# Questions tagged [coherent-sheaves]

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### Resolution property in rigid analytic geometry

I am not a rigid analytic geometrer, so I apologise if the question is trivial, but I can't find an answer anywhere myself. I'm trying to understand in what ways (rigid) analytic geometry compares to ...
• 1,227
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### Trace map on Ext group

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map $$\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.$$ According to the ...
• 617
575 views
+100

### Converses to Cartan's Theorem B

Here is a phrasing of some Cartan Theorem B statements: Consider the following conditions: $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
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408 views

### Smooth analogue of Cartan's Theorem B

Cartan's Theorem B can be stated as follows: Let $X$ be a space let $\mathcal{F}$ be a sheaf on $X$. Consider the following three conditions: $X$ is "simple"; $\mathcal{F}$ is "nice&...
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232 views

### On the bounded derived category of sheaves with coherent cohomology

Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
152 views

### Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
• 737
1 vote
35 views

### (Quasi)-coherence of the weight $\theta$-sheaf

In this paper, the author defined the weight $\theta$-sheaf as follows: Let $A^{k}_{X}$ be the sheaf germs of real smooth $k$ forms on the smooth manifold $X$, we perform a complexification on this ...
• 321
1 vote
67 views

### Trace map for universal bundle of Grassmannian

Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...
• 79
1 vote
104 views

### Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties

Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
616 views

### Coherent sheaves, Serre’s theorem and ext groups

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$. Then, the ...
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233 views

### Compatibility of Beck Chevalley condition: sheaves

Given a (not necessarily Cartesian) square of spaces $$\require{AMScd}\begin{CD} X @>g>> \overline{X} \\ @VVfV @VV\overline{f}V \\ Y @>\overline{g}>> \overline{Y} \end{CD}$$ does the ...
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### Is any "relative support" for (complexes of) quasi-coherent sheaves known?

Let $f:X\to S$ be a morphism of Noetherian schemes; in the case I am interested in $S=\operatorname{Spec}R$ is affine and $f$ is proper. For a complex $C$ a complex of quasi-coherent sheaves on $X$ I ...
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