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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 ...
Tim's user avatar
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3 votes
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116 views

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 ...
Qwert Otto's user avatar
7 votes
1 answer
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 ...
Tim's user avatar
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7 votes
1 answer
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&...
Tim's user avatar
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5 votes
1 answer
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 ...
Fernando Peña Vázquez's user avatar
5 votes
0 answers
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 $\mathscr{X}$ ...
Hajime_Saito's user avatar
1 vote
0 answers
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 ...
Mishkaat's user avatar
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1 vote
0 answers
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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 ...
maxo's user avatar
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1 vote
0 answers
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 ...
Thomas Kurbach's user avatar
4 votes
1 answer
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 ...
Walterfield's user avatar
2 votes
1 answer
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 ...
Pulcinella's user avatar
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2 votes
1 answer
229 views

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 ...
Mikhail Bondarko's user avatar
2 votes
0 answers
293 views

The definition of the determinant of a coherent sheaf

Let $ X $ be a smooth (projective) variety and $ \mathcal{F} $ a torsion-free coherent sheaf of rank $ r $ on $ X $. The determinant $ \det \mathcal{F} $ can be defined by (1) $ \det \mathcal F := ( \...
Cranium Clamp's user avatar
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0 answers
282 views

Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?

Given a scheme $X$ and sum of divisors $D$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...
Pulcinella's user avatar
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2 votes
0 answers
49 views

Pullback of coherent sheaves on Stein manifolds

Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
Doug Liu's user avatar
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1 vote
0 answers
57 views

What is the semistable reduction for sheaves?

Let $\Bbbk$ be an algebraically closed field with characteristic zero. Let $X$ be a projective scheme over $\Bbbk$ and let $L$ be an ample invertible $\mathcal{O}_X$-module. Fix a Hilbert polynomial $...
Display Name's user avatar
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3 votes
0 answers
116 views

Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety

Let $X$ be a smooth projective variety over a field $k$ of characteristic 0, and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$ where $I$ is an ideal such that $I^2=0$. Let $\frak X$ be a ...
Mikhail Borovoi's user avatar
1 vote
0 answers
85 views

Slope-stability, tilt-stability, and Bridgeland stability

Following the definition of slope-stability ($\mu$-stability) and tilt-stability ($\nu$-stability) on page 8 of https://arxiv.org/abs/1410.1585, does an object's tilt-stability imply its slope-...
Ying's user avatar
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2 votes
0 answers
121 views

Canonical basis and perverse coherent sheaves on the nilpotent cone

In the paper of Ostrik, he introduced a canonical basis of $K^{G\times {\mathbb C}^*}(\mathcal N)$, where $\mathcal N$ is the nilpotent cone for the group $G$. Question: does this canonical basis ...
Yellow Pig's user avatar
  • 2,540
2 votes
0 answers
108 views

Unicity of modifications of vector bundle on a regular base

I think that I have overheard the following statement, and would be grateful for either a reference or an explanation about why hearing must be slightly faulty and a clarification about what must ...
Stefan  Dawydiak's user avatar
3 votes
1 answer
125 views

Finer classification of semistable sheaves

Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder ...
Display Name's user avatar
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4 votes
1 answer
299 views

Derived pushforward of a projection

Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of ...
Abel 's user avatar
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4 votes
0 answers
289 views

Are $\mathcal{O}_X$-modules "more actual" then quasicoherent sheaves in some sense?

In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is ...
Mikhail Bondarko's user avatar
9 votes
1 answer
903 views

Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful? If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full ...
Mikhail Bondarko's user avatar
1 vote
1 answer
191 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 ...
Hamed Khalilian's user avatar
1 vote
1 answer
119 views

Families of torsion-free sheaves whose length jumps

For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}_{c_1,c_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its ...
Pulcinella's user avatar
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2 votes
1 answer
294 views

Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797 Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...
angry_math_person's user avatar
1 vote
0 answers
66 views

Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on TCP^1

My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves $$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$ I want to ...
AlgGeoNoob's user avatar
4 votes
2 answers
620 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 $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
rfauffar's user avatar
  • 653
2 votes
1 answer
274 views

Resolving complexes of coherent analytic sheaves

Background Throughout, let $X$ be a smooth complex manifold. It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...
Tim's user avatar
  • 1,227
2 votes
1 answer
140 views

Could certain closed covering determine a coherent sheaf?

