local sky bundle
Let $\mathbb{PT}$ be projective twistor space with its standard real structure, and let $\mathbb{PT}^+$, $\mathbb{PT}^-$, and $\mathbb{PN}$ denote the positive-frequency region, negative-frequency ...
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Action on cohomology of $G$-space reconstructable from monodromy action
Let $G$ be a topological group with $\pi_0(G) \simeq G$ acting on topological space $X$ nicely enough such that induced quotient map $X \to X/G$ is a $G$-principal bundle. Clearly this action induces ...
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Cohomology of formal schemes
Let $\mathfrak{X}=(\mathfrak{X}, \mathcal{O}_\mathfrak{X})$ be a Noetherian formal scheme in sense of EGA I, Def 10.4. Let $I \subset \mathcal{O}_\mathfrak{X}$ be an coherent ideal contained in an (...
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Wikipedia's version of Zeeman's comparison theorem for spectral sequences
Wikipedia states a version of Zeeman's comparison theorem for spectral sequences of flat modules over a commutative ring. For sources, one can look at the following.
Zeeman's original article
...
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Why is there only one Frobenius lift on $\mathbb{Z}_p$
I want to verify that there exists a unique $\delta$-structure on $\mathbb{Z}_p$.
Since $\mathbb{Z}_p$ has no torsion, we have a bijective correspondence between $\delta$-structures and lifts of ...
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Non-nilpotent elements in the cohomology of the mod-2 Steenrod algebra
I have heard that there are interesting non-nilpotent elements on the $E_2$-page of the mod-2 Adams spectral sequence (besides just $h_0$). For example, I have heard that $g$ and some related elements ...
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Connection between Quantum Functional Analysis and Quantum Cohomology in Witten-Gromov theory
The root of this answer stems from the similarity in names between the two subjects. As infantile as that may seem, I think there is some merit to exploring that connection.
To resume the foundations ...
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Ordered Čech nerve
Given a topological space $X$, an abelian sheaf $\mathcal{F}$, and a covering $X = \bigcup_{i \in I}U_i$, we can compute the sheaf cohomology using the Čech complex, whose $k$-th term is given by $$\...
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Computing the action of the Frobenius on $\ell$-adic cohomology
When is it possible to explicitly compute the action of the Frobenius on the $\ell$-adic cohomology (without presupposing the truth of the Weil Conjectures, the Lefschetz formula, or importing any ...
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Computing the negligible ideal of H4 in mod 2 cohomology
Let $G$ be a finite group and write
$$
H^*(G) := H^*(G;\mathbb F_2).
$$
Following Serre, an element $x \in H^d(G)$ is said to be negligible
(over $\mathbb Q$) if for every field extension $L/\mathbb Q$...
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The cycle-class map in Étale Cohomology
In Deligne’s SGA 4 1/2 (p.80), he defines the cycle-class map for a local complete intersection $i : Y\hookrightarrow X$ of codimension $c$ as follows:
He restricts to a complete intersection of ...
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How does Habiro cohomology relate to prismatic cohomology?
Preface: I know that Habiro cohomology is a theory that is very new and still under development, and further there appear to be multiple different approaches to it, and in the post below I will remain ...
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Torelli for divisors in products of projective space
Let $Z_{m,n}$ be a divisor of bidegree $(m,n)$ inside $\mathbb{P}^1 \times \mathbb{P}^2$. Do any Torelli theorems hold for $Z_{m,n}$? I'm particularly interested in the case $(m,n)=(2,4)$.
It would ...
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