Questions tagged [coherent-sheaves]
The coherent-sheaves tag has no usage guidance.
119
questions with no upvoted or accepted answers
17
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716
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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 ...
13
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0
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704
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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_{...
8
votes
0
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256
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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 ...
8
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274
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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
$$
Ext^k(\...
7
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320
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What is the category of coherent sheaves on a logarithmic scheme?
I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...
6
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276
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Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?
Given a scheme $X$ and sum of divisors $D$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...
6
votes
0
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196
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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-...
5
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138
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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 $\mathscr{X}$ ...
5
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245
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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 ...
5
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553
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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} \...
5
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463
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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 ...
5
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0
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174
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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 ...
4
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278
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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 ...
4
votes
0
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131
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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 ...
4
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87
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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 ...
4
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157
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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 $...
4
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0
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135
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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? ...
4
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289
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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-...
4
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0
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127
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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 ...
4
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190
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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 ...
4
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0
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479
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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 ...
4
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234
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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[...
4
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379
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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 ...
4
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0
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339
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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 ...
4
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460
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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 ...
4
votes
0
answers
234
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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 $i$:
...
4
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171
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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 ...
4
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108
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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....
4
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290
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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 ...
3
votes
0
answers
107
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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 ...
3
votes
0
answers
89
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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 $\...
3
votes
0
answers
387
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} \...
3
votes
0
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216
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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 ...
3
votes
0
answers
227
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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)...
3
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0
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148
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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 ...
3
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0
answers
199
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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{...
3
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0
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197
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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 $...
3
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0
answers
116
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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 ...
3
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0
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153
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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 $...
3
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0
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586
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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'|_{...
3
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0
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151
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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) ...
3
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0
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224
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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 $\overline{K}$ ...
3
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0
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275
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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 ...
3
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0
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126
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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 $r+1$ ...
3
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716
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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 $X$.
...
3
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0
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278
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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,...
3
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188
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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 "...
3
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0
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490
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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 ...
3
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0
answers
130
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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. ...
3
votes
0
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503
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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 ...