Questions tagged [coherent-sheaves]

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Is there an approach to Gabber's theorem from the singular support of coherent sheaves?

David BZ told me that the old theory of singular support of $D$-modules fits into the new theory of singular support of coherent sheaves, via the derived loop space. I wonder how to reconcile that ...
David Treumann's user avatar
13 votes
0 answers
704 views

Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?

Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively. $D^b_{...
Zhaoting Wei's user avatar
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8 votes
0 answers
256 views

Direct summands of a pushforward in the derived category of coherent sheaves

For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$. Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object ...
naf's user avatar
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8 votes
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274 views

Def-Obs theory of sheaves with fixed determinant on CY3.

Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map $$ Ext^k(\...
Zheng's user avatar
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7 votes
0 answers
320 views

What is the category of coherent sheaves on a logarithmic scheme?

I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...
hennlu's user avatar
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6 votes
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276 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
  • 5,506
6 votes
0 answers
196 views

Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
Andrei Halanay's user avatar
5 votes
0 answers
138 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
5 votes
0 answers
245 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
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5 votes
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554 views

Torsion-free sheaves over nodal curves

Let $X$ be an irreducible nodal curve (over $\mathbb{C}$) with exactly one node, say at $x$. Let $F$ be a torsion-free, rank $n$ sheaf on $X$. We know that $F_x \cong \mathcal{O}_{X,x}^{\oplus n-a} \...
Ron's user avatar
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5 votes
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463 views

de Rham isomorphism with holomorphic forms

For a non-compact Riemann surface $X$ there is an isomorphism: $$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$ where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
user336494's user avatar
5 votes
0 answers
174 views

object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane. From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf which surveys Gieseker stable sheaves, I have understood that there are ...
user36931's user avatar
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4 votes
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278 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
4 votes
0 answers
131 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
4 votes
0 answers
87 views

Derived category supported in a Serre subcategory of a locally noetherian category

This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
mnm's user avatar
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4 votes
0 answers
157 views

Coherent sheaves and space filling curves

This paper constructs smooth space filling curves for smooth varieties over finite fields. Let's say we are working in char $p$ on the variety $X$ then this means that there is smooth curve $C_i$ in $...
user127776's user avatar
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4 votes
0 answers
135 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?

For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? ...
Mikhail Bondarko's user avatar
4 votes
0 answers
289 views

Which derived categories of coherent sheaves are equivalent (or "$t$-related") to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...
Mikhail Bondarko's user avatar
4 votes
0 answers
127 views

Cancellation property of vector bundles on non-proper varieties

Krull-Schmidt theorem for proper varieties over a field implies that given an isomorphism of vector bundles between $E\oplus F$ and $G\oplus F$ we can deduce that $E$ and $G$ are isomorphic. My ...
user127776's user avatar
  • 5,831
4 votes
0 answers
190 views

Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
Zhaoting Wei's user avatar
  • 8,657
4 votes
0 answers
479 views

Chern classes of torsion-free sheaves

Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by ...
user127776's user avatar
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4 votes
0 answers
234 views

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
Andrea's user avatar
  • 263
4 votes
0 answers
379 views

Question about Corollary II.5.18 in Hartshorne

Corollary II.5.18 in Hartshorne says that if X is a projective scheme over a Noetherian ring, then any coherent sheaf on X is a quotient of a finite direct sum of twisted structure sheaves. The ...
Sarah's user avatar
  • 49
4 votes
0 answers
339 views

Is the relative moduli space of semi-stable sheaves on families of curves fine

Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...
Ron's user avatar
  • 2,116
4 votes
0 answers
460 views

When is a coherent subsheaf determined by its global sections

I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections. The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
user24453's user avatar
  • 333
4 votes
0 answers
234 views

Proper base change for non-quasicoherent sheaves

For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$: ...
Jakob's user avatar
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4 votes
0 answers
171 views

When does a "universal" quot scheme exist?

