All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
1
vote
1
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271
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Second fundamental exact sequence of sheaves of differentials. Sufficient condition to get a splitting s.e.s
$\def\spec{\operatorname{Spec}}$I am trying to understand the proof of Lemma 0474 of the Stacks Project. I'll give some context to its statement before discussing its proof: In commutative algebra, if ...
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 12.9k
5
votes
0
answers
220
views
Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
- 12.9k
3
votes
0
answers
215
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How to read the definition of Grothendieck Pretopology in SGA4?
In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$
sont quarrables. (Rappelons qu’un morphisme ...
- 237
7
votes
2
answers
1k
views
Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
- 6,367
2
votes
0
answers
126
views
Compact generators of $\mathcal{D}\text{-Mod}$ via Mayer Vietoris
Let $X$ be a complex variety (or any space for whom a category of sheaves with the six functors is defined), and
$$i\ :\ Z\ \to\ X\ \leftarrow\ U\ :\ j$$
be complementary open and closed embeddings.
...
- 5,711
1
vote
0
answers
91
views
Quotient sheaf obtained from a quotient of $\mathrm{SL}_2$ having a section
$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $...
- 63.9k
8
votes
0
answers
333
views
Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
- 4,713
2
votes
0
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239
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Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 721
5
votes
1
answer
512
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Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 721
1
vote
1
answer
251
views
Some question about (semi-)stable sheaves
Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:
Question 1. Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan ...
- 26
2
votes
0
answers
100
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Global sections of relative characteristic of log-smooth curves
$\DeclareMathOperator\Spec{Spec}$I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am ...
- 223
11
votes
2
answers
1k
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Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
-1
votes
1
answer
177
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When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 38k
10
votes
2
answers
2k
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Geometric interpretation of sheaf cohomology
Please forgive me for the informal and naïve nature of my question, as I am a beginner in algebraic geometry.
In the famous book by Hartshorne, sheaf cohomology is defined as a certain derived functor....
- 5,857
1
vote
1
answer
226
views
flatness of restriction of structure sheaf over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
- 4,111
2
votes
0
answers
175
views
Is this double quotient of $\operatorname{SL}_2$ representable by an algebraic space or a scheme?
$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ ...
- 12.9k
7
votes
1
answer
255
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Subobject classifier for sheaves on large sites with WISC
Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...
- 15.9k
1
vote
0
answers
79
views
What is $\text{Hom}(\mathcal{F}\times\mathcal{G}, \mathbb{A}^1)$?
Let $S$ be an affine scheme and let $\text{Aff}(S)$ be the site of affine $S$-schemes. Let $\mathcal{F}$ and $\mathcal{G}$ be a pair of sheaves on $\text{Aff}(S)$, and let $\mathbb{A}^1_S$ be the ...
- 2,907
3
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1
answer
284
views
Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?
Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\...
- 156k
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 ...
- 639
1
vote
0
answers
137
views
The stack $\operatorname{GL}_2/B$
Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...
- 12.9k
3
votes
1
answer
345
views
Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
- 39.3k
2
votes
0
answers
122
views
Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$
Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
- 473
4
votes
3
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483
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"Quasi-coherent" vector spaces in Sch/S
$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...
- 63.1k
2
votes
1
answer
218
views
About the support of a holonomic D-module
Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...
- 6,829
1
vote
0
answers
82
views
Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?
I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
- 1,969
6
votes
1
answer
455
views
Subsheaves of Spec K, K a field
$\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\...
- 15.9k
3
votes
1
answer
147
views
What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?
Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
- 39.3k
4
votes
2
answers
642
views
Basic question on projective bundles
Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
- 28.7k
2
votes
0
answers
111
views
Canonicity in split sequence in cotangent spaces
Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...
- 63.1k
3
votes
0
answers
154
views
When the sheaf of principal parts is reflexive?
Following EGA IV, we can construct the sheaf of principal parts of a sheaf $\mathcal{E}$ over a scheme $X$ by $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{P}_X^n \otimes \mathcal{E}$, where $\otimes$ is ...
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
57
votes
3
answers
10k
views
What are the benefits of viewing a sheaf from the "espace étalé" perspective?
I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. ...
- 12.9k
4
votes
2
answers
228
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Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?
Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
- 12.9k
5
votes
1
answer
443
views
Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?
The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
- 149k
1
vote
0
answers
106
views
Joins of (closed) subschemes and Zariski-local Z-functors
$\newcommand\Aff{\mathrm{Aff}}\newcommand\cRing{\mathrm{cRing}}\newcommand\Sch{\mathrm{Sch}}$Equip $\Aff = \cRing^\text{op}$ with the Zariski Grothendieck-topology. There are nested categories:
$$\Aff\...
- 12.9k
0
votes
0
answers
248
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 ...
- 511
7
votes
2
answers
458
views
A technical question about derivations of sheaves on group schemes
Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O_G$ the structural sheaf of $G$.
Let $D_e : O_{G,e} \to k$ a derivation.
I would ...
- 12.9k
1
vote
1
answer
244
views
Chern class of torsion sheaf support on a point
Let $X$ be a smooth projective surface. Let $p$ be a closed point of $X$. Let $k(p)$ be the corresponding skyscraper sheaf, then actually we could use Grothendieck-Riemann-Roch to calculate the Chern ...
- 5,901
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 ...
- 1,110
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, ...
- 37.8k
0
votes
0
answers
343
views
Stalks of Sheaves
I saw a statement in a paper like what follows:
Let $X=\text{Spec} A$ be an affine scheme, and let $\mathscr{F}$ be a sheaf of $\mathscr{O}_X$-modules on $X$. For each geometric point $x$ of $X$ we ...
- 1
6
votes
5
answers
4k
views
Why is the rank of a locally free sheaf well-defined?
In Hartshorne p. 109 he defines a sheaf $\mathcal{F}$ of $O_X$-modules to be locally free if there is an open cover of $X$, s.t. on each $U$, $\mathcal{F}|_U$ is a free $O_X|_U$ module of rank $I$. ...
- 1,446
2
votes
0
answers
114
views
Two natural morphisms of sheaves with the same source and target; do they agree?
Suppose we have a diagram
$\require{AMScd}$
\begin{CD}
A @>a>> B\\
@V b V V @VV c V\\
C @>>d> D @>e>> E \\
@VfVV @VVgV @VVhV \\
F @>>i> G @>>j> H
\end{CD}...
- 125
15
votes
2
answers
616
views
Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about ...
- 15.9k
1
vote
0
answers
226
views
Resolution of the pushforward of a vector bundle
Let $i:Z\hookrightarrow X$ be a subvariety of a compact Kahler manifold. Assume that $Z$ can be realize as the zero locus of a section $s$ of a holomorphic vector bundle $E\to X$ of rank $r$. The ...
- 789
2
votes
0
answers
168
views
Criteria for a sheaf to be locally free over subvariety
Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.
Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:...
- 789
5
votes
0
answers
269
views
Line bundle whose pushforward is a complex of vector bundles
If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...
- 789
3
votes
1
answer
201
views
Is this a true weakening of the quasi-coherence property?
Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition
(#) For all containments $V \subseteq ...
- 19.6k