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Results tagged with ag.algebraic-geometry
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user 6856
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
22
votes
Accepted
Is a locally free sheaf projective in the category of $\mathcal{O}_X$-modules when $X$ is an...
This answer is inspired by the discussion at this question. Let $X$ be an integral affine scheme admitting an open cover $X=U\cup V$ with $U$, $V$ and $X$ all distinct. I claim that $\mathscr O_X$ is …
- 5,759
8
votes
2
answers
965
views
When does the categorical definition of a module work?
$\DeclareMathOperator{\ab}{Ab}\DeclareMathOperator{\qcoh}{QCoh}$
This entry in the nlab shows that for $A$ a (commutative unital) ring, the category $\mathsf{Mod}_A$ of $A$-modules is equivalent to th …
- 5,759
7
votes
Infinitesimal deformations of the formal group of $\mathbb{G}_m$
This isn't a complete answer, but I think the general case is subtle.
If $R\supset\mathbf{Q}$ is (pro-)artinian, then [SGA 3, VIIB 3.2] tells us that formal groups over $R$ are uniquely determined b …
- 5,759
5
votes
Category of sheaves on the topological space X
$\DeclareMathOperator{\sh}{Sh}\DeclareMathOperator{\psh}{PSh}$
A1. Not especially. Essentially, one uses the fact that $\psh(X)$ is abelian (which is essentially trivial to prove) and then the sheafif …
- 5,759
3
votes
0
answers
247
views
K-theory of categories of group schemes and abelian varieties
Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group schem …
- 5,759