Wide Morita contexts, relative injectivity and equivalence results

Abstract. By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring C. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring C or the comodule Σ is finitely generated and projective as an Amodule. That is, we obtain relations between the category of C-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End C (Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

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“…The following lemma extends [15,Proposition 5.2] about Morita contexts between unital rings. Lemma 1.8.…”

Section: Introductionmentioning

confidence: 59%

Morita Theory for Comodules Over Corings

Böhm

Vercruysse

2009

Communications in Algebra

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“…Further we investigate some homological aspects of this construction. Note that a similar construction has appeared in [37]. We mention that this Chapter serves as an introductory step for Chapter 4.…”

Section: The Morita Categorymentioning

confidence: 89%

Representation dimension, Cohen-Macaulay modules and triangulated categories

Psaroudakis¹,

Ψαρουδάκης²

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I declare herewith statutorily that the present dissertation was carried out under the international ethical and academical rules and under the protection of intellectual property. According to these rules, I avoided plagiarism of any kind and I made reference to any source which I used in this thesis.To my teacher Apostolos Beligiannis and to my wife Ioanna Katrakazou AcknowledgementFirst and foremost I would like to express my sincere gratitude to my advisor Apostolos Beligiannis for being a real teacher for me over the last eight years -and not only in mathematics. He has always encouraged and helped me very generously. His guidance was invaluable for learning several different topics in algebra and generally in mathematics. During my PhD I remember several moments of uncertainty. Apostolos Beligiannis was always there helping me with his deep knowledge. I want to thank him for sharing and discussing with me, so kindly, his powerful ideas, his inspiration and his insight. It was a great honor for me to be his student, and without him this thesis would have never been completed.I am also very grateful to Nikolaos Marmaridis and Apostolos Thoma for being the interior members of the evaluation committee and for supporting me strongly over all these years. Further, I would like to thank the other members of the committee: Olympia Talelli and Ioannis Emmanouil from the University of Athens, and Hara Charalambous and Athanassios Papistas from the University of Thessaloniki, for their useful comments and interesting questions.During my PhD I spent a few months in Bielefeld, as well as in Trondheim and in Stuttgart. It is a great pleasure to thank Henning Krause, Øyvind Solberg and Steffen Koenig for the warm hospitality and the excellent working conditions. I would also like to thank several members of their groups for providing a very friendly environment and for useful discussions. Moreover, I want to acknowledge Edward L. Green and Jorge Vitória for the fruitful collaborations and the enjoyable discussions we had.My sincere thanks to Andreas Savas-Halilaj for his continuous encouragement through our everyday conversations and to Ioannis Giannoulis for his big support and advice during the last two years. Moreover, I want to thank Christos Tatakis, my office mate for almost six years, for helping me always in difficult circumstances. Special thanks to Magdalini Lada for her support and for the generous hospitality (with Øyvind) during my visits in Trondheim. I would also like to thank George Vafiadis for his help with L A T E X. Further, I want to mention several friends that helped me a lot all these years:

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“…A Morita context for R and S is a datum (R, S, R M S , S N R , φ, ψ) with the condition that M and N are unital modules to the left and to the right. If the Morita context is strict, then we obtain by [13,Theorem 4.3] an equivalence of categories R − M OD and S − M OD.…”

Section: Morita Contexts and The Central Charactermentioning

confidence: 99%

“…for any m, m ′ ∈ M, n, n ′ ∈ N. A Morita context is strict if both maps φ, ψ are isomorphisms, see [13]. Let R be a ring with local units, i.e.…”

Section: Morita Contexts and The Central Charactermentioning

confidence: 99%

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The Hecke algebra of a reductive p-adic group: a geometric conjecture

Aubert

Baum

Plymen

Noncommutative Geometry and Number Theory

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Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig's conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n) and SO(5).

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Wide Morita contexts, relative injectivity and equivalence results

Cited by 6 publications

References 28 publications

Morita Theory for Comodules Over Corings

Morita Theory for Comodules Over Corings

Representation dimension, Cohen-Macaulay modules and triangulated categories

The Hecke algebra of a reductive p-adic group: a geometric conjecture

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