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Hello,

Maybe this question is trivial, so sorry

Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).

Then we can define the ring module of differential operators $D^{\leq n} (A)$, a subring submodule of $End_k (A,A)$ (endomorphisms of vector spaces). $D^{\leq -1} = 0$, and then inductively $D^{\leq n} = { d | [d,a]\in D^{\leq n-1}}$.

We have a lemma:

Lemma. Let $f \in A$. Then for every $d \in D^{\leq n}(A)$ we can find unique $e \in D^{\leq n}(A_f )$, such that $l\circ d = e \circ l$, where $l: A \to A_f$ is the localization map.

I think that I know how to prove the lemma, by induction on the order of diff. op. (just need to see how to apply operators to fractions). It gives us a map $D^{\leq n}(A) \to D^{\leq n}(A_f)$.

Question 1. Under which assumptions on $A/k$ this map $D^{\leq n}(A) \to D^{\leq n}(A_f)$ is a localization map (i.e. becomes an isomorphism after tensoring (say on the left, it does not matter) with $A_f$)?

Question 2 (my real question). If $A/k$ is finitely generated, or finitely presented, is this a localization map?

Somehow, I am having trouble with the "surjectivness" part. Maybe there is some reference?

Thank you, Sasha

1

# Localizability of differential operators a la Grothendieck

Hello,

Maybe this question is trivial, so sorry

Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).

Then we can define the ring of differential operators $D^{\leq n} (A)$, a subring of $End_k (A,A)$ (endomorphisms of vector spaces). $D^{\leq -1} = 0$, and then inductively $D^{\leq n} = { d | [d,a]\in D^{\leq n-1}}$.

We have a lemma:

Lemma. Let $f \in A$. Then for every $d \in D^{\leq n}(A)$ we can find unique $e \in D^{\leq n}(A_f )$, such that $l\circ d = e \circ l$, where $l: A \to A_f$ is the localization map.

I think that I know how to prove the lemma, by induction on the order of diff. op. (just need to see how to apply operators to fractions). It gives us a map $D^{\leq n}(A) \to D^{\leq n}(A_f)$.

Question 1. Under which assumptions on $A/k$ this map $D^{\leq n}(A) \to D^{\leq n}(A_f)$ is a localization map (i.e. becomes an isomorphism after tensoring (say on the left, it does not matter) with $A_f$)?

Question 2 (my real question). If $A/k$ is finitely generated, or finitely presented, is this a localization map?

Somehow, I am having trouble with the "surjectivness" part. Maybe there is some reference?

Thank you, Sasha