Finitistic dimensions over commutative DG-rings

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 projdimA(M)≤dim(H0(A))-inf(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{proj}\\,\ extrm{dim}}_A(M) \\le \\dim (\ extrm{H}^0 (A)) - \\inf (M)$$\\end{document}. 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))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dim (\ extrm{H}^0 (A)) - {\ ext {amp}}(A) \\le \ extsf{FPD}(A) \\le \\dim (\ extrm{H}^0(A))$$\\end{document}. 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.