2023/06/25 by Mitsuyasu Hashimoto, Hashimoto, Mitsuyasu
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary 13E05 #Secondary 13A50
paper · pdf · doi:10.48550/arxiv.2306.14366
openalex publication_date 2023/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (A,\mathfrak m)→ (B,\mathfrak n) be a pure homomorphism between Noetherian commutative rings. If B/\mathfrak m B is an Artinian ring, then we have dim A=dim B and \mathopdepth A≥ \mathopdepth B. Using this version of Hochster-Eagon theorem, we prove the following: Let A→ B be a pure homomorphism between Noetherian commutative rings. Assume that the fiber ring κ(\mathfrak p)⊗A B is Artinian for each \mathfrak p∈\mathopSpec A, and B satisfies Serre's (Sn) condition. Then A also satisfies Serre's (Sn) condition. In particular, if a finite group G acts on B and the order |G| of G is invertible in B, and if B is Noetherian with the (Sn) condition, then the ring of invariants A=BG also satisfies the (Sn) condition.