Finitistic dimensions over commutative DG-rings

2022/04/14 by Isaac Bird, Liran Shaul, Bird, Isaac +5 · 2 citations
Mathematics · #13H10 #16E45 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Primary: 13D05 #Rings and Algebras (math.RA) #Secondary: 13D09

paper · pdf · doi:10.48550/arxiv.2204.06865

openalex publication_date 2022/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module M of finite flat dimension over such a DG-ring satisfies projdim_A(M) ≤ dim(H⁰ (A)) - inf(M). We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds dim(H⁰ (A)) - amp(A) ≤ FPD(A) ≤ dim(H⁰(A)). Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras.

Cited by

• Existence of balanced dualizing dg-modules
• Openness with respect to levels in triangulated categories

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