Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?

Question

Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-action below: \begin{equation*} a \mapsto \sum_{j=0}^{\infty} \frac{\delta^{j}(a)t^{j}}{j!}. \end{equation*} The sheaf $ \Omega_{A/k} $ parameterizes $ A $-derivations of $ A $ over $ k $.

If $ k $ is a field of positive characteristic then actions of $ \widehat{\mathbb{G}_{a}} $ are parameterized by iterative, higher derivations. A collection of $ k $-homomorphisms $ \phi_{i}: A \to A $ is a higher derivation if $ \phi_{0} = \operatorname{id}_{A} $ and $ \phi_{n}(a_{1}a_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \phi_{i}(a_{1}) \phi_{j}(a_{2}) $. A higher derivation is called iterative if $ \phi_{i} \circ \phi_{j} = \binom{i+j}{i} \phi_{i+j} $. One obtains an action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spec}(A) $ from an iterative, higher derivation $ \{\phi_{i}\}_{i \in \mathbb{N}_{0}} $ via the co-action below: \begin{equation*} a \mapsto \sum_{j=0}^{\infty} \phi_{j}(a)t^{j}. \end{equation*}

One could generalize the notion of a $ B $-derivation of $ M $ over $ A $ as follows. Let $ J_{B/A} $ be the kernel of the ring homomorphism $ B \otimes_{A} B \to B $ which sends $ b_{1} \otimes b_{2} $ to $ b_{1}b_{2} $. Define $ \delta_{i}(b) $ to be the image of $ b \otimes_{A} 1 $ in $ J_{B/A}^{i}/J_{B/A}^{i+1} $ so that the image of $ b $ in $ \prod_{i \in \mathbb{N}_{0}} J_{B/A}^{i}/J_{B/A}^{i+1} $ is $ \{\delta_{i}(b)\}_{i \in \mathbb{N}_{0}} $. A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \prod_{i \in \mathbb{N}_{0}} J_{B/A}^{i}(B \otimes_{A} M)/J_{B/A}^{i+1}(B \otimes_{A} M) $ such that if $ \phi_{i}(b) \in J_{B/A}^{i}(B \otimes_{A} M)/J_{B/A}^{i+1} $ and $ \phi(b) = \{\phi_{i}(b)\}_{i \in \mathbb{N}_{0}} $, then $ \phi_{n}(b_{1}b_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}, i+j=n} \delta_{i}(b_{1})\phi_{j}(b_{2}) $. However, there might be a more optimal way to generalize the notion of a $ B $-derivation of $ M $ over $ k $ to higher derivations so that one could define a sheaf $ \Xi_{A/k} $ with a similar universal property to $ \Omega_{A/k} $.

Has anyone tried to generalize the notion of higher differentials so that one could create a sheaf $ \Xi_{A/k} $ with a similar universal property to that of $ \Omega_{A/k} $ for derivations? Specifically are there any references in the literature of such a generalization and sheaf $ \Xi_{A/k} $?

Answer 1

I have this strange feeling I have commented on this or an identical question recently, but can't place it.

Here is one solution which may not be what you are looking for and you have mentioned it in your question.

Let $A$ be a $B$- algebra (all commutative) and let $M$ be an $A$-module. A $B$- module homomorphism $D:A\to M$ is a differential operator of order $\leq n$, if $[D,a]$, the Lie bracket, is a differential operator of order $\leq n-1$ for all $a\in A$. Of course differential operator of order zero is just an $A$-module homomorphism.

Then the universal object is got as follows. Let $J$ (as in question) be the kernel of the multiplication map $A\otimes_B A\to A$. For any $n$, one has the $A$- module $D_n=A\otimes_B A/J^{n+1}$ where the module structure is from the left factor, say. Then one has a $B$-module map $d:A\to D_n$ given by $d(a)=1\otimes a$ and one can check that this is a differential operator of order $n$ and has the required universal property.