Projective lattices of tiled orders
DOI:
https://doi.org/10.17721/1812-5409.2018/4.2Abstract
Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.
Key words: tiled order, exponent matrix, projective lattice.
Pages of the article in the issue: 16-19
Language of the article: Ukrainian
References
HAZEWINKEL, M., GUBARENI, N. and KIRICHENKO, V.V. (2004) Algebras, Rings and Modules. Series: Mathematics and Its Applications, 575, Kluwer Acad. Publish. 1. p. 380.
HAZEWINKEL, M, GUBARENI, N. and KIRICHENKO, V.V.(2007) Algebras, Rings and Modules. Series: Mathematics and Its Applications, 586, Kluwer Acad. Publish. 2. p. 400.
ZAVADSKIJ, A.G., KIRICHENKO, V.V. (1976) Torsion-free modules over prime rings. Zap. Nauch. Seminar. Leningrad. Otdel. Mat. Steklov Inst. (LOMI). v. 57, p. 100-116.
TARSY, R.B. (1970) Global dimension of orders. Trans. Amer. Math. Soc. vol. 151. P. 335-340.
JATEGAONKAR, V.A. (1974) Global dimension of tiled orders over a discrete valuation ring. Trans. Amer. Math. Soc. vol. 196. P. 313-330.
FUJITA, H. (1990), (1991) Tiled orders of finite global dimension. Trans. Amer. Math. Soc., v.322, 1990, pp. 329–342; Erratum to "Tiled orders of finite global dimension". Trans. Amer. Math. Soc., v.327, No.2 pp. 919–920.
ZHURAVLEV, V.N., ZHURAVLYOV, D.V. (2012) Projective resolution of irreducible modules over tiled order. Algebra and discrete mathematics. Volume 14. 2. pp. 323 – 336.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
