Algebraic differential operators/Introduction/Exercise sheet

Exercise

Prove for the polynomial ring ${}K[X_{1},\ldots ,X_{n}]$ over an arbitrary field ${}K$ that the formal partial derivatives commute.

Exercise

Show that the ring of differential operators on ${}K[X_{1},\ldots ,X_{n}]$ is not commutative.

Exercise

Let ${}H\in K[X_{1},\ldots ,X_{n}]$ be a homogeneous polynomial of degree ${}e$ . Show the equality

{}eH=X_{1}{\frac {\partial H}{\partial X_{1}}}+\cdots +X_{n}{\frac {\partial H}{\partial X_{n}}}\,.

Exercise

Let ${}A$ be a commutative ${}R$ -algebra over a commutative ring ${}R$ . Let

\mu _{f}\colon A\longrightarrow A,x\longmapsto fx,

denote the ${}R$ -linear multiplication map for ${}f\in A$ . For ${}R$ -linear maps

\varphi _{1},\varphi _{2}\colon A\longrightarrow A

set

{}[\varphi _{1},\varphi _{2}]=\varphi _{1}\circ \varphi _{2}-\varphi _{2}\circ \varphi _{1}\,.

Suppose that a ${}R$ -derivation ${}\delta \colon A\rightarrow A$ is given. Show that for all ${}g\in A$ the map ${}[\delta ,\mu _{g}]$ is multiplication by some element.

Exercise

Let  ${}R$  denote a commutative  ${}K$ -algebra and let

${}W\subseteq R$ denote a multiplicative system. Let ${}D\colon R\rightarrow R$ denote a ${}K$ -derivation. Show that we get via

{}D{\left({\frac {f}{g}}\right)}:={\frac {gD(f)-fD(g)}{g^{2}}}\,

a derivation on the localization ${}R_{W}$ which extends ${}D$ .

Exercise

Let ${}R$ denote a commutative ${}K$ -algebra and let ${}E\colon R\rightarrow R$ denote a ${}K$ -linear map. Show that the following statements are equivalent.

${}E$ is a differential operator of order ${}\leq n$ .
For arbitrary elements ${}f_{0},f_{1},\ldots ,f_{n}\in R$ we have
${}{\begin{aligned}E(f_{0}\cdots f_{n})&=\sum _{s=1}^{n}(-1)^{s+1}\sum _{I\subseteq \{0,\ldots ,n\},\,{\#\left(I\right)}=s}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}\\&=\sum _{I\subseteq \{0,\ldots ,n\},\,I\neq \emptyset }(-1)^{{\#\left(I\right)}+1}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}.\end{aligned}}$

Exercise

Describe the derivations on ${}K[X,Y,Z]/(Z^{2}-X^{2}-Y^{2})$ and show that there are no unitary derivations on it.

We consider the twodimensional cone ${}C$ given by the edges ${}\mathbb {R} _{\geq 0}{\begin{pmatrix}7\\2\end{pmatrix}}$ and ${}\mathbb {R} _{\geq 0}{\begin{pmatrix}5\\6\end{pmatrix}}$ and the corresponding monoid ${}C\cap \mathbb {Z} ^{2}$ . Determine the describing integral linear forms and the signatures of the cone.

Exercise

Let ${}C\subseteq \mathbb {R} ^{n}$ denote a positve rational polyhedrial cone and ${}F$ a facet of the cone. Let

\ell \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R}

be a linear form, such that its kernel contains the facet. Suppose that the linear form is given by integers which are coprime. Show that ${}\ell$ or ${}-\ell$ is the canonical integral linear form of ${}F$ .