Lifts of endomorphisms of Weyl algebras modulo $p^2$

Abstract: Let $\varphi$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $\varphi$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $\varphi$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(\varphi) < p$. In particular, we conclude that $\varphi$ is injective if ${\rm deg}(\varphi) < p$.

• The paper identifies a key liftability condition: Endomorphisms of the $n$-th Weyl algebra over a field of characteristic $p$ can be lifted to $W_2(k)$-algebra endomorphisms if and only if they are Poisson morphisms on the center.
• The authors offer enhanced bounds on the degree of endomorphisms which must not exceed a certain limit to qualify for additional properties like injectivity, flatness, and birationality no longer generalized to degree bins and apwrkviously holds
• A structural theorem linking the $lift$ability criterion and It links deformation of a Poisson structure to zero whch summarizes the resulting consequences with the new degree bounds in characteristic $p$; complementary to existing results.

Overview

This paper, by Niels Lauritzen and Jesper Funch Thomsen (2601.23110), studies $k$-algebra endomorphisms $\varphi$ of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of characteristic $p>0$. The central result is a clean equivalence: $\varphi$ admits a lift to a $W_2(k)$-algebra endomorphism of the Weyl algebra over the length-two Witt vectors if and only if the induced map $\varphi_Z$ on the center is a Poisson morphism. The paper also strengthens a degree bound of Tsuchimoto from $\deg(\varphi) < p/2$ to a pairwise condition implying $\varphi$0, and derives consequences for injectivity, flatness, and birational endomorphisms in positive characteristic.

The setting exploits the fact that in characteristic $\varphi$1 the Weyl algebra $\varphi$2 is Azumaya, with center $\varphi$3 where $\varphi$4, and that $\varphi$5 carries the standard Poisson bracket induced by the symplectic form $\varphi$6. Any endomorphism $\varphi$7 preserves $\varphi$8, but $\varphi$9 need not be a Poisson morphism — counterexamples exist even among automorphisms (the Belov-Kanel–Kontsevich example on $n$0).

The lifting obstruction and its cohomological interpretation

Writing $n$1 as Teichmüller lifts in $n$2, the commutators take the form $n$3 with uniquely determined $n$4. A lift of the form $n$5 exists exactly when the system

$n$6

is solvable (Lemma potentiallift). The authors identify this as a cohomological obstruction: under the identification of $n$7 with a polynomial ring via the basis adapted to $n$8, where $n$9 acts as $A_n(k)$0, the class of the closed $A_n(k)$1-form $A_n(k)$2 in de Rham cohomology encodes the failure to lift. Using Katz's computation of de Rham cohomology of a polynomial ring in characteristic $A_n(k)$3, they show the form decomposes into an exact part plus a canonical representative supported on the monomials $A_n(k)$4. The coefficients are the elements

$A_n(k)$5

which are shown to be central and to satisfy $A_n(k)$6. The main structural theorem states that $A_n(k)$7 lifts to $A_n(k)$8 if and only if all $A_n(k)$9 vanish.

Equivalence with the Poisson condition

The bridge between the obstruction matrix $k$0 and the Poisson structure is the identity

$k$1

where $k$2 is the Jacobian of $k$3. This follows from the fact that the Poisson bracket on $k$4 is realized inside $k$5 by $k$6. Consequently, $k$7 is a Poisson morphism precisely when $k$8, which by the previous section happens precisely when $k$9 lifts. This confirms and sharpens the observation of Belov-Kanel and Kontsevich that liftability to characteristic $p>0$0 implies the Poisson property: here it is an if-and-only-if statement.

Notably, the paper also gives an intrinsic criterion (Proposition diffeqpois): solving Tsuchimoto's differential equations $p>0$1-st derivative identities for functions $p>0$2, the lift exists if and only if the Jacobian matrix $p>0$3 is symmetric.

Refinement of Tsuchimoto's degree bound

Tsuchimoto proved that $p>0$4 is a Poisson morphism whenever $p>0$5 for all $p>0$6. The authors improve this to the condition

$p>0$7

which in particular covers all endomorphisms with $p>0$8. The proof uses the differential equations characterizing the correction terms $p>0$9: under the degree hypothesis, the left-hand side has degree less than $\varphi$0, forcing each $\varphi$1 to be constant, hence $\varphi$2 and the symplectic form is preserved. The bound is stated to be optimal in a precise sense: an explicit family of endomorphisms of $\varphi$3 with $\varphi$4 shows that the conclusion can fail just outside the hypothesis (the case $\varphi$5 yields a non-étale map on the center). The authors note the strengthened theorem holds in even greater generality than the degree formulation suggests.

An additional simplification is recorded for $\varphi$6: there the single obstruction element $\varphi$7 vanishes automatically under the degree bound, without invoking the differential equations at all.

Applications: injectivity, flatness, and birationality

The final section applies these results to long-standing questions about endomorphisms of Weyl algebras in positive characteristic:

• Injectivity: Bavula conjectured that every endomorphism of $\varphi$8 is injective; Tsuchimoto confirmed this for $\varphi$9, but Makar-Limanov constructed a non-injective endomorphism of $W_2(k)$0 of degree $W_2(k)$1. Since injectivity of $W_2(k)$2 follows from étaleness of $W_2(k)$3 (as $W_2(k)$4 is a domain), the strengthened degree bound implies $W_2(k)$5 is injective whenever $W_2(k)$6.
• Flatness and birationality: The authors' earlier work established that over fields of characteristic zero, every endomorphism of a Weyl algebra is flat and every birational endomorphism is an automorphism; those arguments relied on reductions modulo large primes inducing étale maps on centers. Theorem etalecenter supplies the required explicit characteristic bounds, so both properties now hold verbatim over fields of characteristic $W_2(k)$7.

These are, to the authors' knowledge, the first positive-characteristic analogues of these two results, previously known only in characteristic zero.

Limitations and open questions

Several caveats are explicit in the paper. The degree bound, while improved, does not cover all endomorphisms: Makar-Limanov's counterexample shows injectivity fails in general for $W_2(k)$8, and the Belov-Kanel–Kontsevich automorphism of $W_2(k)$9 shows the Poisson property itself can fail for automorphisms, so no unconditional statement is possible along these lines. The optimality example indicates the boundary $\varphi_Z$0 cannot be relaxed uniformly. The paper leaves open whether Bavula's injectivity conjecture holds beyond the degree-bounded regime, and whether the equivalence between liftability and the Poisson property extends to Witt vectors of greater length or to more general Azumaya settings. The relationship between the obstruction matrix $\varphi_Z$1 and the symmetric part of $\varphi_Z$2 is fully characterized, but computing the solutions $\varphi_Z$3 of the defining differential equations remains nontrivial in general.

Conclusion

The paper establishes a precise dictionary among three conditions on an endomorphism of the Weyl algebra in characteristic $\varphi_Z$4: liftability to $\varphi_Z$5, vanishing of an explicitly computed de Rham obstruction class, and preservation of the Poisson structure on the center. Combined with a sharpened version of Tsuchimoto's degree estimate, this yields injectivity, flatness, and the birationality-to-automorphism property for endomorphisms of degree below $\varphi_Z$6, extending results previously confined to characteristic zero. The remaining gap between the degree-bounded regime and arbitrary endomorphisms delineates the open territory for subsequent work.