Suppose $M$ is a moduli space of semistable sheaves on a projective variety $X$. Let $v$ be some the discrete invariants. I would like to form a space $\mathcal Q(v) \rightarrow M$, where the fiber ...
Drew's user avatar
  • 1,469
4 votes
0 answers
108 views

Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push forward....
operadical's user avatar
4 votes
0 answers
290 views

The support of a finite type module on an algebraic space

I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type. In here the schematic ...
Jacob Bell's user avatar
  • 1,275
3 votes
0 answers
107 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
3 votes
0 answers
89 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
388 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
3 votes
0 answers
216 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
3 votes
0 answers
227 views

Stability and simplicity of tangent sheaf of Grassmannian

Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $. (1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $? (2)...
Cranium Clamp's user avatar
3 votes
0 answers
148 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
Mikhail Bondarko's user avatar
3 votes
0 answers
199 views

When a reflexive sheaf is flat over its base

Let $X$ be a projective and smooth variety with a codimension 2 closed subvariety $Z$. Let $Y=X\times \mathbb{A}^2$ and $E$ a reflexive sheaf on $Y$ that is a vector bundle outside of $Z\times \mathbb{...
user127776's user avatar
  • 5,831
3 votes
0 answers
197 views

A question about extending vector bundles from formal neighborhood to a coherent sheaf

I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $...
user127776's user avatar
  • 5,831
3 votes
0 answers
116 views

Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
Andy Sanders's user avatar
  • 2,890
3 votes
0 answers
153 views

Bass theorem on non-affine scheme

A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free. Now, consider a connected noetherian scheme $...
prochet's user avatar
  • 3,432
3 votes
0 answers
586 views

Gluing for derived category of coherent sheaves

Let $X$ be a scheme and assume $X=U \cup V$ for two affine schemes $U_0$ and $U_1$. If $\mathcal F'$ and $\mathcal F''$ are some (coherent) sheaves on $U$ and $V$ respectively such that $\mathcal F'|_{...
Hang's user avatar
  • 2,719
3 votes
0 answers
151 views

Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$. I would like to show (at least when $X$ is a surface) ...
Andrea's user avatar
  • 263
3 votes
0 answers
224 views

Obstruction to lifting coherent sheaves on discrete valuation ring

Let $R$ be a discrete valuation ring with algebraically closed residue field $k$. Let $K:=\mathrm{Frac}(R)$ the fraction field of $R$. Suppose $K$ is of characteristic zero. Denote by $\overline{K}$ ...
Chen's user avatar
  • 1,583
3 votes
0 answers
275 views

Examples of varieties with every stable sheaf simple

Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some ...
user43198's user avatar
  • 1,949
3 votes
0 answers
126 views

Minimum number of generators for a gloablly generated sheaf over a curve

Let $X$ be a smooth, projective curve over an algebraically closed field and $E$ be a globally generated locally free sheaf of rank $r$. Is it always possible to write $E$ as the quotient of $r+1$ ...
user45397's user avatar
  • 2,195
3 votes
0 answers
717 views

G-equivariant coherent sheaves and their quotients

Let $X$ be a smooth complex quasiprojective variety and let $G$ be a finite group acting on $X$. The quotient $X/G$ is then a variety. Let $\mathcal{F}$ be a $G$-equivariant coherent sheaf on $X$. ...
Sarah's user avatar
  • 31
3 votes
0 answers
279 views

Direct limit of coherent sheaves and semi-stability

Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective scheme,...
Ron's user avatar
  • 2,116
3 votes
0 answers
188 views

K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?

For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the "...
Mikhail Bondarko's user avatar
3 votes
0 answers
490 views

Can Kuranishi families glue together to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a K3 surface through a theorem that asserts the existence of a ...
User3773's user avatar
  • 401
3 votes
0 answers
130 views

K-theory of ringed spaces (including henselian and formal schemes); excision and Mayer-Vietoris

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...
Mikhail Bondarko's user avatar
3 votes
0 answers
503 views

Global sections of a coherent sheaf in terms of a presentation

Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an ...
Martin Brandenburg's user avatar