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

Technical Summary: Lifts of Endomorphisms of Weyl Algebras Modulo $p^2$

Problem Statement
Let $k$ be a perfect field of positive characteristic $p > 0$, and let $A_n(k)$ denote the $n$-th Weyl algebra over $k$. In positive characteristic, $A_n(k)$ is an Azumaya algebra with a center $Z \cong k[x_1, \dots, x_{2n}]$, where $x_i = z_i^p$. The center $Z$ carries a natural Poisson algebra structure defined by $\{x_i, x_j\} = -\omega_{i,j}$.

The central problem addressed is determining the conditions under which a $k$-algebra endomorphism $\phi: A_n(k) \to A_n(k)$ can be "lifted" to an endomorphism $\Phi: A_n(W_2(k)) \to A_n(W_2(k))$, where $W_2(k)$ is the ring of Witt vectors of length two over $k$ (a ring of characteristic $p^2$). A secondary, related question concerns the properties of the induced endomorphism $\phi_Z$ on the center $Z$, specifically whether $\phi_Z$ preserves the Poisson bracket (i.e., is a Poisson morphism).

Prior work by Tsuchimoto established that if the degree of $\phi$ is less than $p/2$, then $\phi_Z$ is a Poisson morphism. Belov-Kanel and Kontsevich observed that if a lift to characteristic $p^2$ exists, then $\phi_Z$ must be a Poisson morphism. However, the precise equivalence between the existence of a lift and the Poisson property, as well as the optimal degree bounds for these properties, remained to be fully characterized.

Methodology
The authors employ a combination of non-commutative algebra, cohomological obstruction theory, and differential equations in characteristic $p$.

1. Lifting Conditions: The authors analyze the commutator relations in $A_n(W_2(k))$. If $\phi(z_i) = u_i$, lifting requires finding elements $v_i \in A_n(k)$ such that the perturbed elements $u_i + p v_i$ satisfy the Weyl relations modulo $p^2$. This leads to a system of equations involving the commutators $[u_i, v_j] - [u_j, v_i]$ and obstruction terms $u_{ij}$ derived from the expansion of $[u_i, u_j]$.
2. Cohomological Interpretation: The obstruction to lifting is interpreted via the Chevalley–Eilenberg complex of $A_n(k)$ viewed as a module over an abelian Lie algebra. Using a specific trivialization of $A_n(k)$ over its center, the authors map the problem to the de Rham complex of a polynomial ring $S = k[y_1, \dots, y_{2n}]$. The obstruction is shown to correspond to a class in the second de Rham cohomology group $H^2_{dR}(S/k)$.
3. Tsuchimoto's Framework and Refinement: The paper utilizes Tsuchimoto's approach involving the trivialization of $A_n(k)$ as a matrix algebra over an extension $S$. By studying $A_n(k)$-linear endomorphisms of a specific module $M$, the authors derive differential equations governing the "error terms" ($\gamma_i$) that arise when comparing the lifted operators to the standard ones.
4. Degree Analysis: The authors perform a detailed degree analysis on the solutions to these differential equations. They utilize the fact that in characteristic $p$, the operator $\partial^{p-1}/\partial y^{p-1}$ acts as a projection onto specific monomials, allowing them to bound the degrees of the obstruction terms.

Key Contributions and Results

• Equivalence of Lifting and Poisson Morphisms: The primary result (Theorem 2.7) establishes a necessary and sufficient condition: A $k$-algebra endomorphism $\phi$ of $A_n(k)$ lifts to an endomorphism of $A_n(W_2(k))$ if and only if the induced endomorphism $\phi_Z$ on the center is a Poisson morphism. This is proven by relating the Jacobian matrix of $\phi_Z$ to a matrix $C$ of central obstruction elements, showing that $C=0$ (the condition for lifting) is equivalent to the Poisson condition $J_\phi \omega^{-1} J_\phi^T = \omega^{-1}$.
• Improved Degree Bounds: The paper refines Tsuchimoto's result (Theorem 3.10). It proves that if $\deg(\phi(z_l)) + \deg(\phi(z_{n+l})) < 2p$ for all $l=1, \dots, n$, then $\phi_Z$ is a Poisson morphism. Consequently, such a $\phi$ admits a lift to $W_2(k)$. This improves the previous bound of $\deg(\phi) < p/2$ to a condition effectively allowing $\deg(\phi) < p$.
• Injectivity and Flatness: Under the condition $\deg(\phi) < p$, the authors conclude that $\phi$ is injective. This resolves the injectivity question for this degree range, extending known results for $n=1$ and characteristic zero. Furthermore, if $\phi$ is birational and $\deg(\phi) < p$, it is shown to be an automorphism.
• Differential Criterion for Lifting: The paper provides an explicit criterion (Proposition 3.15) for $\phi_Z$ to be a Poisson morphism. This involves solving specific differential equations of the form $\frac{\partial^{p-1}}{\partial x_i^{p-1}} (\dots) = f^p + \frac{\partial^{p-1}f}{\partial x_i^{p-1}}$. The lifting exists if and only if the solutions $f_i$ to these equations satisfy the symmetry condition $\frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i}$.

Significance
The paper claims significance in unifying the algebraic problem of lifting endomorphisms to characteristic $p^2$ with the geometric property of preserving the Poisson structure on the center. By establishing the "if and only if" relationship, the authors provide a structural characterization of liftable endomorphisms.

The improvement of the degree bound from $p/2$ to $p$ (specifically $\deg(\phi) < p$) is highlighted as a significant refinement of Tsuchimoto's work. This bound is shown to be optimal in certain senses (via Example 3.11). The results allow the authors to transfer properties known for Weyl algebras in characteristic zero (such as injectivity and the implication that birational endomorphisms are automorphisms) to the positive characteristic setting, provided the degree of the endomorphism is sufficiently small relative to the characteristic $p$. The work thus bridges the gap between the theory of Weyl algebras in characteristic zero and positive characteristic, offering new tools for analyzing endomorphisms in the latter context.