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      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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      Use of Steenrod's higher cup product and the graded-commutativity

      In Steenrod's ``Products of Cocycles and Extensions of Mappings (1947),'' which derives [Theorem 5.1] $$ \delta(u\cup_{i} v)=(-1)^{p+q-i}u\cup_{i-1}v+(-1)^{pq+p+q}v\cup_{i-1}u+\delta u\cup_{i}v+(...
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      Non-abelian cohomologies

      Let A be a non-commutative algebra and let X be some geometric space (such as a topological space or an algebraic variety or scheme). Is there a notion of cohomology ring of X with coefficients in A? ...
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      Applications of one of Serre's Theorems

      This theorem is due to Serre: Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no element of order $p$ and let $H \leq G$ be an open subgroup. Then $cd_p(G) = cd_p(H)$. Where $cd_p(...
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      Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$

      I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...
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      Atiyah class and coboundary map

      Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $ be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...
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      133 views

      Generalized cohomology of CW complex is direct limit?

      Let $E$ be a (pre)spectrum (in the most classical sense, i. e. the sequence of CW complexes $E_n$ and maps $SE_n \to E_{n+1}$). Then we have the generalized cohomology theory $E^*$. For finite CW ...
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      On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

      In The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513 Woodward proposed a classification of $\mathrm{PU}...
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      Lie groupoid cohomology

      Given a Lie grouopid $\mathcal{G}=(\mathcal{G}_1\rightrightarrows \mathcal{G}_0)$ I am trying to understand what should be "the groupoid cohomology of this Lie groupoid $\mathcal{G}$". There are some ...
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      non zero differential in a spectral sequence

      This is the situation: Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
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      A question about Poincare duality

      Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}...
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      Second Bounded Cohomology of a Group: Interpretations

      Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...
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      About Kan-Thurston theorem

      The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...
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      118 views

      Mixed Hodge Polynomial for Algebraic Stacks

      Let $X$ be a complex algebraic variety. The numerical invariants associated with the Mixed Hodge Structure of $X$ can be encoded in a polynomial in three variables called the mixed Hodge polynomial $H(...
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      Cohomology of a chain complex over a polynomial ring

      I asked this on SE but I did not get any answer; I got some progress but I hope here I can find some help to finish the problem out. Let $R = F[x_1, \ldots, x_n]$ be a polynomial ring over a field $...
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      Iwasawa theory and cohomological $p$-dimension of Inertia

      Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...

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