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I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question:

Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra.

Show that the $A_n$ module $K[X,p^{-1}] \otimes _{K[X]} K[\partial _1 , \ldots , \partial _n] = 0$.

So far I've been asked to prove that $K[X,p^{-1}] \otimes _{K[X]} K[X] \simeq K[X,p^{-1}]$ which was fine, and I'm pretty sure all that I need to do is show that $ p^{-k} \otimes \partial_1 ^{m_1}\cdots \partial_n ^ {m_n} = 0 $ for any $k \in \mathbb{N}$ and $m_i \in \mathbb{N}$ but I'm really struggling with this so any help would be really appreciated, thanks!

CameronJWhitehead
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  • What does $p^{-1}$ stand for here? For any polynomial in $K[x_1,\ldots,x_n]$ you can find a polynomial in the $\partial_i$ that kills it. – Pedro Jul 17 '14 at 17:20
  • $p^{-1}$ is just the inverse of $p$ in the function field. And yeah I agree but I'm not so sure thats helpful here? – CameronJWhitehead Jul 17 '14 at 18:15
  • Oh actually, for any operator in $K[\partial_1, \ldots \partial_n]$ there is a polynomial killed by it? – CameronJWhitehead Jul 17 '14 at 18:16
  • Yes. Suppose $X^{\alpha}$ is the monomial of highest degree in your polynomial. Then $\partial^{\alpha+1}$ kills it. – Pedro Jul 17 '14 at 20:09

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