Half–exact coherent functors over Noetherian rings Adson Banda Department of Mathematics and Statistics, University of Zambia Department of Mathematics, Link¨ oping University First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017 1 / 10
Other information Sponsor: ISP through EAUMP Year I came to Sweden : February 2013 Year of PhD studies: Semester Five (5) or 3rd year Estimate to graduate: In three (3) semesters Areas of Specialization: Category theory and homological algebra. 2 / 10
My Advisors Prof. Milagros Izquierdo Dr. Leif Melkersson Main advisor Assistant advisor Link¨ oping University Link¨ oping University 3 / 10
Research Topic First I give definitions of the terms in the topic. 4 / 10
Research Topic First I give definitions of the terms in the topic. Definition 1) A ring R is Noetherian if it satisfies the ascending chain condition; i.e, any chain · · · ⊆ I 0 ⊆ I 1 ⊆ I 2 ⊆ . . . of ideals of a ring R stops, i.e, there exist and n such that I n = I n +1 = . . . . 4 / 10
Research Topic First I give definitions of the terms in the topic. Definition 1) A ring R is Noetherian if it satisfies the ascending chain condition; i.e, any chain · · · ⊆ I 0 ⊆ I 1 ⊆ I 2 ⊆ . . . of ideals of a ring R stops, i.e, there exist and n such that I n = I n +1 = . . . . 2) Informally, a category C consists of a set of objects, and a set of all maps between two objects in which composition is defined and is associative. For example, given a ring R the set of all R–modules and R–module homomorphisms form a category. 4 / 10
Research Topic First I give definitions of the terms in the topic. Definition 1) A ring R is Noetherian if it satisfies the ascending chain condition; i.e, any chain · · · ⊆ I 0 ⊆ I 1 ⊆ I 2 ⊆ . . . of ideals of a ring R stops, i.e, there exist and n such that I n = I n +1 = . . . . 2) Informally, a category C consists of a set of objects, and a set of all maps between two objects in which composition is defined and is associative. For example, given a ring R the set of all R–modules and R–module homomorphisms form a category. 3) Given categories C and D , a functor is a map F : C → D such that (a) F (1 A ) = 1 F ( A ) , where for A ∈ C , 1 A : A → A is the identity; (b) for any composable pair f : A → B and g : B → C in C , we have F ( g ◦ f ) = F ( g ) ◦ F ( f ). 4 / 10
Research Topic Definition 4) A functor F : C → D is half–exact if given a short exact sequence 0 → M ′ → M → M ′′ → 0 in C , then the sequence F ( M ′ ) → F ( M ) → F ( M ′′ ) is exact in D . 5 / 10
Research Topic Definition 4) A functor F : C → D is half–exact if given a short exact sequence 0 → M ′ → M → M ′′ → 0 in C , then the sequence F ( M ′ ) → F ( M ) → F ( M ′′ ) is exact in D . 5) Let A be a noetherian ring and let M fg A be the category of finitely generated A –modules. A covariant functor F : M fg A → M fg A is said to be coherent if, for some morphism f : M → N of finitely generated A -modules, there is an exact sequence Hom ( N , − ) → Hom ( M , − ) → F → 0 , where Hom ( X , Y ) is the set of all maps from X to Y . 5 / 10
Research Topic What is the problem? 6 / 10
Research Topic What is the problem? We aim to give the necessary and sufficient conditions for a coherent functor on finitely generated modules to be half–exact over a Noetherian ring. Precisely we are trying to answer the following research question: 6 / 10
Research Topic What is the problem? We aim to give the necessary and sufficient conditions for a coherent functor on finitely generated modules to be half–exact over a Noetherian ring. Precisely we are trying to answer the following research question: Do all half–exact coherent functors on finitely generated modules over Noetherian rings arise from a complex of projective modules? 6 / 10
Research Topic What is the problem? We aim to give the necessary and sufficient conditions for a coherent functor on finitely generated modules to be half–exact over a Noetherian ring. Precisely we are trying to answer the following research question: Do all half–exact coherent functors on finitely generated modules over Noetherian rings arise from a complex of projective modules? The research question was formulated based on the result in Robin Hartshorne’s paper titled ’Coherent Functors’, where he showed that all half–exact coherent functors on finitely generated modules over a discrete valuation ring (DVR) arise from a complex of projective modules. 6 / 10
