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 ...

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}...

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 ...

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 ...

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 ...

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 $\...

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}...

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 ...

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 ...

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

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 ...

Community wiki

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: ...

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/\...

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 $Ext^i(M,A/\mathfrak{m}) = 0$ ...

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 ...

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 ...

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 $\...

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 \...

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, ...

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{...

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 ...

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 ...

Community wiki

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 ...

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 $...

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 ...

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 (...

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 ...

Community wiki

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 ...

Community wiki

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

coherent-sheaves × 247ag.algebraic-geometry × 223

derived-categories × 39

vector-bundles × 35

sheaf-theory × 25

complex-geometry × 21

reference-request × 20

ac.commutative-algebra × 15

moduli-spaces × 15

deformation-theory × 15

schemes × 14

flatness × 14

sheaf-cohomology × 13

homological-algebra × 10

algebraic-stacks × 8

cohomology × 7

derived-functors × 7

ct.category-theory × 6

cv.complex-variables × 6

algebraic-curves × 6

kt.k-theory-and-homology × 6

duality × 6

triangulated-categories × 5

dg.differential-geometry × 4

projective-geometry × 4