Questions tagged [sheaf-cohomology]
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364 questions
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maps between two Leray spectral sequences based on maps on cochain complexes
Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
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Maps between Leray spectral sequences
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
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Leray spectral sequence for continuous functions on pairs of topological spaces
Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.
The Leray spectral sequence (with complex ...
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On the Leray spectral sequence and sheaf cohomology
I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
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exact sequence of fundamental groups associated to "almost" smooth families of curves
Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
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An algorithm to compute coherent sheaf cohomology in projective space over a ring [closed]
EDIT: As the article was put on hold, because it was unclear what I am asking, here I put again my two questions:
1) Is the argument I used to derive the algorithm valid?
The second question is a ...
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Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $...
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Independence of embedding for higher sheaf cohomology of local cohomology on projective space
Suppose $Y$ is a projective variety over a field $k$. Fix an embedding $\iota: Y \hookrightarrow \mathbb{P}^n_k$ for some $n$, and consider the local cohomology sheaves $\mathcal{H}^j_Y(\omega_{\...
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Canonical map in the direct image of $\mathscr{D}_X$
Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...
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Sheaf cohomology with support vanishes
I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...
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Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
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The cohomology of meromorphic functions
Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...
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Reference for the Koszul--Malgrange Theorem
The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
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Cohomology of sheaf of Schwartz distributions with support in a submanifold
Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...
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Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
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Cartier Divisor generated by Global Sections
Let $X$ be an integer curve of (arithmetic) genus $g=0$. (the arithmetic genus $g$ is defined by $g:= 1 -\chi_k(\mathcal{O}_X)$ where $\mathcal{O}_X$ is the structure sheaf of $X$ and $\chi_k(\mathcal{...
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Question on de Rham complex with distributional coefficients
Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
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Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?
Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$.
For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed ...
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What are the uses of coefficient systems for arithmetic cohomology theories?
In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...
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Defining the Euler class in different ways
Let $\pi: E\to M$ be a rank two real vector bundle over a manifold $M$. Bott and Tu defines the Euler class by:
giving $M$ a Riemannian structure,
taking a trivializing chart $U_\alpha$ of $M$,
...
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Theta divisor on compactified jacobian of nodal curve
Let $X$ be a Nodal curve. Let $\bar{J}(X)$ be compactified Jacobian (rank one torsion free sheaf of degree one) and $\Theta$ denote the theta divisor in $J$.
How to compute $H^0(\bar{J}(X);\Theta^k)$, ...
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Countability assumption for good covers in Bott-Tu
In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
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Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
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Can we use sheaf cohomology to say anything interesting for vector bundles with non-flat connections?
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...
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Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
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Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
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Is this Sequences of Complexes of Sheaves Exact?
So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ \underline{...
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Cohomology of boundary of locally symmetric space
Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...
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Cohomology of adelic locally symmetric spaces
I am most probably wrong in asserting as follows.
Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be ...
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$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$
Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$
Let $C = \mathbb{P}^1$.
By Kunneth $H^2(\...
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What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?
The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
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In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
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Generalities on sheaves - Where can I find the technology that can make this "proof" of Atiyah duality precise?
Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct).
Let $X$ ...
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Dolbeault cohomology of $\text{sl}(2,\mathbb{C})$
Consider the complex Lie group $G=\text{SL}(2,\mathbb{C})$ and let us denote $\Omega$ the sheaf of top holomorphic forms of this group. Are the cohomology spaces $H^{*}(G,\Omega)$ known ? I am ...
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de Rham isomorphism with holomorphic forms
For a non-compact Riemann surface $X$ there is an isomorphism:
$$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$
where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
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Different definition of sheaf cohomology
It could be related to my previous question here.
Let $\mathcal F$ be a sheaf on a topological space $X$. Hartshorne in his book on Algebraic geometry defines the sheaf cohomology by
$$
H^i(X, \...
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Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
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Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
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Exactness of $j_!$ in abelian category recollement
Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...
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An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology
Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$.
Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as ...
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The sheaf propagation is open in the zero section
Let $X$ a smooth manifold and $F$ a sheaf (let's say of abelian groups) on $X$. We will say that $F$ propagates at $x\in X$ in the (co)-direction $p \in T_x^*X$ if for all $C^1$-function $\phi$ ...
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Global section of line bundle on anti-canonical rational surface
Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\...
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Relative version of de Rham cohomology with local coefficients
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...
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Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
CommunityBot
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Sheaf Cohomology as Glueing of Local Data
For some time I've been trying to find an answer to the question "why do we care about or compute sheaf cohomology". As far as I can tell books like Hartshorne treat this as something we already want ...
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difference between the small and big étale/flat/... site
What is the difference between the small and the big étale (or flat or syntomic or ...) site? How does the cohomology vary? When should I use which one? Up to now, I have always used the small sites.
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When does the sheaf cohomology of a topological space vanish?
The question is in the title. A more precise formulation is:
Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?
The obvious example is a ...
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Algebraic groups without torsors
If $G$ is an algebraic group such that $H^1(S, G) = 0$ for all schemes $S$, must $G$ be the trivial group?
My original motivation for the question is the rationale I always give students for studying ...
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Cohomology of fiber bundles with non constant coefficients
Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the ...
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