We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started ...
user avatar
3 votes
0 answers
91 views

Explicit example of wall-crossing for sheaves

I would like to see an explicit example of a coherent sheaf $\mathcal{E}$ on a projective complex threefold $X$ which crosses the wall of stability. That is, I would like some $1$-parameter family $\...
Quaere Verum's user avatar
3 votes
0 answers
93 views

Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
Invariance's user avatar
1 vote
0 answers
91 views

Chern class of rank one sheaves supported on subvarieties

Let $X$ be a smooth, quasi-projective variety of dimension $n$ and $\mathcal{F}$ be a globally generated coherent sheaf supported on a codimension two subvariety $V \subset X$. Is $c_2(\mathcal{F}) \...
Chen's user avatar
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9 votes
1 answer
314 views

Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?

On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same ...
Jason Starr's user avatar
  • 4,101
0 votes
0 answers
197 views

Sheaves of abelian groups over a smooth projective variety

Can someone point some good reference (books or lecture notes) for these topics: Let $X$ a smooth projective variety over an algebraically closed field Sheaves of abelian groups over $X$ Quasi-...
Abel 's user avatar
  • 61
3 votes
1 answer
318 views

Existence of rigid objects in the derived category of a smooth projective variety

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a ...
Libli's user avatar
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0 votes
1 answer
240 views

Stability of sheaves of non-constant rank

Let $E\to X$ be a coherent sheaf over a compact (projective) Kahler manifold. The definition of stability of sheaves as stated in Huybrechts-Lehn (Definition 1.2.12) says that $E$ is stable if for all ...
BinAcker's user avatar
  • 767
3 votes
0 answers
433 views

Does a torsion-free coherent sheaf embed into a locally free sheaf?

Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
Cranium Clamp's user avatar
2 votes
1 answer
239 views

Understanding spaces is the same as understanding (sheaves of) functions on the space

I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written: [...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and understanding vector bundles (...
Abel 's user avatar
  • 61
3 votes
1 answer
193 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 ...
Neil Epstein's user avatar
  • 1,752
3 votes
1 answer
279 views

Can we recover the sheaf from the functor?

Let $k$ be an algebraically closed field of characteristic zero. Let $S$ be a scheme of finite type over $k$. Let $\mathrm{Sch}/S$ be the category of schemes of finite type over $S$. Let $\mathcal F$ ...
Display Name's user avatar
  • 1,000
4 votes
0 answers
132 views

Connectedness of moduli spaces of semistable sheaves on K3

In 1987/88, Mukai described the moduli spaces of (semi-)stable sheaves on a K3 surface $X$, showing that they consist of smooth pieces $M(v)$ of dimension $\langle v , v \rangle + 2$, for every ...
Urs Schreiber's user avatar
1 vote
0 answers
174 views

Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...
BinAcker's user avatar
  • 767
3 votes
0 answers
233 views

For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

This question has been asked in SE. 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 ...
Display Name's user avatar
  • 1,000
4 votes
1 answer
195 views

Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$ be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...
KKD's user avatar
  • 463
2 votes
0 answers
246 views

Projectivization of a coherent sheaf using resolution by vector bundles

Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles. Is there a way to ...
BinAcker's user avatar
  • 767
3 votes
1 answer
265 views

When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?

This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question. Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
PrimeRibeyeDeal's user avatar
3 votes
1 answer
266 views

Is local freeness open for curves?

Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (...
nolatos's user avatar
  • 151
5 votes
1 answer
361 views

Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...
Mikhail Bondarko's user avatar

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