Gelfand-Fuks cohomology of vector fields on algebraic varieties

Technical Summary: Gelfand-Fuks Cohomology of Vector Fields on Algebraic Varieties

Problem Statement
The study of Lie algebras of vector fields has a long history, yet the computation of their general cohomology remains intractable beyond simple examples. While Gelfand and Fuks introduced a cohomology theory for $C^\infty$ manifolds in the analytic setting, and subsequent work (e.g., by Tsujishita) explored diagonal complexes in the complex setting, standard approaches have historically been computationally difficult. Recent progress includes Hennion and Kapranov's proof of finite-dimensionality for Lie algebra cohomology on smooth affine varieties using factorization algebras, and Williams' work connecting local cohomology to the diagonal complex.

This paper addresses the gap in a purely algebraic framework. The authors aim to develop an algebraic Gelfand-Fuks cohomology for polynomial vector fields on affine algebraic varieties. The specific challenge is to define this cohomology with coefficients in differentiable $AV$-modules (modules where the action of vector fields is a differential operator in the sense of Grothendieck) and to compute it explicitly for varieties admitting uniformizing parameters.

Methodology
The authors employ a multi-step algebraic approach leveraging recent developments in representation theory and the theory of jets:

1. Algebraic Definition: They define the algebraic Gelfand-Fuks cohomology $H^*_{GF}(V, M)$ for a Rinehart pair $(A, V)$ (where $A$ is the coordinate ring and $V$ is the Lie algebra of vector fields) and a differentiable $AV$-module $M$. The cochain complex consists of cochains that are differential operators in the sense of Grothendieck, satisfying a specific vanishing condition involving finite differences of the arguments.
2. Jet Isomorphism: A central technical tool is the identification of the Gelfand-Fuks complex with the complex of finite cochains on the Lie algebra of polynomial jets, denoted $A^\#V$. They prove that for a differentiable $AV$-module $M$, the cohomology $H^*(V, M)$ is isomorphic to the $A$-linear cohomology $H^*_A(A^\#V, M)$.
3. Completion and Decomposition: By considering the completion of the jet algebra, $\hat{A}^\#V$ (the Lie algebra of $\infty$-jets), they utilize a structural result from [BI22]. For varieties with uniformizing parameters (covered by a single étale chart), $\hat{A}^\#V$ decomposes as a semi-direct product:
   $$\hat{A}^\#V \cong V \ltimes (\hat{A} \otimes L_+)$$
   where $L_+$ is the Lie algebra of vector fields on affine space vanishing at the origin (the positive part of the graded Lie algebra of derivations).
4. Künneth Formula: The authors establish a Künneth formula for the semi-direct product $V \ltimes (A \otimes \mathfrak{g})$. They prove that the $A$-linear cohomology of such a product with coefficients in a tensor module $S \otimes W$ decomposes as:
   $$H^*_A(V \ltimes (A \otimes \mathfrak{g}), S \otimes W) \cong H^*_A(V, S) \otimes H^*(\mathfrak{g}, W)$$
   This relies on the Hochschild-Serre spectral sequence, which they show stabilizes at the $E_2$ term.
5. De Rham Identification: They demonstrate that the $A$-linear cohomology of vector fields with coefficients in the algebra of functions $A$ is isomorphic to the de Rham cohomology of the variety: $H^*_A(V, A) \cong H^*_{dR}(X)$.

Key Contributions and Results

• Definition of Algebraic Gelfand-Fuks Cohomology: The paper formally introduces and validates the cohomology theory for differentiable $AV$-modules, showing it forms a well-defined subcomplex of the Chevalley-Eilenberg complex.
• Main Decomposition Theorem (Theorem 1.1 & 6.1): For an affine algebraic variety $X$ with uniformizing parameters, and a tensor module $M = A \otimes W$ (where $W$ is a finite-dimensional module for $L_+$), the Gelfand-Fuks cohomology decomposes as a tensor product:
  $$H^*_{GF}(V, A \otimes W) \cong H^*_{dR}(X) \otimes H^*(L_+, W)$$
  This result aligns the algebraic computation with known results in the $C^\infty$ setting.
• Explicit Computations: The authors apply this framework to compute cohomology for specific cases:
  • Affine Space ($A^n$): The cohomology reduces to $H^*(L_+, W)$, as the de Rham cohomology of affine space is trivial in positive degrees. This recovers known results for polynomial vector fields $W_n$.
  • Torus ($T^n$): The cohomology is expressed as a sum involving the binomial coefficients of the torus dimension and the cohomology of $L_+$.
  • Krichever-Novikov Algebras: For meromorphic vector fields on a punctured Riemann sphere, the cohomology is computed as a direct sum involving the de Rham cohomology of the punctured sphere and the cohomology of $L_+$.

Significance and Claims
The authors position this work as a development of a "new algebraic approach" to Gelfand-Fuks cohomology. They explicitly state that the results obtained are "not surprising" in that they agree with those obtained in the analytic ($C^\infty$) setting. The significance lies in:

1. Purely Algebraic Framework: Moving the theory from analytic manifolds to affine algebraic varieties using algebraic tools (jets, $AV$-modules, Rinehart pairs).
2. Computational Tractability: Providing a method to compute these cohomologies for a broad class of varieties (those with uniformizing parameters) by reducing the problem to the de Rham cohomology of the variety and the known cohomology of the Lie algebra $L_+$.
3. Unification: Connecting the algebraic theory of vector fields with the de Rham cohomology and the representation theory of infinite-dimensional Lie algebras (specifically $L_+$ and its modules).

The paper does not propose new experimental applications or future implications beyond the explicit computations provided for affine space, the torus, and Krichever-Novikov algebras. It serves as a theoretical bridge, validating that the algebraic machinery yields results consistent with the established analytic theory.