Questions tagged [coherent-cohomology]
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11 questions
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Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
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Can one prove vanishing of higher direct images fiber-wise?
Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?
The derived direct image of $O_X$ is $O_Y$.
...
- 2,639
6
votes
1
answer
911
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English translation of Borel-Serre, Le théorème de Riemann-Roch?
Would be happy to receive a translation in English of Borel and Serre's Le théorème de Riemann-Roch, Bulletin de la Société Mathématique de France, Tome 86 (1958) pp. 97-136, doi:10.24033/bsmf.1500, ...
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Steenrod operations in algebraic geometry
What are some applications of Steenrod operations (or similar constructions) in algebraic geometry?
I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be ...
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6
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Is $h^{0,k}$ a topological invariant?
Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ such that $X(\mathbb{C})$ is homeomorphic to $Y(\mathbb{C})$. Is it true that $\dim_{\mathbb{C}} H^k(X,\mathcal{O}_X)=\dim_{\mathbb{...
- 796
4
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1
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Reference request: category of sheaves of O-modules with coherent cohomology
Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-...
- 7,952
3
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1
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393
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Cohomological base change
$\require{AMScd}$
Consider the Cartesian diagram of Noetherian schemes and commutative rings $R$, $R'$:
\begin{CD}
N' @>{h'}>> N\\@VV{g'}V @VVgV\\ M' @>h>> M \\ @VVV @VVV \\ \mathbf{...
- 349
3
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Is it possible to use the Cech complex to compute coherent cohomology in practice?
Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute the coherent cohomology of $V$ with coefficients in a line ...
- 2,964
2
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0
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141
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Computing the coherent cohomology of a quasiprojective variety
I have a quasiprojective variety given by some explicit quations. How do I compute its coherent cohomology (with coefficients in the structure sheaf)? Do I use the Cech complex for an open affine ...
- 2,964
1
vote
1
answer
320
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Criteria for acyclicity
Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally,
a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is,
for the property $H^i(X,E)=0$ ...
- 2,639
1
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0
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Quasicoherent analogue of a theorem on fiberwise acyclicity for etale cohomology
I am interested in knowing what (if any) is the quasicoherent analogue of the following result that I have paraphrased from SGA 4, exposé xv, Théorème 1.15:
Let $g \colon X \to ...
- 7,273