31 votes

Does module Hom commute with tensor product in the second variable?

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only ...
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26 votes
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Does module Hom commute with tensor product in the second variable?

The example $A=M=\mathbb{Z}$, $L=N=\mathbb{Q}$ shows that the answer is negative: we have $\text{Hom}(L,M)=0$ so $\text{Hom}_A(L,M)\otimes_AN=0$, but $\text{Hom}_A(L,M\otimes_AN)=\text{Hom}_{\mathbb{Z}...
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23 votes
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Lemma 1 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition?

I think some of the commenters have forgotten the time when they found vector space linear algebra understandable, but triangulated categories confusing. For someone in such a state, a useful tool ...
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  • 8,468
23 votes
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Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?

Take $A$ to be Brian Conrad's universal counter example, i.e., the infinite countable product of copies of $\mathbf{F}_2$. Then $A$ is absolutely flat and every finitely presented module is finite ...
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  • 371
15 votes
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When do real analytic functions form a coherent sheaf?

For real-analytic manifolds, coherence of the structure sheaf always holds. The 1-sentence reason is that one can pass to real and imaginary parts on Oka's coherence theorem in several complex ...
13 votes
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Different definition of sheaf cohomology

Taking global sections is the same thing as computing the hom from $O_X$. In other words, there is an isomorphism of functors $\Gamma(X,-)\cong\hom(O_X,-)$, so both functors have the same derived ...
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12 votes
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Grothendieck group generated by classes of invertible sheaves

Here is a different, perhaps more elementary example. Consider the Grassmannian $\mathrm{Gr}(2,4)$ of lines in $\mathbf{P}^3$, and let $\mathscr{Q}$ be the universal quotient bundle. The line bundle $\...
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10 votes

Grothendieck group generated by classes of invertible sheaves

The answer is "no" in general, even if one asks if the classes of invertible sheaves generated $K_0(X)$ as a ring; K3 surfaces provide a counterexample. For a K3 surface $X$ over $\mathbb{C}...
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  • 20.8k
9 votes
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Cohomology of real analytic coherent sheaves

In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's Analytic cycles on complex manifolds. For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri ...
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8 votes

H. Cartan's "Variétés analytiques complexes et cohomologie"?

In addition to the Myshkin's answer. An electronic version of the paper can be found here: http://www.inp.nsk.su/~silagadz/Cartan.pdf
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8 votes
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Locally ringed space with noetherian stalks and a non-coherent structural sheaf

Welcome new contributor. It is best to think about these kinds of examples on one's own. Thus, please try this for yourself before reading the following. Let $(Y,Z)$ be any pair of a Noetherian ...
8 votes
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Grothendieck-Verdier duality without the noetherian condition

Does one have the duality in this setting? Yes, we do have duality in a very general setting. Your question is equivalent to asking for the existence of a right adjoint to the derived pushforward ...
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  • 2,726
7 votes

Hilbert polynomial for any invertible sheaf

Prove that $\chi (\mathcal F \otimes L_1^{n} \otimes L_2^m)$ is a polynomial function of $n$ and $m$ of degree $d$ when $L_1$ and $L_2$ are very ample. You can do this by exactly the same induction ...
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  • 116k
7 votes
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Question on condition for a sheaf to be locally free in Orlov 2004

The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ ...
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  • 31.8k
7 votes
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A non-rational variety with a full exceptional collection?

Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay ...
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  • 31.8k
7 votes

Heart of a bounded $t$-structure on the derived category of coherent sheaves

One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the ...
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7 votes
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Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

Please find below a short argument in the case of schemes. The answer is positive for algebraic spaces too; in that case it can be proven using the characterization: $X$ has the resolution property $\...
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  • 799
7 votes

When are two resolutions of a coherent sheaf homotopic

No. The simplest example is given by the following two resolutions of the structure sheaf of a point $P \in \mathbb{P}^1$: $$ 0 \to \mathcal{O}_{\mathbb{P}^1}(-1) \to \mathcal{O}_{\mathbb{P}^1} \to \...
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  • 31.8k
7 votes
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When are two resolutions of a coherent sheaf homotopic

If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E_1 \to \mathscr E_2 \to \mathscr E_3 \to 0$ is a short exact sequence of vector bundles, ...
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7 votes
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Is local freeness open for curves?

To elaborate on Jason's comment: consider a morphism of schemes $f:\mathrm{X}\rightarrow\mathrm{S}$, and an $f$-flat coherent sheaf $\mathscr{F}$ on $\mathrm{X}$. Then (1) the singular locus $\mathrm{...
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6 votes
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Euler characteristic on flat families of quasi-projective schemes

I am not aware of such results in full generality, but I know that working without the properness assumtpion was in part the main motivation for Grothendieck to write SGA 2. Let me focus on a ...
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  • 6,191
6 votes
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Is locally freeness of a sheaf (of fixed rank) around a divisor detectable from a first order neighbourhood?

Let $X$ be a connected reduced noetherian scheme and $\mathscr F$ a coherent sheaf on $X$. Let $\varrho(x)=\dim_{\kappa(x)}\mathscr F_x\otimes \kappa(x)$ where $x\in X$ is a point and $\kappa(x)$ is ...
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6 votes
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Zero locus of a family of morphisms of vector bundles

I am just writing my comment above as an answer. This follows from Corollaire 7.7.8, EGA III. Grothendieck, Alexander Éléments de géométrie algébrique (rédigés avec la collaboration de Jean ...
6 votes

Progress on Bondal–Orlov derived equivalence conjecture

A lot of work has been done to prove the Bondal-Orlov conjecture for (stratified)-Mukai flops. To quote a few names : Namikawa (Mukai flops), Kawamata (one stratified Mukai flop), Cautis and collab (...
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  • 6,191
5 votes
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Torsion free sheaves in flat families

Not in general. For example, take $X=\mathbb{P}^1_R$ and let $x\in X$ be a closed point on the special fiber. Then $I_x$, the ideal sheaf of $x$ is torsion free (and thus flat over $R$), but the ...
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  • 5,552
5 votes
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modular forms, invertible sheaves, and quotients

You have to be a bit careful what "vanish" means in this context. For $k = 2 \bmod 4$, at an elliptic point of order 4, "vanishing as a section of the sheaf" and "vanishing as a function on the upper ...
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5 votes

Strong form of Grothendieck's algebrization theorem

No, that is not true, for essentially the same reason that I explained a couple of days ago in a different question. For simplicity, assume that $R$ contains its residue field $k$. Let $(X,x)$ be a ...
5 votes
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Do finite flat sheaves define families of $0$-cycles?

I quickly reviewed the following article of David Rydh. David Rydh. Families of Cycles. 2008. https://people.kth.se/~dary/famofcycles20080518.pdf Rydh extends to positive characteristic the ...
5 votes
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Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$

Theorem 2.6 (page 9) from Hartshorne Lectures on Deformation Theory (it seems that Hartshorne uses the same notation both for an affine scheme $D$ and its function algebra): Let $X$ be a scheme over $...
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  • 1,930

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