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, ...
1,354
questions
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Higher direct image for rational singularities
Let $X$ be a normal, projective (complex) variety with at worst rational singularities. Let $\pi:Y \to X$ be the resolution of singularities obtained by blowing-up the singular points. Is $R^1 \pi_*\...
2
votes
1
answer
236
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Computing (relative) cohomology classes on quotient (vector) space via Hodge theorem
I am working on a graded vector space $V = \bigoplus_{i\in \mathbb{N}}V_i$ (which is a parabolic Verma module in the sense of [1], but let's ignore such specifics) with a positive definite inner ...
10
votes
1
answer
907
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Is every additive cohomology operation stable?
To start, let's work with mod $p$ cohomology $H\mathbb F_p$ where $p$ is a prime. Consider the following three things:
The bigraded abelian group of all unstable cohomology operations, comprising all ...
10
votes
1
answer
1k
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The (current) obstructions for a cohomological interpretation of the Riemann zeta function
I am interested in the idea of a cohomological interpretation of the Riemann hypothesis (suggested by Deninger/Connes).
I am a beginner in étale cohomology, and I would like to ask the following
...
16
votes
1
answer
575
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Multiplicative Brown representability?
The Brown representability theorem can be convenient way to construct a spectrum. But to get a ring spectrum of even a very unstructured form seems to be harder. There's even currently a statement on ...
4
votes
1
answer
134
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Compactness as a consequence of the adjunction formula for genus second homology class
Recall the adjunction formula $$ g(\alpha) = 1 + \frac{1}{2}\left( \alpha^2 -c_1(X)\cdot \alpha \right)$$ where $g(\alpha)$ is the genus of a pseudoholomorphic representative of the Poincaré dual of $\...
5
votes
1
answer
401
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Using HoTT, why is twisted cohomology of BG group cohomology?
I've been reading Michael Shulman's blog posts defining cohomology in homotopy type theory, and I'd like to understand (using HoTT) why cohomology of BG is group cohomology.
if I understand correctly, ...
6
votes
1
answer
519
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Representable cohomology theories in motivic homotopy theory
I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question:
Which cohomology theories on $Sm/k$ are representable, i.e. ...
3
votes
0
answers
181
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Splitting vector bundles on $\mathbb{P}^n$
There are results by Kleiman and Sumihiro that claim you can split an algebraic vector bundles (in the sense that it admits filtration that quotients are given by line bundles) by applying ...
3
votes
0
answers
162
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Obstruction to delooping
Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
7
votes
1
answer
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Motivation for the definition of complex orientable cohomology theory
PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...
4
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0
answers
219
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Relative Thom isomorphism
Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
6
votes
2
answers
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Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack
Let $G$ be an affine group scheme over a field $k$ of characteristic zero.
I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) ...
2
votes
0
answers
264
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Killing cohomology of structure sheaf by pullback along Frobenius and finite etale covers
On a smooth projective variety $X$ over a finite field, you can pullback any element in $H^1(X,\mathcal{O}_X)$ by a combination of Frobenius and finite etale cover so it gets killed. In order to prove ...
5
votes
2
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Complement to a union of spheres in a sphere
Take $S^n$ and consider the union $Z$ of $k_1$ circles, $k_2$ 2-dimensional spheres, ..., $k_{n-2}$ $(n−2)$-dimensional spheres, embedded in $S^n$ in an unknotted way, with no mutual intersection and ...
2
votes
1
answer
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How to compute cup product of derived limits / presheaf cohomology
I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
5
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1
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What is the meaning of this coboundary homomorphism for group hypercohomology?
$\require{AMScd}$
Let $\Gamma=\{1,\gamma\}$ be a group of order 2.
In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules
(abelian groups ...
2
votes
0
answers
250
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Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
5
votes
1
answer
197
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Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
19
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2
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864
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A non-Abelian de Rham complex?
This question is inspired by this physics stack exchange post, which is recent and has not received an answer yet, nontheless I feel that there is a better way to ask this question here with a larger ...
7
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0
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257
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Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
6
votes
0
answers
388
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Yoneda product on Ext
Let $M$ be a closed manifold and consider the constant sheaf $\mathbb{R}_M$. I have heard that the Yoneda product on $\operatorname{Ext}(\mathbb{R}_M, \mathbb{R}_M)= H^*(M; \mathbb{R})$ coincides with ...
2
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1
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Non-trivial example of $H^2(G,M)$ where $M$ is a non-trivial G-representation
Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$.
Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism.
Let M be the $G$ representation $\mathbb{Z}...
2
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0
answers
135
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Definition of odd topological K-theory using circles
I wanted to check whether the following characterization of odd complex topological $K$-theory is correct (reposted from Math.SE).
Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined ...
23
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3
answers
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Next steps for a Morse theory enthusiast?
I don't know if this question is really appropriate for MO, but here goes: I quite like Morse theory and would like to know what further directions I can go in, but as a complete non-expert, I'm ...
1
vote
1
answer
263
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Cohomology with local coefficients in homotopy type theory
I was just reading Mike Shulman's blog post on how to define cohomology in homotopy type theory (HoTT), and I was curious if we can similarly define cohomology with local coefficients in HoTT as well?
...
1
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0
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57
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Sequence in local cohomology for multiple closed subsets
Let $X$ be topological space with closed subsets $A,B,C \subset X$ and $\mathcal{F} \in Sh(X)$.
I'm trying to understand
\begin{equation*}
H^i_{A\cap B}(X,\mathcal{F}) \oplus H^i_{A\cap C}(X,\mathcal{...
2
votes
0
answers
150
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Rigid \'etale cohohomology of flag variety minus its rational points e.g $p$-adic Drinfeld half plane
Let $Fl=G/B$ over $\mathbb Q_p$ be the flag variety of a quasi-split reductive group $G$ over $\mathbb Q_p$, then $X=Fl-Fl(Q_p)$ shall exist as a rigid analytic variety over $\mathbb Q_p$, how to ...
9
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An overcomplex of the Cartan model in equivariant cohomology?
Given any graded vector space $V^\bullet$ and any degree 1 linear operator $d\colon V^\bullet\to V^{\bullet+1}$, one gets a complex $(\ker(d^2),d)$. Moreover, if $V^\bullet$ is a graded algebra and $d$...
4
votes
1
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270
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A cochain complex using degeneracy maps
In constructing singular homology for a topological space, the boundary operator for the singular chain complex is given as an alternating sum of face maps. The degeneracy maps seem to be discarded in ...
5
votes
1
answer
356
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triviality of homology with local coefficients
Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ...
5
votes
1
answer
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Finding the right map between cohomology with local coefficients and Čech cohomology
Let $X$ be a space which is paracompact, Hausdorff, and sufficiently nice that it has a universal covering space (and map) $p:\tilde{X}\to X$. Also, let $\pi:=\pi_1(X)$ and $A$ some $\mathbb{Z}[\pi]$-...
4
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0
answers
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Have mod $p^k$ Dyer Lashof operations been studied?
Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...
3
votes
0
answers
240
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Evaluating the Euler class of a circle bundle on fibers
I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle.
This might be completely obvious, but I don't see how to answer the ...
6
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0
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339
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Cohomology without comonad?
TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be?
For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). ...
2
votes
1
answer
258
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can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C_m(\tilde X)$ ...
2
votes
0
answers
205
views
Are Milnor K-groups algebraic groups?
Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is,
$$
K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I,
$...
4
votes
1
answer
278
views
Conjugation action on relative homology
Let $G$ be a group and $K$ be a subgroup. Suppose $g \in G$ commutes with every element of $K$. Is it true that conjugation by $g$ will act trivially on $H_*(G,K)$?
6
votes
1
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318
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Comparing Hochschild (co)homology for algebras and coalgebras
Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...
4
votes
0
answers
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Continuity property for Čech cohomology
Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
4
votes
1
answer
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Homology with local systems
Let $X$ be a connected topological space with abelian fundamental group. Let $\mathcal{L}$ be a $\mathbb{Z}$-valued local system on $X$.
Suppose that I know the full homology $H_*(X;\mathbb{Z})$. Are ...
3
votes
1
answer
306
views
Is every middle exact functor a derived functor?
Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...
6
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0
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250
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If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?
This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general?
For instance, McMullen's polytope algebra is ...
4
votes
1
answer
249
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Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
3
votes
0
answers
186
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Comparing long exact sequences for local cohomology
Let $X$ be a topological space, $Z_1,Z_2 \subset X$ closed subsets and $\mathcal{F} \in Sh(X)$.
Then we have, for example by Hartshorne Excercise III 2.4, the Mayer Vietoris sequence for local ...
3
votes
1
answer
240
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Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
0
votes
1
answer
200
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Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$
Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$
I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
2
votes
0
answers
91
views
lie algebra bundle and underlying vector bundle
Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$.
As a vector bundle it is trivial, ...
8
votes
0
answers
337
views
Proving faithful flatness of a K-theoretic map without the moduli stack of formal groups
I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....
3
votes
0
answers
95
views
Cohomology of higher codimensional arrangements
Hyperplane arrangements are classical objects of study and there is a large literature on this subject, e.g. dealing with computing the cohomology of the complement. I am looking for similar results ...