On the subalgebra of invariant elements: finiteness and immersions

2022/02/20 by Jesús Martín Ovejero, Ovejero, Jesús Martín, Ángel Luis Muñoz Castañeda +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2202.09829

openalex publication_date 2022/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be an algebra over a ring \Bbbk, T an R-algebra, M a finitely generated projective R-module, and N a T-module. Let G be a linearly reductive group scheme over \Bbbk equipped with a representation ρ:\underlineG_R→ \underline\textrmAut_\textrmMod(R)(M). For the graded T-algebra A, defined as A := ( S_T^\bullet (M^\vee ⊗_RN ))^G, we determine the conditions under which the graded T-algebra A is finitely generated, finitely presented, or flat. Furthermore, we establish the conditions under which a closed embedding of \textrmProj A into a projective space exists. Since we do not impose any Noetherian hypotheses, our results generalize those in the literature, providing new powerful tools regarding moduli problems.

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