34
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 ...
- 114k
26
votes
Accepted
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}...
- 54.6k
23
votes
Accepted
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 ...
- 8,643
20
votes
Accepted
Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?
No, not always.
In
Positselski, Leonid; Schnürer, Olaf M., Unbounded derived categories of small and big modules: is the natural functor fully faithful?, J. Pure Appl. Algebra 225, No. 11, Article ID ...
- 32.9k
16
votes
Accepted
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
Accepted
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 ...
- 46.4k
12
votes
Accepted
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 $\...
- 2,729
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}...
- 21.9k
9
votes
Accepted
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 ...
- 1,234
9
votes
Accepted
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 ...
- 2,841
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
- 16.2k
8
votes
Accepted
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
Accepted
Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?
There are counterexamples by Stefan Schröer. One of them is not locally separated (a bug-eyed cover, as Kollár calls it), another is a (non-normal) proper surface. See the paper here.
About the link: ...
- 19.2k
8
votes
Accepted
Basic question on projective bundles
This can fail when $\mathscr E$ has torsion:
Example. Let $k$ be a field, let $R = k[t]$ with maximal ideal $\mathfrak m = (t)$, and let $S = \operatorname{Spec} R$. Take $\mathscr E = R\oplus R/\...
- 26.4k
7
votes
Accepted
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 $Ext^i(M,A/\mathfrak{m}) = 0$ ...
- 36.2k
7
votes
Accepted
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 ...
- 36.2k
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 ...
- 2,250
7
votes
Accepted
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 $\...
- 1,059
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 \...
- 36.2k
7
votes
Accepted
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, ...
- 26.4k
7
votes
Accepted
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{...
- 2,729
6
votes
Accepted
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 ...
- 7,010
6
votes
Accepted
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
Which category of sheaves on a manifold remembers the manifold?
Assuming the manifold is Hausdorff, the category of sheaves of modules over the sheaf of smooth functions do the trick.
This category is equivalent to the category of non-degenerate module over the ...
- 39.4k
6
votes
Accepted
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 $...
- 1,980
6
votes
Accepted
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 ...
- 42.1k
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 (...
- 7,010
5
votes
Vanishing of some Ext groups of coherent sheaves
I am posting my comment as an answer. Perhaps the result that you are looking for is Ischebeck's Theorem.
Ischebeck's Theorem. For finitely generated modules $F$ and $G$ over a Noetherian local ...
5
votes
Accepted
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 ...
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