کلیه ي حقوق مادي مترتب بر نتایج مطالعات ابتکارات و نوآوري هاي ناشی از تحقیق موضوع این پایان نامه متعلق به دانشگاه اصفهان است.

Transcription

1

2 کلیه ي حقوق مادي مترتب بر نتایج مطالعات ابتکارات و نوآوري هاي ناشی از تحقیق موضوع این پایان نامه متعلق به دانشگاه اصفهان است.

3

4

5

6 ModR R R ModR

7 C(modA)

8 [1] S. T. Aldrich, E. Enochs, J. R. García Rozas, L. Oyonarte, Covers and envelopes in Grothendieck categories: flat covers of complexes with applications, J. Algebra 243 (2001), [2] L. Alonso, A. Jeremíoas, J. Lipman, Duality and flat base change on formal schemes, in Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes, Contemp. Math. 244, Amer. Math. Soc., 1999, [3] F. W. Anderson, K. R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics 13, Springer-Verlag, Berlin-Heidelberg-New York, [4] J. Asadollahi, H. Eshraghi, R. Hafezi, Sh. Salarian, On the homotopy categories of projective and injective representations of quivers, J. Algebra 346 (2011),

9 Evaluation Tilting Bounded above Bounded below Left rooted Right rooted Ray Left adjoint Right adjoint

10 Adjoint isomorphism Adjoint pair Almost split sequence Arrow Auslander-Reiten formula Auslander-Reiten theory Auslander-Reiten translation Auslander-Reiten triangle

11 C(modA) ModR

12 Auslander Reiten

13 Abstract This thesis is devoted to study the category of representations of a guiver. In the first part of the thesis, our aim is to investigate the Auslander-Reiten theory in the category of bounded complexes of modules over artin algebras as well as Cohen-Macualy rings. To this end, we first describe explicitly the Auslander-Reiten translation in this category. Then the Auslander-Reiten formula is generalized for complexes and we prove the existence theorem of almost split sequences. As an application of our results, we investigate the existence of Auslander-Reiten triangles in the category of perfect complexes as a full triangulated subcategory of bounded derived category of modules. Note that, in case a ring R is regular, D b (mod-r) coincides with the category of perfect complexes. So this result can be viewed as a version of Happel s theorem for regular rings. In the second part, we study bounded derived categories of the category of representations of infinite quivers over a ring R. Let R be a commutative noetherian ring with a dualising complex. We intent to investigate equivalences similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. Moreover, we turn our attention to reflection functors to get equivalences between bounded derived categories of representations of quivers. We also generalize a result of Happel to artin algebras, instead of fields. Key Words. Auslander-Reiren theory, almost split sequences for complexes, homotopy category, derived category, Grothendieck duality, representation of quivers, reflection functors.

14 University of Isfahan Faculty of Sciences Doctoral Thesis Submitted in Partial Fulfillment of The Requirements For The Degree of Doctor of Science in Pure Mathematics Title Supervisors... and... Advisor... by September 2014