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Touzé Conjecture on CFG for Reductive Group Schemes over Noetherian Bases

Establish that every reductive algebraic group scheme G over a commutative Noetherian ring k satisfies the cohomological finite generation (CFG) property; that is, for every finitely generated G-algebra A over k, the graded cohomology algebra H^*(G, A) is finitely generated over k.

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Background

Over a field, the cohomological finite generation (CFG) property is equivalent to the finite generation of invariants (FG) for reductive algebraic groups. Van der Kallen’s work proves CFG for finite group schemes over arbitrary Noetherian bases, and CFG is established for GL_n over Noetherian rings containing a field.

The natural next step is to understand CFG for general reductive group schemes over Noetherian bases. Evidence in support of this conjecture includes verification for specific cases such as SL_2 and SL_3.

References

Antoine Touzé has conjectured that this is indeed the case, and there is supporting evidence: for example, the conjecture has been verified for the groups \mathrm{SL}_2 and \mathrm{SL}_3 (see Section 4). Let $G$ be a reductive algebraic group scheme over a commutative Noetherian ring $k$. Then $G$ satisfies the cohomological finite generation property.

— Notes on Cohomological Finite Generation for Finite Group Schemes (2510.02908 - Goméz et al., 3 Oct 2025) in Further directions and applications (Introduction), immediately preceding and including Conjecture [Touzé]