Finitistic dimensions over commutative DG-rings

Isaac Bird; Liran Shaul; Prashanth Sridhar; Jordan Williamson

doi:10.1007/s00209-024-03617-2

Finitistic dimensions over commutative DG-rings

Isaac Bird, Liran Shaul, Prashanth Sridhar, Jordan Williamson

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	3
Journal	Mathematische Zeitschrift
Volume	309
Issue number	1
DOIs	https://doi.org/10.1007/s00209-024-03617-2
State	Published - 1 Jan 2025
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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10.1007/s00209-024-03617-2

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N2 - 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 projdimA(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.

AB - 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 projdimA(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.

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Finitistic dimensions over commutative DG-rings

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