On Some Homological Dimensions Of Artin Algebras

The chief objective of the representation theory of Artin algebras is to characterise such an algebra by properties of its module category.One of the advantages of the module theoretic approach is that the language and machinery of both category theory and homological algebra become available.Homological algebra has been used extensively in Mathematics and Physics since the forties of 20th century.In mathematics there is a famous Auslander-Buchbaum-Serre Theorem,which states that an algebraic variety over an algebraically closed field is smooth if and only if its coordinate ring has finite global dimension.Thus we can use homological properties of algebras to characterise the smoothness of algebraic varieties.However there may exist some algebraic variety whose global dimension is infinite.To get more information,people use finitistic dimension instead of global dimension.To some degree finitistic dimension can reflex the homological complexity of algebras better.Finitistic dimension is one of homological invariants,which is defined to be the supremum of projective dimensions of finitely generated modules having finite projective dimensions.The famous finitistic dimension conjecture says that the finitistic dimension of an arbitrary Artin algebra is finite.The conjecture is very famous in representation theory of Artin algebra and homological algebra.It is nearly 50 years old and is far from to be proven.The finitistic dimension conjecture is not an isolated conjecture,for it is connected with some other famous conjectures.In the following we introduce some open conjectures and their relationship with finitistic dimension conjecture.(1)The finitistic dimension conjecture implies the Strong Nakayama conjecture. (2)The Strong Nakayama conjecture implies the Generalized Nakayama conjecture.(3)The Generalized Nakayama conjecture implies the Nakayama conjecture.(4)The finitistic dimension conjectures implies the Gorenstein symmetry conjecture.The study on the finitistic dimension conjecture is helpful to comprehend the above conjectures,and during the process many new concepts and methods arise,so it attracts much attention.In this paper,we use finitary subcategories,representation dimension and global dimension to investigate the finitistic dimensions of an Artin algebra and its subalgebras, and obtain some classes of algebras whose finitistic dimensions are finite.The importance of the representation finite Artin algebras for the whole representation theory of Artin algebras is well understood.There is a bijective correspondence between the class of representation finite Artin algebras and that of Artin algebras with global dimension of at most 2 and with dorminant dimension of at least 2.This result was established by Auslander in[1].Motivated by this correspondence,Auslander introduced the concept of representation demension for Artin algebras."It is hoped that this notion gives a reasonable way of measuring how far an Artin algebra is from being representation finite type."In 1998,Reiten asked whether any Artin algebra has a finite representation dimension or not.In 2002,Iyama gave a positive answer to this question,see[2].In 2005,Igusa and Todorov proved in[3]if the representation dimension of an Artin algebra is at most 3,then the finitistic dimension of A is finite.The interest in the representation dimension is recently revived by its relationship with the finitistic dimension conjecture.In this paper,we use relative global dimension to investigate representation dimension.In the first chapter,we give the introduction and preliminaries.In the second chapter,we define relative global dimension,discuss the relationship between relative global dimension and representation dimension,characterize some properties of x and F_x,and construct F^x-cotilting module and F_x-tilting module. We obtain the following main results:Theorem 2.2.6 Let A be an Artin algebra,V=A⊕D(A),x=add V=add (A⊕D(A)).Then rep.dim A≤gl.dim_FxA+2.Theorem 2.2.11 Let A be a hereditary algebra.Then gl.dim_FxA≤1.Thoerem 2.2.15 Let A be a quasi-tilting algebra.Then gl.dim_FxA≤2.Theorem 2.3.6 Let M be in A-mod.Then M is isomorphic to a summand of some module N where N has a F_x-filtration in x if and only if M∈x.Theorem 2.4.6 Let A be an Artin algebra,V=A⊕D(A),x=add V=add (A⊕D(A)).Then V is an F^x-cotilting module.In the third chapter,we prove that the finitistic dimension of A is finite if A-mod has some finitary subcategories.We also study the subalgebras of an Artin algebra A with certain conditions and obtain some classes of algebras whose finitistic dimensions are finite.We obtain the following main results:Theorem 3.2.1 Let A be an Artin algebra.If gen DA is of finite type,then fin.dim A is finite.Theorem 3.2.6 Let A be a weakly stable hereditary algebra in the sense that each indecomposable submodule of a projective module is either projective or simple.Then fin.dim A is finite.Theorem 3.2.7 Let A be an Artin algebra,x be a contravariantly finite subcategory of A-mod.If cogen x is of finite type,and x(?)P,then fin.dim A is finite.Theorem 3.3.7 Let B be a subalgebra of an Artin algebra A such that rad B is an ideal in A.If gl.dim A≤2,then fin.dim B is finite.Theorem 3.3.9 Let C(?)B(?)A be a chain of subalgebras of an Artin algebra A such that rad C is a left ideal in B,rad B is a left ideal in A.If gl.dim A≤1,then fin.dim C is finite. Theorem 3.4.2 Let A,B be Artin algebras,φ:B→A an epimorphism,and kerφ(?)soc_BB.If cogen A is of finite type,then fin.dim B is finite.Corollary 3.4.4 Let A,B be Artin algebras,φ:B→A an epimorphism,and kerφ(?)soc_BB.If A is weakly stable hereditary,then fin.dim B is finite.In the fourth chapter,let A be an Artin algebra and e be an idempotent element in A.We prove that if A has representation dimension at most three,then the finitistic dimension of eAe is finite,and deduce that if all quasi-hereditary algebras have representation dimensions at most three,then the finitistic dimension conjecture holds. The main results are as follows:Theorem 4.2.1 Let A be an Artin algebra,e an idempotent element in A,and B=eAe.If rep.dim A≤3,then fin.dim B is finite.Theorem 4.2.2 If rep.dim A≤3 for any quasi-hereditary algebra A,then the finitistic dimension conjecture holds.Theorem 4.2.3 Let A be an Artin algebra,e an idempotent element in A,and B=eAe.If add {Ω_A³(X)|X∈A-mod} is of finite type,then fin.dim B is finite.Theorem 4.2.4 Let A be an Artin algebra,e an idempotent element in A,and B=eAe.If gl.dim A≤3,then fin.dim B is finite.In the fifth chapter,we characterize some properties of homological stratifying systems,provide some sufficient conditions to the question of when the Homological Stratifying System conjecture holds,and discuss the relationship among stratifying systems,finitistic dimensions and global dimensions.The main results are as follows:Theorem 5.3.2 Let(θ,≤)be a stratifying system of size t,(θ,Q,≤)be the associated Ext-projective stratifying system.If F(θ)∩cogen Q is of finite type,then pfd F(θ)is finite.Theorem 5.3.6 Let(θ,≤)be a canonical stratifying system of size t.If F(θ)is closed under submodules,then we have the following results:(1)All the torsionless modules are in F(θ). (2)If pd Y＜∞,then gl.dim A=fin.dim A.Theorem 5.3.7 Let A be a quasi-hereditary algebra and T a characteristic tilting module.If F(Δ)(?)(add T)^A,then gl.dim A=pd T.