Classifying Serre subcategories via atom spectrum. (English) Zbl 1255.18011
The paper extends Herzog’s one-to-one correspondence between localizing subcategories of a locally coherent Grothendieck category and the open sets of the Ziegler spectrum. A first instance of such a correspondence was found in Gabriel’s thesis, and similar topics were studied intensively in H. Krause’s habilitation thesis.
An object of an abelian category \(\mathcal A\) is called monoform if it is a rational extension of every non-zero subobject. Monoform objects are “strongly uniform” in the sense of H. Storrer. The author introduces the atom spectrum of \(\mathcal A\), points of which are the classes of monoform objects with a common non-zero subobject. With a suitable topology, the atom spectrum generalizes the Ziegler spectrum of a locally noetherian Grothendieck category.
An object of an abelian category \(\mathcal A\) is called monoform if it is a rational extension of every non-zero subobject. Monoform objects are “strongly uniform” in the sense of H. Storrer. The author introduces the atom spectrum of \(\mathcal A\), points of which are the classes of monoform objects with a common non-zero subobject. With a suitable topology, the atom spectrum generalizes the Ziegler spectrum of a locally noetherian Grothendieck category.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
Serre subcategory; atom spectrum; monoform object; localizing subcategory; Ziegler spectrumReferences:
| [1] | Benson, D.; Iyengar, S. B.; Krause, H., Stratifying modular representations of finite groups, Ann. of Math. (2), 174, 3, 1643-1684 (2011) · Zbl 1261.20057 |
| [2] | Dlab, V., Rank Theory of modules, Fund. Math., 64, 313-324 (1969) · Zbl 0192.38001 |
| [3] | Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France, 90, 323-448 (1962) · Zbl 0201.35602 |
| [4] | Garkusha, G.; Prest, M., Classifying Serre subcategories of finitely presented modules, Proc. Amer. Math. Soc., 136, 3, 761-770 (2008) · Zbl 1133.13007 |
| [5] | Herzog, I., The Ziegler spectrum of a locally coherent Grothendieck category, Proc. Lond. Math. Soc. (3), 74, 3, 503-558 (1997) · Zbl 0909.16004 |
| [6] | Krause, H., The spectrum of a locally coherent category, J. Pure Appl. Algebra, 114, 3, 259-271 (1997) · Zbl 0868.18003 |
| [7] | Lambek, J.; Michler, G., The torsion theory at a prime ideal of a right Noetherian ring, J. Algebra, 25, 364-389 (1973) · Zbl 0259.16018 |
| [8] | Matlis, E., Injective modules over Noetherian rings, Pacific J. Math., 8, 511-528 (1958) · Zbl 0084.26601 |
| [9] | Reyes, M. L., A one-sided prime ideal principle for noncommutative rings, J. Algebra Appl., 9, 6, 877-919 (2010) · Zbl 1219.16003 |
| [10] | Storrer, H. H., On Goldman’s primary decomposition, (Lectures on Rings and Modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I). Lectures on Rings and Modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Lecture Notes in Math, vol. 246 (1972), Springer: Springer Berlin), 617-661 · Zbl 0227.16024 |
| [11] | Takahashi, R., Classifying subcategories of modules over a commutative Noetherian ring, J. Lond. Math. Soc. (2), 78, 3, 767-782 (2008) · Zbl 1155.13008 |
| [12] | Ziegler, M., Model theory of modules, Ann. Pure Appl. Logic, 26, 2, 149-213 (1984) · Zbl 0593.16019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
