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existence of a coherent sheaf

I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
KAK's user avatar
  • 613
3 votes
1 answer
240 views

Cohomology of the complement of a subvariety

Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map $$ H^i(X,\mathbb Q)\to H^i(U,\mathbb Q) $$ is an ...
fgh's user avatar
  • 178
8 votes
0 answers
644 views

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 ...
MChocko's user avatar
  • 69
1 vote
0 answers
141 views

Homeomorphic endomorphism of schemes inducing equivalence of sheaves

Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
user267839's user avatar
  • 6,038
0 votes
0 answers
116 views

How can I calculate $\chi(\mathscr{O}(P))$

Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set \begin{gather*} \overline{X}=L \otimes X. \end{gather*} Assume $\overline{X}$ also irreducible. Now, ...
MChocko's user avatar
  • 69
1 vote
0 answers
126 views

Local freeness of dualizing sheaf

I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following. When does the dualizing sheaf of a projective scheme is an locally ...
KAK's user avatar
  • 613
0 votes
0 answers
57 views

Lifting of quadrics containing hyperplane section for projectively normal curves

Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
user267839's user avatar
  • 6,038
16 votes
1 answer
448 views

Zorn's lemma for Grothendieck sites

In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
cat man's user avatar
  • 163
62 votes
8 answers
14k views

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 ...
user avatar
6 votes
1 answer
441 views

Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
Jakob Werner's user avatar
  • 1,153
6 votes
2 answers
631 views

Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$

Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$. I ...
gigi's user avatar
  • 1,343
0 votes
0 answers
247 views

Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off? Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
locally trivial's user avatar
1 vote
0 answers
355 views

Global section of pullback of an ideal sheaf

For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
Takagi Benseki's user avatar
1 vote
0 answers
215 views

Artin-Winters proof of semi-stable reduction theorem: details

I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail— Let $\...
BelowAverageIntelligence's user avatar
4 votes
2 answers
1k views

Sheaf cohomology commutes with colimits of sheaves

Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...
user267839's user avatar
  • 6,038
0 votes
0 answers
91 views

Why does the associated sheaf vanish?

I am learning local cohomology from Hartshorne’s book Local Cohomology. My question is about understanding a line in the proof of proposition 1.11 in this book. The set-up for proposition 1.11 is that ...
Boris's user avatar
  • 639
3 votes
1 answer
331 views

Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism

If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
The Thin Whistler's user avatar
3 votes
1 answer
258 views

Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263. Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
The Thin Whistler's user avatar
1 vote
1 answer
2k views

Pullback map on global sections surjective

Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism! Let $\mathcal{L}$ ...
user267839's user avatar
  • 6,038
10 votes
0 answers
958 views

intuition about perverse sheaves

firstly, I would know if my very basic intuition on perverse sheaves is correct . secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves . my intuition ...
Amos Kaminski's user avatar
6 votes
1 answer
761 views

The Yoneda pairing, hypercohomology, and cup product

Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
Svinto's user avatar
  • 294
4 votes
1 answer
289 views

Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3) and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON $\...
user267839's user avatar
  • 6,038
2 votes
1 answer
1k views

Cohomology of tangent sheaf of a hypersurface

Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
user avatar
1 vote
0 answers
103 views

$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. ...
XT Chen's user avatar
  • 1,168
6 votes
1 answer
728 views

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 ...
Lilolance's user avatar
  • 233
5 votes
0 answers
268 views

Coherent cohomological dimension and affine morphisms

For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$. The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
Linda's user avatar
  • 59
2 votes
1 answer
460 views

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{...
user267839's user avatar
  • 6,038
2 votes
0 answers
139 views

Defineing a Sheaf of rings over a topological space

Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
A. R. Magid's user avatar
8 votes
2 answers
684 views

Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?

Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\...
Daniel Pomerleano's user avatar
1 vote
0 answers
104 views

A sheaf for factorization

Let $R$ be a commutative ring with $1$ and let $X$ be the space of connected componens of $Spec (R) $ with Zariski topology ( The boolean spectrum of $R $ )and let for each $x\in X$ there exists a ...
Ros...'s user avatar
  • 11
3 votes
2 answers
488 views

Application of sheaves theory in ring theory

Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
Alex's user avatar
  • 49
7 votes
0 answers
574 views

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). ...
ಠ_ಠ's user avatar
  • 6,025
6 votes
1 answer
752 views

Sections of the conormal bundle

Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$. Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user avatar
3 votes
1 answer
3k views

Cohomology of tangent bundles

Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up $$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$ of $X$ along $Z$. What is the relation between the cohomology of the ...
Puzzled's user avatar
  • 8,998
4 votes
1 answer
601 views

Explicit examples presheaves associated to higher direct images which fail to be sheaves

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$...
Hugo Chapdelaine's user avatar
4 votes
1 answer
409 views

Does the nearby cycle functor commute with the Verdier duality?

I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
asv's user avatar
  • 21.8k
1 vote
2 answers
276 views

Sections of a sheaf of differentials on a weighted complete intersection

Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$. Is it true that if $q\geq 1$ then $H^0(X,\...
user avatar
3 votes
1 answer
515 views

For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?

Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$. Now we consider a similar ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
0 answers
126 views

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 ...
Avi Steiner's user avatar
  • 3,079
5 votes
2 answers
331 views

Sheaf cohomology on non paracompact topological spaces

I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me. My reference is Godement's book "Topologie algebrique et theorie dex faisceaux"....
User28341's user avatar
  • 609
3 votes
1 answer
159 views

Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?

Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules: $$ \cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
Timothy's user avatar
  • 355
6 votes
2 answers
385 views

cohomology and $j_!$

I have a projective variety $X$ and an open immersion $j : U \to X$. Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,...
Andy B's user avatar
  • 758
1 vote
0 answers
236 views

Canonicity of Čech cohomology

For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$. For a sheaf $F$ on $X,$ the cohomology $H^...
Pierre MATSUMI's user avatar
0 votes
0 answers
191 views

First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$

Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{...
user avatar
1 vote
3 answers
845 views

Higher cohomology of sheaves on a projective space

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf $$\mathcal{F}_k:=\mathcal{O}_{\mathbb{P}^n}(kd)\...
user avatar
4 votes
1 answer
263 views

Relating deformations of a scheme to deformations of its singular locus

Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
Puzzled's user avatar
  • 8,998
0 votes
0 answers
239 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$...
User3773's user avatar
  • 401
2 votes
1 answer
293 views

global sections of higher direct images of étale sheaves

Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?
user avatar
0 votes
1 answer
382 views

The behavior of pure sheaves under functor Hom( F, -)

We know that a submodule A of B is pure if and only if the functor $Hom(M, -)$ is exact on the sequence $ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ for every finitely presented module ...
Gholam's user avatar
  • 55
5 votes
0 answers
374 views

Sheaf Cohomology on Zariski-Riemann Spaces

Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
Jizhan Hong's user avatar