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Jan
6
comment Defining a sheaf of differential operators
Actually, I'm not sure I understand why those two are the same for basis opens... sorry for for being so confused! It seems to me that the first should be much larger, as its sections on U need not be ad-nilpotent (on $U$), which is not true for the latter.
Jan
6
comment Defining a sheaf of differential operators
I'm a bit confused what with all the open sets flying about, so let me try to be more precise. Let $U$ be open, and $\{U_i\}$ an open cover, with $s_i$ on $U_i$ compatible and ad-nilpotent above $U_i\cap V$, $V$ any basis open. We obtain $s$ on $U$ restricting to $s_i$ from the sheaf-ness of $\mathscr{H}$; it remains to check that $s$ is ad-nilpotent above $U\cap V$, $V$ any basis open. But I don't think this follows immediately from ``sections on [$U\cap V$] are precisely those that live in ... for each $i$,'' as the $U_i$ that we are additionally restricting to are not arbitrary basis opens.
Jan
4
comment Defining a sheaf of differential operators
The obvious thing to do would be to use the sheaf condition on $\mathcal{N}$ together with the commutativity of some diagram to conclude that ad action by any $\max_{i\in I} k_i$ functions on $s$ yields 0, where the $k_i$ are the orders of the $s_i$. Unfortunately, this $\max_{i\in I} k_i$ is not well-defined -- the open cover might yield sections with orders following an infinite arithmetic progression. Does (quasi-)coherence of $\mathcal{M}$,$\mathcal{N}$ come into play? Or is there something that I'm missing?
Jan
4
comment Defining a sheaf of differential operators
This cleared up a lot of things, thanks! One remaining question I have is the following. In proving that $\operatorname{Diff}(\mathcal{M},\mathcal{N})$ is a subsheaf of $\operatorname{\mathscr{H}\!om}_\mathbb{C}(\mathcal{M},\mathcal{N})$ one needs to show (more or less) that derivations $s_i$ over restrictions $V_i$ glue to a derivation $s$ over $V$ (for $V$ open affine, say). The sheaf condition on $\mathscr{H}$ yields a homomorphism that restricts to the $s_i$, but how does one show that there exists $k$ such that for any $k+1$ functions, repeated ad action on $s$ yields 0?
Jan
3
accepted Defining a sheaf of differential operators
Jan
2
asked Defining a sheaf of differential operators
Jun
11
revised Explicit Weierstrass Preparation Theorem decomposition
added relevant tags
Jun
9
accepted Is the zero set of a holomorphic function nowhere dense?
Jun
9
comment Is the zero set of a holomorphic function nowhere dense?
Oh, of course! Well that was very silly of me -- thanks!
Jun
9
asked Explicit Weierstrass Preparation Theorem decomposition
Jun
9
asked Is the zero set of a holomorphic function nowhere dense?
Mar
23
revised Closed orbits of complete flags in $\mathbb{C}^n$
edited tags
Mar
12
revised $\dim (A/I) \le \dim (A)$
added a reference
Mar
12
answered $\dim (A/I) \le \dim (A)$
Mar
12
asked Closed orbits of complete flags in $\mathbb{C}^n$
Sep
11
awarded  Enthusiast
Aug
29
answered Partial derivatives, don't know how to solve it
Aug
27
awarded  Citizen Patrol
Aug
27
revised Find all intermediate fields of extension $\mathbb{Q}(\sqrt{2} , \sqrt{3}) : \mathbb{Q}$ without using Galois correspondence.
Minor typos
Aug
27
answered Find all intermediate fields of extension $\mathbb{Q}(\sqrt{2} , \sqrt{3}) : \mathbb{Q}$ without using Galois correspondence.