Alonso Tarrío, Leovigildo; Jeremías López, Ana; Saorín, Manuel Compactly generated \(t\)-structures on the derived category of a noetherian ring. (English) Zbl 1237.14011 J. Algebra 324, No. 3, 313-346 (2010). Let \(R\) be a commutative noetherian ring and \({\mathbf D}(R)\) the derived category of the modules category. In this paper, the authors study the \(t\)-structure in the sense of A. A. Beǐlinson, J. Bernstein and P. Deligne [“Faisceaux pervers”, Astérisque 100, 172 p. (1982; Zbl 0536.14011)] of \(D(R)\) generated by complexes in \({\mathbf D}_{fg}^-(R)\). They are interested in the problem of classifying \(t\)-structures on \({\mathbf D}(R)\) by filtrations of subsets of \(\mathrm{Spec}(R)\), the spectrum of \(R\), extending the result, shown by A. Neeman [Topology 31, No. 3, 519–532 (1992; Zbl 0793.18008)], that Bousfield localizations are classified by subsets of \(\mathrm{Spec}(R)\). A filtration support of \(\mathrm{Spec}(R)\) is a decreasing map \(\varphi\) from \({\mathbb Z}\) to \(P(\mathrm{Spec}(R))\) such that \(\varphi (i)\subseteq \mathrm{Spec}(R)\) is a stable under specialization subset. The \(t\)-structures on \(D(R)\) generated by complexes in \({\mathbf D}_{fg}^-(R)\) are described by filtration supports. In Section 4, the weak Cousin condition for a filtration support is introduced, and it is shown that this condition is satisfied by the compactly generated \(t\)-structures on \(D(R)\) which restrict to a \(t\)-structure on \({\mathbf D}_{fg}(R)\) and the converse is true when \(R\) has a pontwise dualizing complex. Finally, when \(R\) has a dualizing complex all the \(t\)-structures on \({\mathbf D}^b_{fg}(R)\) are described. Reviewer: Blas Torrecillas (Almeria) Cited in 5 ReviewsCited in 25 Documents MSC: 14B15 Local cohomology and algebraic geometry 18E30 Derived categories, triangulated categories (MSC2010) 16D90 Module categories in associative algebras Keywords:t-structures; supports; derived categories; Cousin condition Citations:Zbl 0536.14011; Zbl 0793.18008 PDFBibTeX XMLCite \textit{L. Alonso Tarrío} et al., J. Algebra 324, No. 3, 313--346 (2010; Zbl 1237.14011) Full Text: DOI arXiv References: [1] Alonso Tarrío, L.; Jeremías López, A.; Lipman, J., Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup., 30, 1-39 (1997) · Zbl 0894.14002 [2] Alonso Tarrío, L.; Jeremías López, A.; Souto Salorio, M. J., Localization in categories of complexes and unbounded resolutions, Canad. J. Math., 52, 2, 225-247 (2000) · Zbl 0948.18008 [3] Alonso Tarrío, L.; Jeremías López, A.; Souto Salorio, M. J., Construction of \(t\)-structures and equivalences of derived categories, Trans. Amer. Math. 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