IDEALS AND I-SEQUENCES IN THE CATEGORY OF MODULES Ahmed A. Khammash Department of Maths , Umm Al-Qura University, Makkah, Saudi Arabia E-mail : aakhammash@uqu.edu.sa AAKHAMMASH I-SEQUENCES KSU 2016

OUTLINE 0. INTRODUCTION 1. IDEALS AND QUOTIENT CATEGORY 2 OUTLINE 0. INTRODUCTION 1. IDEALS AND QUOTIENT CATEGORY 2. I-SEQUENCES 3. EXAMPLES 4. GENERALIZED HELLER FUNCTOR 5. CONNECTION WITH VERTEX THEORY 6. POINTS FOR FURTHER INVESTIGATION AAKHAMMASH I-SEQUENCES KSU 2016

INTRODUCTION In representation theory of groups and algebras, the two most important examples of short exact sequences (extensions) are almost split sequences of modules (introduced by M. Auslander and I. Reiten[AR1],[AR2],[AR3]) and the relative projective sequences of modules (defined by R. Knorr [RK]). This work is an attempt to unify those two sequences using the notion of I-sequences where I is an ideal in the category of modules. AAKHAMMASH I-SEQUENCES KSU 2016

IDEALS AND QUOTIENT CATEGORY If is an abelian category and , we write for the set of morphisms . is an ideal in (notation: ) if is a subgroup of for all such that and means ; i.e. . THE QUOTIENT CATEGORY Objects: Morphisms: for all ; AAKHAMMASH I-SEQUENCES KSU 2016

I-SEQUENCES (Green) If is a finite dimensional algebra over a field k , write for the category of (left) -modules. Let A short exact sequence in is right -sequence if: for all and , factors through iff ; i.e. Iff where This means that (1) is a rt -sequence iff the sequence is exact for all . AAKHAMMASH I-SEQUENCES KSU 2016

I-Sequence AAKHAMMASH I-SEQUENCES KSU 2016

Equivalently (1) is rt -sequence iff is exact in the category of contravariant functors from to the category of k-spaces. … The functors and are finitely presented. The sequence (1) is said to be minimal if where denotes the Jacobson radical. AAKHAMMASH I-SEQUENCES KSU 2016

EXAMPLES Left -sequences are defined analogously EXAMPLES Left -sequences are defined analogously. Sequence (1) is bilateral (two sided) if it is both left and right -sequence. Properties of f. presented functors implies that if -sequences exist then minimals exist. AAKHAMMASH I-SEQUENCES KSU 2016

GENERALIZED HELLER FUNCTOR If is minimal bilateral right -sequence and is indecomposable then we write . The operator is called generalized Heller operator ( a generalization of the Heller operator when ). In fact is an equivalence of categories between two quotient categories. Denote the full subcategory of consists of such by AAKHAMMASH I-SEQUENCES KSU 2016

If and two bilateral sequences as follows Then: such pairs are called compatible (2) (**) is k-isomorphism . This paves the way to the following AAKHAMMASH I-SEQUENCES KSU 2016

THEOREM [AK]: There is a k-additive functor which is an equivalence of categories and maps according to In case , it is known that and share many properties concerning indecomposability , vertices , sources, extensions …etc. To give a slight generalization, we shall introduce a hypothesis on the ideal . AAKHAMMASH I-SEQUENCES KSU 2016

CONNECTION WITH VERTEX THEORY Let and let be the full subcategory of with ob( )= modules which are restrictions to of -modules; i.e. objects of are Ob( ) but . Note that DEFINITIONS: (1) Define to be the ideal of generated by ; that is for is the k-subspace of spanned by all product ,where and at least one of lies in . AAKHAMMASH I-SEQUENCES KSU 2016

(2) Let , . Define by , where is a transversal of the set of cosets in and . Now we introduce the following hypothesis for HYPOTHESIS A. For all there holds ; i.e. THEOREM: If satisfies hypothesis A , then and have the same vertices. AAKHAMMASH I-SEQUENCES KSU 2016

APPLICATIONS It is encouraging that hypothesis A is satisfied by The following result is a sample of how to deduce common properties of the start and end modules of sequences such as almost split sequences and relative projective sequence in a more general setting using the concept of I-sequences. THEOREM ([AK]): Let be an seq., where , then and have the same vertices. AAKHAMMASH I-SEQUENCES KSU 2016

POINTS FOR FURTHER INVESTIGATION (1) Improve HYPOTHESISA in order to prove the existence of an I-sequence in this general setting (in analogue with the proof given by Auslander-Reiten[MA], [R. Knorr [RK]] for the existence of almost split sequences [relative projective sequences]. (2) The effect of on irreducible maps in (A-R)-sequences. (3) The composite of with the Green correspondence (4) Connection of with the extension groups (5) Existence of I-sequences for certain (popular ideals!) such as Ghost ideal [GW] , Phantom ideal [GC] and the power ideals for the radical ideal. [6] It is worth proving directly (if true!) that the radical ideal satisfies hypothesis A (Such a proof was given for the ideal .) AAKHAMMASH I-SEQUENCES KSU 2016

REFERENCES [AR] M. Auslander and I REFERENCES [AR] M. Auslander and I. Reiten, Representation theory of Artin Algebras I , Comm. Algebra 1 (1974),177-268,4 (1974),269-310, 3 (1977),239-294 . [PG] P. Gabriel, Auslander-Reiten Sequences and Representation-finite Algebras, in " Representation Theory I " LNM, Vol. 831, Springer-Verlag, Berlin/New York, 1980 [JG] J. A. Green , Functors On Categories for Finite Group Representations, J. Pure Appl. Algebra 37(1985), 265-298. [AK] A. Khammash , I-sequences in the Category of Modules , J. Algebra 250, 450-472 (2002). [RK] R. Knorr, Relative Projective Covers , in " Proc. Of symp. Mod. Representations of Finite Groups " , Arhus University , 1978. [PL] P. Landrock," Finite Group Algebras and Their Modules", LMS 84, Cambridge Univ Press, Cambridge , UK, 1983. [GW] G. Wang, Ghost numbers of group algebras, PhD Thesis , Univ. of Western Ontarion 2014 [JC] J. D. Christensen, Ideals in triangulated categories: Phantoms, Ghosts and Skeleta, Advances in Maths 136, 284-339(1998) AAKHAMMASH I-SEQUENCES KSU 2016

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