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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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        0answers
        133 views

        What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?

        Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line. What is the ...
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        205 views

        $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

        First of all I want to apologize for the much too long post. A Lie group $G$ is acting on a smooth manifold $M$, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
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        1answer
        239 views

        Differential forms of a Lie group giving cohomology of the Lie group

        Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{...
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        80 views

        cohomology of nilpotent matrices of fixed $m$-th power

        Let $k$ be an algebraically closed field, $\mathcal{N}$ is the variety of $n \times n$ nilpotent matrices over $k$, and consider the natural $m$-power map $\mathcal{N} \rightarrow \mathcal{N}$ given ...
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        1answer
        173 views

        De Rham cohomology of Lie groupoid

        Let $G$ be a Lie group acting on a manifold $M$. Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...
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        180 views

        A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

        Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
        5
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        1answer
        131 views

        Deformations of Vertex Algebras

        As the title suggests, I'm interested in deformation theory of vertex algebras and their representations. In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
        7
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        1answer
        366 views

        Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product

        Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$. ...
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        25 views

        An exact complex of tensor families

        Given a field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, some natural numbers $q,n,m$ with $q \leq n$ and $m < n$ and a polynomial algebra $S := \mathbb{K}[\eta_1, \dots, \eta_n]$. We can consider $...
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        510 views

        Conjectures of Peter Scholze about q-de Rham complex: examples

        Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
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        2answers
        575 views

        Homology of the universal cover

        $k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous ...
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        187 views

        A cohomology associated to a vector field on a Riemannian manifold

        Edit: Accoring to the comment of Asura Path I revise the question. Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
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        145 views

        Comparing cohomology using homotopy fibre

        I have a question, which might be very basic, but I don't know enough topology to answer. Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by ...
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        126 views

        Cokernel of section of a general coherent sheaf

        Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...
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        172 views

        Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$

        I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...

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