Questions tagged [coherent-sheaves]
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252
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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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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 ...
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'|_{...
2
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264
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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 ...
2
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1
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674
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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 ...
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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 ...
2
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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, ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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763
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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 "...
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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 ...
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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^...
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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 ...
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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 $P_G$, ...
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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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336
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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 ...
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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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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 ...
4
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680
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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 ...
4
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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 ...
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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 (...
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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 ...
4
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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 ...
5
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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 ...
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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 $\overline{K}$ ...
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461
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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 $k$. ...
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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 ...
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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 ...
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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 $k$...
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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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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} \...
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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 ...
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2
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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 $\...
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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 ...
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2
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901
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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
$$
H^i(X, \...
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105
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Stable restrictions of sheaves
Let $X$ be a projective variety and $Y$ a subvariety.
If $E$ is a stable sheaf on $X$, then under certain circumstances (e.g. the theorems of Flenner, Mehta-Ramanathan, Bogomolov) the restriction $E|...
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When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
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Vanishing of some Ext groups of coherent sheaves
We call a coherent sheaf 'of pure support' if it has no subsheaves with support of smaller dimension.
Now, let $X$ be a smooth projective variety, $F$ and $G$ coherent sheaves of pure support on $X$....
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Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
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Minimum number of generators for a gloablly generated sheaf over a curve
Let $X$ be a smooth, projective curve over an algebraically closed field and $E$ be a globally generated locally free sheaf of rank $r$. Is it always possible to write $E$ as the quotient of $r+1$ ...
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Is sheaf stability an open condition?
Let $X$ be a smooth projective variety. If $E$ is a coherent sheaf on $X$, we write its Hilbert polynomial:
$$P_E(m) = \alpha^E_dm^d + O(m^{d-1}).$$
We say $E$ is Gieseker stable if $E$ is pure and $...
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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 $F$. ...
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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 ...
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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{...
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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 $\...
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283
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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 ...
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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 $\mathcal{E}$...