Questions tagged [coherent-sheaves]

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Devissage lemma (Mumford's & Oda's AG II)

This question is part II of my proof reading of Lemma of devissage from Mumford's & Oda's Algebraic Geometry II, findable on page 81; Theorem 6.12: Theorem 6.12 (“Lemma of devissage”). Let $K$...
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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 ...
Linda's user avatar
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3 votes
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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
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2 votes
0 answers
264 views

Derived category of coherent sheaves with a codimension $\geq$ 1 support

Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories: $D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the ...
Arkadij's user avatar
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2 votes
1 answer
674 views

Push-forward of flat module under a finite, flat morphism

Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
user45397's user avatar
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565 views

Direct image functor commuting with infinite direct sum of sheaves

Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial. Let $f: X \rightarrow Y$ be a ...
Luke's user avatar
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2 votes
1 answer
219 views

Glueing modules over $\{x\}\times \operatorname{Spec} R$

Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...
Stabilo's user avatar
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8 votes
1 answer
1k views

Progress on Bondal–Orlov derived equivalence conjecture

In their 1995 paper, Bondal and Orlov posed the following conjecture: If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
mathphys's user avatar
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3 votes
1 answer
736 views

Which complexes of coherent sheaves are dual to perfect ones?

Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of ...
Mikhail Bondarko's user avatar
5 votes
1 answer
371 views

Normal Cones for Complex Spaces

Suppose $U\subset\mathbb C^n$ is an open subset and $f_1,\ldots,f_k$ are analytic functions on it, generating the coherent ideal sheaf $\mathcal I$ which defines a closed complex subspace $Z\...
Mohan Swaminathan's user avatar
2 votes
0 answers
112 views

When is a locally bounded complex of sheaves globally bounded

Let $X,Y$ be projective varieties over $\mathbb{C}$ with $Y$ smooth. Suppose $\mathcal{F} \in D(X \times Y)$, the unbounded derived category of coherent sheaves on $X \times Y$. Suppose further that ...
STCJ's user avatar
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4 votes
3 answers
430 views

Locally ringed space with noetherian stalks and a non-coherent structural sheaf

I am looking for a locally ringed space the stalks of which are noetherian and such that the structural sheaf is not coherent over itself. Can you provide me an example of this? Notice that one may ...
Gaussian's user avatar
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17 votes
1 answer
763 views

Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "...
Misha Verbitsky'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
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0 answers
147 views

Is this construction with stacks a blow-up?

Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...
Yellow Pig's user avatar
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Atlas for a stack of sheaves of rank 1 with torsion

I would like to construct an atlas for the stack of sheaves E of rank 1 and degree b on an elliptic curve C such that E has torsion of length at most 1. Am I allowed to fix both the determinant L of ...
Yellow Pig's user avatar
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1 vote
0 answers
267 views

Segre embedding and Hilbert polynomial of coherent sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, ...
Chen's user avatar
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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
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3 votes
1 answer
336 views

A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for ...
user avatar
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
1 answer
324 views

Gluing finitely presented quasi coherent sheaves

Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover. Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...
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4 votes
1 answer
680 views

Proper mapping theorem

Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules. In particular, the coherent ...
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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
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0 answers
337 views

Singularities of reflexive sheaves

I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...
User43029's user avatar
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6 votes
1 answer
301 views

Question on condition for a sheaf to be locally free in Orlov 2004

In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free: If for all closed points $t:x ...
Marc Besson's user avatar
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
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5 votes
1 answer
1k views

Pushforward maps for cohomology of coherent sheaves

Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map. For ...
David Loeffler's user avatar
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
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1 vote
1 answer
461 views

Pushforward of coherent sheaves and field extensions

This is a continuation of the discussion in the mathoverflow, Pushforward of semi-stable sheaves. Let $X$ be a smooth projective variety over a field $k$ and $L$ be a finite field extension of $k$. ...
user43198's user avatar
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1 vote
1 answer
260 views

Isomorphism of sheaves in families of projective varieties

Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in ...
Jana's user avatar
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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
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2 votes
0 answers
174 views

