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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 \otimes_{A} 1 $ in $ \operatorname{gr}_{J_{B/A}}(B \otimes_{A} B) $ is $ \{\delta_{i}(b)\}_{i \in \mathbb{N}_{0}} $. A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \operatorname{gr}_{J_{B/A}}(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} $?

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} $?

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} $. A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \operatorname{gr}_{J_{B/A}}(B \otimes_{A} M) $ such that if $ \phi_{i}(b) \in J_{B/A}^{i}(B \otimes_{A} M)/J_{B/A}^{i+1} $, then $ \phi_{n}(b_{1}b_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \delta_{i}(b_{1})\phi_{j}(b_{2}) $.

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} $?

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 \otimes_{A} 1 $ in $ \operatorname{gr}_{J_{B/A}}(B \otimes_{A} B) $ is $ \{\delta_{i}(b)\}_{i \in \mathbb{N}_{0}} $. A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \operatorname{gr}_{J_{B/A}}(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} $?

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*} 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?

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} $. A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \operatorname{gr}_{J_{B/A}}(B \otimes_{A} M) $ such that if $ \phi_{i}(b) \in J_{B/A}^{i}(B \otimes_{A} M)/J_{B/A}^{i+1} $, then $ \phi_{n}(b_{1}b_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \delta_{i}(b_{1})\phi_{j}(b_{2}) $.

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} $?