Auslander-Reiten Components, Tilted Algebras And Representation Type Of An Artin Algebra

The thesis is devoted to the study of representationtheory of Artin algebras.The thesis consists of four parts.In the first part,we mainly constructure of Auslander-Reiten components without oriented cycles.We generalizeZhang's and Liu's constructure theorems of componentswithout oriented cycles ([zh],[L3])We obtain thefollowing result.Theorem 2.3.10.Let G be a component ofΓ_A.If L without orientedcycles contains no pathes from injectives to projectives,then there existsa valued locally finite quiverâ–³without oriented cycles such that L isisomorphic to a full subquiver of Zâ–³,which is closed under predecessors(successors)when L contains no projectives(injectines).And (?) when L is stable.A component containing oriented cycles is Very differentfrom a component containing no oriented cycles.If we candetermine if a component contains oriented cycles or not,thenwe be able to determine roughly the shape of the component. At the same time,in the first part,we give some cristerions and characters of components containing oriented cycles.We obtain the following results.Theorem 2.2.1.Let L be a comprnents ofΓ_A.If there are M,N∈L such that for any n∈|N∪{0},L contains pathes from M toÏ„ⁿN,then L contains oriented cycles.Theorem 2.2.3. Let L be a component ofΓ_A.If there are M,N∈L such that for any n∈|N∪{0},L contains.pathes fromÏ„^-nM to N,then L contains oriented cycles.Theorem 2.2.4. Let L be a component ofΓ_A.i) If L is left stable,then L contains oriented cycles if and only if there exist M,N∈L such that L contains pathes from MtoÏ„ⁿN for any n∈|N∪{0};ii) If L is right stable,then L contains oriented cycles if and only if there exist M,N∈L such that L contains pathes fromÏ„^-nMto N for any n∈|N∪{0},Corollary 2.2.7. Let L be a component ofΓ_A without orientedcycles.If there is a path from M to N in L,for M,N∈L,thenthere exists n∈|N∪{0} such that L contains pathes from M toÏ„ⁿNbut L contains no pathes from M toÏ„ⁿ⁺¹N.Theorem 2.2.8. Let L be a component ofΓ_A and arrowM→N in L.If M and N belong to the sameÏ„-orbit,then L contains criented cycles.In the second part,we mainly study preprojective componentsandÏ„-preprojective components.We depict essential distinction of the two kinds of components.And give some criterions for a preprojective component beingÏ„-preprojective component.Ï„-preprojectivecomponents are familiar to us.At the same time,wededuce the algebras to the tilted algebras by considering someproperties of preprojective component of algebras.The class of tilted algerbas is closest to the class of hereditary algebras.Hereditary algebras are very much familiar to us. We obtainthe following results.Theorem 3.2.1. Let A be an Artin algebra,L be a preprojective component ofΓ_A.Then L isÏ„-preprojective if and only if L contains no oriented cycles.Theorem 3.2.2. Let A be an Artin algebra,L be a preprojective component ofΓ_A.If L contains nopathes prom injective modulesto projective modules,then L isÏ„-preprojective component.Theorem 3.2.3. Let L be a preprojective component ofΓ_A.If L contains no non-sectional pathes from injectivemodules to projective modules,then L isÏ„-preprojectivecompinent.Theorem 3.2.4. Any preprojective component of an almost hereditary artin algebra must beÏ„-preprojective component.Corollary 3.2.5.Any preprojective component of a quasitilted Artin algebra must beÏ„-preprojective component.Theorem 3.3.1. Any preprojective component of an Artin is generalized standard component.Theorem 3.3.3. Let A be a connected Artin algebra and letA satisfy Hom A [X,A)=0 for almost all indecomposable A-modules X.If there is a component L containing at least a projective module such that L contains no pathes from injective modules to projective modules,then A is a tilted algebra.Theorem 3.3.4. If there is a preprojective component containing all projective modules ofΓ_A such that L contains nopathes from injective modules to projective modules , then A is a tilted algebra.In the third part,we partly solve open problem 6 in [sk3].And we deduce the algebras to be tilted algebras by considermgsome properties of a stable component of algebras.We obtain the following results.Theorem 4.2.2.Let L be a stable component without periodicmodules ofΓ_A.Then the following are equivalent.(i) L is generalized standard;(ii) L contains only a finsite number ofÏ„-orbits and L contains no short cycles;(iii) L contains only a finite number ofÏ„-orbits and L contains no short chains.Open problem 6 in [sk3] is partly solved here.Theorem 4.2.7. Let A be an Artin algebra. Let L be a stablecomponent without periodic modules ofΓ_A and L contain only a finite number ofÏ„-orbits.If there exists M∈L such thatζ(→M) is faithful and any point ofζ(→M) is not on a short cycle of L, then A is tilted algebra.Corollary 4.2.8. Let L be a stable component withoutperiodic modules ofΓ_A and L contain only a finite number ofÏ„-orbits.If there exists M∈L such thatζ(→M) is faithfuland any point ofζ(→M) is not the middle of a short chain of L,then A is a tilted algebra.In the fourth part,we deduce the representation type of the algebra by studying a small of a component ofΓ_A.We obtain the following results.Proposition 5.2.3. Let X₀→X₁→X₂→…→X_n be a sectionalpath inΓ_A(n≥1).If there are a indecomposable module→Y with Y(?)X₁and a injective irreducible map f:X₀→Y,thenÏ„-X₀,Ï„-X₁,…,Ï„-X_n all exist,and there exist injective irreduciblemaps f_k:X_k→τ-x_k-1,k=1,2,…,n. Proposition 5.2.4.Let X₀→X₁→…→X_n be a sectional pathinΓ_A(n≥1).If there are a indecomposable module Y with Y(?)X_n-1and a durjective irreducible map f:Y→X_n,thenÏ„X₀,Ï„X₁,…,Ï„X_n all exist,and there exist surjective irreducible mapsf_k:Ï„X_k→X_k-1,k=1,2,…n.Theorem 5.2.5. If in AR-quiverΓ_A of an artin algebra A there are a sectional path X₀→X₁→…→X_n-1→X_n and anirreducible mpa f:X₀→X_n with X₀(?)X_n-1 and X₁(?)X_n,then thealgebra A is of representation-infinite type.We reporve theorem 5.2.6.Theorem 5.2.6. If in AR-quiverΓ_A of an artin algebra Athere is chain of irreducible maps (?), then the algebra Ais of representation-infinite type.Theorem 5.2.7. If in AR-quinerΓ_A of an Artin algebra Athere are a path X=X₀→X₁→…X_n-1→X_n=Z (A) and a chain of irreducible maps (?) (B)with (A) and (B) being all sectional and (?),thenthe algebra A is of represintation-infinite type whenever f₁,f₂,…,f_m are all injective or all surjective.