# Questions tagged [coherent-sheaves]

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space.

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### How can I prove this interpretation of right derived functor of the composition of internal hom followed by direct image?

The question comes from the following paper Lange, Herbert, Universal families of extensions, J. Algebra 83, 101-112 (1983). ZBL0518.14008. Let $f:X\to Y$ be a flat projective morphism of Noetherian ...
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### “Usual argument for a bicomplex” in order to extend a cocycle

I was reading this article and I got stuck in the proof of Proposition 2.1. In particular I don't get the last paragraph of the proof when it talks about extending to a cocycle and using a similar ...
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### Integral closure of the ideal sheaf is coherent

Demailly asserts on page 9 of Analytical Methods in Algebraic Geometry that the integral closure of an ideal sheaf is coherent. I don't know how to prove this. Is there any reference? We introduce ...
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### Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$ for any $Z$ is a consensus which I agree with, but then regarding the hilbert ...
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### For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent sheaves on $X$. Suppose we have a family ...
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### A condition for the validity of an identity in the derived category of bounded complexes of modules.

Let $\mathcal{R}$ be a sheaf of (not necessarily commutative) rings on a topological space $X$ and let $M$ and $N$ be $\mathcal{R}-$modules such that $M$ is quasi-isomorphic to a finite complex of ...
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### What is the minimal number of generators for $\mathcal{O}_{\mathbb{P}^n}(1)$?

With generators for $\mathcal{O}_{\mathbb{P}^n}(1)$ I mean a set $a_0,...,a_m\in \mathcal{O}_{\mathbb{P}^n}(1)(\mathbb{P}^n)$ such that $a_0,...,a_m$ generate $\mathcal{O}_{\mathbb{P}^n}(1)_P$ at all ...
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### Explicit correpsondence between ${\cal O}_{{\Bbb P}_k^1}(1) = {\cal O}_{{\Bbb P}_k^1}(x)$.

For a projective curve ${\Bbb P}_k^1$, we have a very ample line bundle ${\cal O}_{{\Bbb P}_k^1}(1)$. Geometrically, I prefer to consider ${\cal O}_{{\Bbb P}_k^1}(1)$ as the fibration \begin{equation*}...
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### Linear fiber space of ideal sheaf and normal fiber space

Suppose that $\left(M, \mathcal{O}\right)$ is an analytic space and $A$ is a subspace with defining coherent ideal sheaf $J$. One can construct from any coherent sheaf $\mathcal{F}$ on $M$ the linear ...
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### Chern character of a coherent sheaf over a projective manifold [closed]

I am looking for a reference describing the definition/construction of the Chern character of a coherent sheaf over a projective manifold (or variety) using a resolution by vector bundle.
My understanding of Serre's twisting sheaves is quite shaky. If $B = \bigoplus_{d\geq 0}B_d$ is a graded module, $X = \operatorname{Proj}B$, and $f\in B_1$, then I know that \$\mathcal{O}_X(n)|_{D_+(f)}...