Journal of the Mathematical Society of Japan

Good tilting modules and recollements of derived module categories, II

Hongxing CHEN and Changchang XI

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Abstract

Homological tilting modules of finite projective dimension are investigated. They generalize both classical and good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are involved. To decide whether a good tilting module is homological, a sufficient and necessary condition is presented in terms of the internal properties of the given tilting module. Consequently, a class of homological, non-trivial, infinitely generated tilting modules of higher projective dimension is constructed, and the first example of an infinitely generated $n$-tilting module which is not homological for each $n \ge 2$ is exhibited. To deal with both tilting and cotilting modules consistently, the notion of weak tilting modules is introduced. Thus similar results for infinitely generated cotilting modules of finite injective dimension are obtained, though dual technique does not work for infinite-dimensional modules.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 2 (2019), 515-554.

Dates
Received: 14 July 2017
Revised: 8 November 2017
First available in Project Euclid: 8 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1552035634

Digital Object Identifier
doi:10.2969/jmsj/78477847

Mathematical Reviews number (MathSciNet)
MR3943449

Zentralblatt MATH identifier
07090054

Subjects
Primary: 18E30: Derived categories, triangulated categories 16G10: Representations of Artinian rings 13B30: Rings of fractions and localization [See also 16S85]
Secondary: 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 13E05: Noetherian rings and modules

Keywords
derived category Gorenstein ring homological subcategory recollement weak tilting modules tilting modules

Citation

CHEN, Hongxing; XI, Changchang. Good tilting modules and recollements of derived module categories, II. J. Math. Soc. Japan 71 (2019), no. 2, 515--554. doi:10.2969/jmsj/78477847. https://projecteuclid.org/euclid.jmsj/1552035634


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