Research Topic What is the problem? We aim to give the necessary and sufficient conditions for a coherent functor on finitely generated modules to be half–exact over a Noetherian ring. Precisely we are trying to answer the following research question: Do all half–exact coherent functors on finitely generated modules over Noetherian rings arise from a complex of projective modules? The research question was formulated based on the result in Robin Hartshorne’s paper titled ’Coherent Functors’, where he showed that all half–exact coherent functors on finitely generated modules over a discrete valuation ring (DVR) arise from a complex of projective modules. We hope to falsify or confirm the hypothesis that all half–exact coherent functors on finitely generated modules over a Noetherian ring arise from a complex of projective modules. 6 / 10
Approach Given a coherent functor F on finitely generated modules over a noetherian rings A , there is an exact sequence of functors � F � F 0 � 0 , α F ( A ) ⊗ · (1) where F 0 = coker α = F / im α . 7 / 10
Approach Given a coherent functor F on finitely generated modules over a noetherian rings A , there is an exact sequence of functors � F � F 0 � 0 , α F ( A ) ⊗ · (1) where F 0 = coker α = F / im α . We investigate noetherian rings A such that F is half–exact, F 0 is left–exact and hence the sequence (1) splits, i.e, F ∼ = F ( A ) ⊗ − ⊕ F 0 . 7 / 10
Approach Given a coherent functor F on finitely generated modules over a noetherian rings A , there is an exact sequence of functors � F � F 0 � 0 , α F ( A ) ⊗ · (1) where F 0 = coker α = F / im α . We investigate noetherian rings A such that F is half–exact, F 0 is left–exact and hence the sequence (1) splits, i.e, F ∼ = F ( A ) ⊗ − ⊕ F 0 . As stated above, it is known that for the discrete valuation ring (DVR), the sequence (1) splits. 7 / 10
Approach Given a coherent functor F on finitely generated modules over a noetherian rings A , there is an exact sequence of functors � F � F 0 � 0 , α F ( A ) ⊗ · (1) where F 0 = coker α = F / im α . We investigate noetherian rings A such that F is half–exact, F 0 is left–exact and hence the sequence (1) splits, i.e, F ∼ = F ( A ) ⊗ − ⊕ F 0 . As stated above, it is known that for the discrete valuation ring (DVR), the sequence (1) splits. We have also established in our work that for the principal ideal domain (PID) and the dedekind domain, the sequence (1) splits. 7 / 10
Approach Given a coherent functor F on finitely generated modules over a noetherian rings A , there is an exact sequence of functors � F � F 0 � 0 , α F ( A ) ⊗ · (1) where F 0 = coker α = F / im α . We investigate noetherian rings A such that F is half–exact, F 0 is left–exact and hence the sequence (1) splits, i.e, F ∼ = F ( A ) ⊗ − ⊕ F 0 . As stated above, it is known that for the discrete valuation ring (DVR), the sequence (1) splits. We have also established in our work that for the principal ideal domain (PID) and the dedekind domain, the sequence (1) splits. We are investigating other rings, like the regular ring, and the graded ring. 7 / 10
References 1 S. MacLane; Categories for the working mathematician, Second edition. Springer, 1998. 2 M. Auslander; Coherent Functors. Proc. Conf. Categorical Algebra.” La Jolla 1965, 189–231. Springer 1966. 3 R. Hartshorne; Coherent Functors. Advances in Mathematics 140(1998); 44–94. 8 / 10
Impact and Applications of My Research 9 / 10
Impact and Applications of My Research In pure mathematics, research is done mainly to just satisfy our curiosity and not necessarily to find solutions to society’s problems. 9 / 10
Impact and Applications of My Research In pure mathematics, research is done mainly to just satisfy our curiosity and not necessarily to find solutions to society’s problems. The importance of this research is to add new knowledge to the fields of category theory, commutative algebra and homological algebra. 9 / 10
Impact and Applications of My Research In pure mathematics, research is done mainly to just satisfy our curiosity and not necessarily to find solutions to society’s problems. The importance of this research is to add new knowledge to the fields of category theory, commutative algebra and homological algebra. The results of this research will have applications to the theory of associated primes of finitely generated modules over Noetherian rings. 9 / 10
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