Base change, descent theory and coherent sheaves

Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$...
Ron's user avatar
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1 vote
1 answer
320 views

Pushforward of semi-stable sheaves under finite field extension

Let $k$ be a field of characteristic zero and $X$ be a non-singular rationally connected variety over $k$. Let $L$ be a finite field extension of $k$. This induces a proper morphism $p:X_L \to X_k$. ...
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5 votes
0 answers
552 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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2 votes
1 answer
766 views

Galois descent for absolute Galois group

Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X_L$ for a ...
user45397's user avatar
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1 vote
2 answers
277 views

Is relative torsion freeness an open condition?

Let $S$ be an integral scheme and $X \to S$ be a smooth scheme of finite type over $S$. Let $\mathcal{E}$ be a coherent sheaf on $X$, and $\eta$ be the generic point of $S$. Assume that restriction $\...
user avatar
5 votes
0 answers
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
4 votes
2 answers
901 views

Different definition of sheaf cohomology

It could be related to my previous question here. Let $\mathcal F$ be a sheaf on a topological space $X$. Hartshorne in his book on Algebraic geometry defines the sheaf cohomology by $$ H^i(X, \...
Hang's user avatar
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1 vote
0 answers
105 views

Stable restrictions of sheaves

Let $X$ be a projective variety and $Y$ a subvariety. If $E$ is a stable sheaf on $X$, then under certain circumstances (e.g. the theorems of Flenner, Mehta-Ramanathan, Bogomolov) the restriction $E|...
Kopper's user avatar
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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
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4 votes
1 answer
1k views

Vanishing of some Ext groups of coherent sheaves

We call a coherent sheaf 'of pure support' if it has no subsheaves with support of smaller dimension. Now, let $X$ be a smooth projective variety, $F$ and $G$ coherent sheaves of pure support on $X$....
user106336'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
  • 8,657
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
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1 vote
0 answers
152 views

Is sheaf stability an open condition?

Let $X$ be a smooth projective variety. If $E$ is a coherent sheaf on $X$, we write its Hilbert polynomial: $$P_E(m) = \alpha^E_dm^d + O(m^{d-1}).$$ We say $E$ is Gieseker stable if $E$ is pure and $...
Kopper's user avatar
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1 vote
0 answers
372 views

Is quotient by maximal destabilizing sheaf, torsion-free?

Let $k$ be an infinite field (not necessarily algebraically closed), $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. ...
user45397's user avatar
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2 votes
0 answers
169 views

Classifying length $4$ modules over $\mathbb C[x,y]$

I am trying to classify all modules of length $4$ over the ring $A=\mathbb C[x,y]$, supported at the origin $0\in \mathbb C^2$, up to ($A$-linear) isomorphism. Let $\mathfrak m=(x,y)$ be the ideal of ...
Andrea Ricolfi's user avatar
2 votes
0 answers
92 views

Singularities of $Spec(Sym^* E^{\vee})$ for $E$ a coherent sheaf

Let $X$ be a smooth complex algebraic variety, and $\mathscr{E}$ a torsion-free coherent sheaf on $X$. Which type of singularities can the total space $\mathrm{Tot}(\mathscr{E}):=\underline{\mathrm{...
Qfwfq's user avatar
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0 votes
0 answers
185 views

Fiberwise injective resolution of coherent sheaf

Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
user45397's user avatar
  • 2,195
3 votes
1 answer
283 views

coherent ring whose nilradical is not finitely generated

Let $A$ be a commutative ring with $1$.We say that $A$ is coherent if and only if every finitely generated ideal of $A$ is finitely presented. Does there exist a coherent ring such that nil-radical ...
Anoop singh's user avatar
5 votes
1 answer
284 views

Is the dual of a compact generator also a compact generator of the derived category of a variety?

Let $X$ be a variety (or more generally a quasi-compact, separated scheme) and $D(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomologies. Let $\mathcal{E}$...
Zhaoting Wei's user avatar
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