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, ...

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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_*\...
user45397's user avatar
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2 votes
1 answer
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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 ...
EdRich's user avatar
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10 votes
1 answer
907 views

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 ...
Tim Campion's user avatar
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10 votes
1 answer
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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 ...
kindasorta's user avatar
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16 votes
1 answer
575 views

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 ...
Tim Campion's user avatar
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4 votes
1 answer
134 views

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 $\...
Luigi M's user avatar
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5 votes
1 answer
401 views

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, ...
ಠ_ಠ's user avatar
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6 votes
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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. ...
Nikolai Opdan's user avatar
3 votes
0 answers
181 views

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 ...
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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 ...
Student's user avatar
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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 ...
Tommaso Rossi's user avatar
4 votes
0 answers
219 views

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}^...
FKranhold's user avatar
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6 votes
2 answers
308 views

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) ...
Patrick Elliott's user avatar
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264 views

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 ...
user127776's user avatar
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5 votes
2 answers
703 views

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 ...
iou's user avatar
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1 answer
190 views

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},...
Mr. Palomar's user avatar
5 votes
1 answer
264 views

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 ...
Mikhail Borovoi's user avatar
2 votes
0 answers
250 views

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 ...
Student's user avatar
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5 votes
1 answer
197 views

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 ...
EJAS's user avatar
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19 votes
2 answers
864 views

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 ...
Bence Racskó's user avatar
7 votes
0 answers
257 views

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 ...
qqqqqqw's user avatar
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0 answers
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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 ...
user155668's user avatar
2 votes
1 answer
353 views

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}...
A_Physicist.'s user avatar
2 votes
0 answers
135 views

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 ...
geometricK's user avatar
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23 votes
3 answers
2k views

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 views

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? ...
ಠ_ಠ's user avatar
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1 vote
0 answers
57 views

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{...
KKD's user avatar
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2 votes
0 answers
150 views

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 ...
sawdada's user avatar
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9 votes
0 answers
122 views

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 $d$...
domenico fiorenza's user avatar
4 votes
1 answer
270 views

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 ...
Mathemologist's user avatar
5 votes
1 answer
356 views

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 ...
Shiquan Ren's user avatar
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5 votes
1 answer
384 views

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 $\mathbb{Z}[\pi]$-...
Xindaris's user avatar
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4 votes
0 answers
152 views

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 ...
elidiot's user avatar
  • 283
3 votes
0 answers
240 views

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 ...
BrianT's user avatar
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6 votes
0 answers
339 views

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). ...
Student's user avatar
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2 votes
1 answer
258 views

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 $C_m(\tilde X)$ ...
Shiquan Ren's user avatar
  • 1,970
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, $...
M masa's user avatar
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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 $H_*(G,K)$?
qqqqqqw's user avatar
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6 votes
1 answer
318 views

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 $$\...
Aidan's user avatar
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4 votes
0 answers
311 views

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(\...
Xindaris's user avatar
  • 255
4 votes
1 answer
452 views

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 ...
user155668's user avatar
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 ...
Adi Ostrov's user avatar
6 votes
0 answers
250 views

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 ...
Geva Yashfe's user avatar
  • 1,356
4 votes
1 answer
249 views

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\...
stupid_question_bot's user avatar
3 votes
0 answers
186 views

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 ...
KKD's user avatar
  • 463
3 votes
1 answer
240 views

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 \...
LeechLattice's user avatar
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0 votes
1 answer
200 views

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 ...
user90041's user avatar
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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, ...
prochet's user avatar
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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....
Michael Klyachman's user avatar
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 ...
Najib Idrissi's user avatar

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