Classifying subcategories of modules over a commutative Noetherian ring.(English)Zbl 1155.13008
Let \(R\) be a quotient ring of a commutative regular ring by a finitely generated ideal. Taking advantage of the Hopkins-Neeman-Thomason Theorem [M. J.Hopkins, Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73–96 (1987;Zbl 0657.55008);A.Neeman, Topology 31, No. 3, 519–532 (1992;Zbl 0793.18008) andR. W.Thomason, Compos. Math. 105, No. 1, 1–27 (1997;Zbl 0873.18003)],M.Hovey [Trans. Am. Math. Soc. 353, No. 8, 3181–3191 (2001;Zbl 0981.13006)] showed a bijection between the set of coherent subcategories of the category of finitely presented \(R\)-modules and the set of thick subcategories of the derived category of perfect \(R\)-complexes.
The author proves that this holds whenever \(R\) is a commutative noetherian ring. A module version of the bijection between the set of localizing subcategories of the derived category of \(R\)-modules and the set of subsets of \(\text{Spec}R\) proved by A.Neeman [loc. cit.] for such a ring \(R\), is presented as well.
The author proves that this holds whenever \(R\) is a commutative noetherian ring. A module version of the bijection between the set of localizing subcategories of the derived category of \(R\)-modules and the set of subsets of \(\text{Spec}R\) proved by A.Neeman [loc. cit.] for such a ring \(R\), is presented as well.
Reviewer: Marek Golasiński (Toruń)
MSC:
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 16D90 | Module categories in associative algebras |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
