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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

10 votes
Accepted

$\mathcal{D}$-modules of level m

I find it helpful to first work through the definition of multiplication on $\mathcal{D}^{(m)}$ when $m = \infty$, in which case it reduces to the "classical" ring of differential operators in the sen …
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  • 3,702
5 votes
3 answers
1k views

Tensor products of Weyl modules in positive characteristic

Let $G$ be a simple algebraic group over a field $k$, and let $U$ be the unipotent radical of a Borel subgroup $B$. Because $B$ normalises $U$, the group $H = B/U$ acts on the coordinate ring $\mathca …
user91132's user avatar
  • 3,702
3 votes
Accepted

Cohomology vanishing for formal completions of modules?

Yes --- this follows from EGA3 I, Chapter 0, Proposition 13.3.1 . This general result gives conditions under which it is possible to conclude that $H^i( X, \lim\limits_\longleftarrow \mathcal{F}_k )$ …
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  • 3,702
21 votes

How to memorise (understand) Nakayama's lemma and its corollaries?

I usually find the statement of Nakayama's Lemma easy to remember because of its proof, which is really nothing more than the definition of the Jacobson radical plus the existence of maximal left idea …
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  • 3,702