0
votes
1answer
229 views

A Version of Nullstellensatz for Rings of DÄ°fferential Operators

Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then ...
2
votes
0answers
243 views

Cyclic vector theorem

Could be proved the cyclic vector theorem for a differential module (over a differential field of characteristic zero) by using the fact that the ring of differential operators (over the same ...
0
votes
0answers
185 views

Does regularity of a D-module for an unusual filtration imply regularity for the usual one?

One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...
22
votes
1answer
656 views

Idempotents in Rings of Differential Operators

Differential Operators on General Commutative Rings Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator ...
7
votes
2answers
726 views

What is the smallest $C^*$-algebra containing the “standard” pseudodifferential operators?

Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying $ |\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|} $ ) a $C^*$-algebra? In other ...
10
votes
1answer
378 views

Does the image of a differential operator always contain an ideal?

Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum $$ \delta = \sum_i f_i\partial_x^i$$ where there $f_i$ are complex ...