35
votes
Accepted
Does every sheaf embed into a quasicoherent sheaf?
That already fails for $X$ equal to $\text{Spec}\ R$, where $R$ is a DVR with generic point $\eta = \text{Spec}\ K$. Since there are only two nonempty open subsets of $X$, namely all of $X$ and $\{\...
11
votes
Accepted
Is Qcoh(X) locally presentable?
Zariski descent tells us that
$$\operatorname{QCoh}(X)=\lim_{U\subseteq X} \operatorname{QCoh}(U)$$
where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. ...
- 16.5k
5
votes
Accepted
Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms?
I never liked the definition of formal completion (as defined in EGA or Hartshorne), so I'll use the following definition from Brian's formal GAGA paper.
Definition: Let $X$ be a locally Noetherian ...
- 2,607
5
votes
Accepted
Enough injectives in the category of quasi-coherent sheaves on a stack
In the paper
A functorial formalism for quasi-coherent sheaves on a geometric stack, Expo. Math. 33 (2015) 452–501,
in Corollary 5.10 it is shown that $\mathrm{Qco}(X)$ is a Grothendieck category for ...
- 9,229
4
votes
Enough injectives in the category of quasi-coherent sheaves on a stack
Proposition 4.2 in Ethan Pribble's 2004 Ph.D. thesis is the result you asked for, under the hypothesis that $X$ is an algebraic stack with affine diagonal: Pribble shows that $Mod_{qcoh}(O_X)$ then ...
- 1,335
4
votes
flatness of restriction of structure sheaf over ring of global sections
That is not true. Let $k$ be a field, and let $X$ be $\text{Spec}(R)$ for the following $k$-algebra, $$R=k[p,q,s,t]/I, \ \ I=\langle ps,pt,qs,qt\rangle.$$ Let $U$ be the open complement of the ...
4
votes
Katz's proof of Cartier's (descent) theorem
Cartier descent is historically important, since together with Galois descent, it was Grothendieck's source of inspiration for fppf descent.
As far as I remember, and with all due respect, Katz's ...
- 3,998
4
votes
Katz's proof of Cartier's (descent) theorem
I learned this from Victor Ginzburg. Let $S = \mathbb A^1_{k}$, with coordinate $x$ and corresponding vector field $\partial = \frac{d}{dx}$, where $k$ is a field of characteristic $p$. Let $(E,\nabla)...
- 2,338
3
votes
Compact quasi-coherent sheaves
$\DeclareMathOperator\colim{colim}\DeclareMathOperator\Qch{Qch}\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\Hom{Hom}\newcommand\Ab{\mathrm{Ab}}\newcommand\Id{\mathrm{Id}}$In a category admitting ...
- 615
3
votes
Obstructions to abelian sheaf being quasi-coherent
Here's an answer to the modified question in the comments:
Question: Let $X$ be an affine scheme, and $\mathcal F$ an $\mathcal O_X$-module. Under what conditions is $\mathcal F$ quasicoherent?
As ...
- 63.9k
3
votes
Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated
Edit. The argument below applies only to Lindelöf schemes.
Proposition. For every Lindelöf non-quasi-compact scheme $X$, there exists a quasi-coherent $\mathcal{O}_X$-module that is globally generated ...
2
votes
Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
Thanks to David Benjamin Lim's comment above, I have found an answer in the main case of interest to me, detailed below. However, I will leave this question open, since I am also interested in the ...
- 1,349
2
votes
Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated
Warning. This does not answer the question. See the comments.
Let $k$ be a field. For each $n\in\mathbb{N}$, let $X_n$ be a copy of $\mathrm{Spec}(k)$. Put $X:=\coprod_{n\in\mathbb{N}}X_n$. We view $\...
- 19.6k
1
vote
Accepted
Is any "relative support" for (complexes of) quasi-coherent sheaves known?
Perhaps what you need is an action of of a tt-category (in your case $D(R)$) on the category $D_{qc}(X)$ which gives a support in $S$ to quasi-coherent complexes over $X$. This is discussed with ...
- 9,229
1
vote
When are free modules on sheaves of sets quasicoherent?
Let $X$ be a scheme over a field $k$ and $x \in X$ be a point with residue field $k$. Consider the skyscraper $\mathcal{E} = \mathrm{Sky}_x \{*\}$. Then $\mathcal{F} := \mathcal{O}_X\langle \mathcal{E}...
- 639
1
vote
Obstructions to abelian sheaf being quasi-coherent
1) In a similar spirit to your own example, here is a necessary condition for a sheaf of abelian groups $\mathscr{F}$ to be an $\mathcal{O}_X$-module for some scheme structure on $X$:
For each ...
- 1,042
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