Complements To Some Almost Complete Tilting Modules

Tilting theory plays a central role in the development of the representation theory of Artin algebras. It was introduced in the early eighties of 20th century in the context of finitely generated modules over Artin algebras by Brenner and Bulter [BB], Bongartz [Bo] and Happel and Ringel [HR]. Tilting theory studied the equivalences and dualities between the subcategories of module categories and so may be viewed as a generalization of Morita equivalence theory. Later, Miyashita [Mil] and Colby and Fuller [CF] investigated finitely generated tilting modules over arbitrary rings. The notion of infinitely generated tilting modules of finite projective dimension over arbitray rings was also introduced by Colpi and Trlifaj [CT] and Angeleri-Hiigel and Coelho [AC]. In [Wa1], Wakamatsu studied the tilting modules of infinite projective dimension over Artin algebras i.e., Wakamatsu-tilting modules. Wei Jiaqun [We1] investigated the tilting modules of infinite projective dimension over arbitrary rings i.e.,∞-tilting modules. Miyashita in [Mi2] introduced the notion of tilting pairs over Artin algebras constructing tilting modules associated with a series of idempotent ideals.On the other hand, for an Artin algebra A, the theory of complements to the partial tilting A-modules and especially theory of complements to the almost complete tilting A-modules is an important aspect of tilting theory. Up to now, the problem how many complements the almost complete tilting module has attracts much attention. Happel and Unger [HU1] gave an equivalent condition that the almost complete tilting A-module admits a complement. In [CHU], Coelho, Happel and Unger studied the bound of the number of non-isomorphic indecomposable complements to the almost complete tilting modules and their homological properties under some assumptions.In the chapter 1 of this thesis, we give introduction and preliminaries, which introduce recent developments related to this dissertation and make a systemic exposition of our main results.In chapter 2, for a finite dimensional hereditary algebra A, using the exact sequence about complements, we discuss the number of non-isomorphic indecomposable complements to a faithful almost complete tilting A^(m)-module with projective dimension at most m. Moreover, a distribution of these complements is given. So. the complements to such almost complete tilting A^(m)-modules are clear. The main results are as follows:Theorem 2.2.5 Let T be a faithful almost complete tilting A^(m)-module with pd_A^(m)T≤m. Then T has exactly m+1 non-isomorphic indecomposable complements with projective dimensions at most m.Theorem 2.2.6 Let T be a faithful almost complete tilting A^(m)-module.(1) T has a finite number of non-isomorphic indecomposable complements {X_i}_i=0ⁿ. where each X_i is given by Theorem 2.2.5.(2) Moreover, if pd(_A^(m))T = t≤m, the complements to T have the distribution as follows:(1°) If pd_A^(m)X₀ = 0, then pd_A^(m)X_i = i for each 0≤i≤n.(2°) If pd_A^(m)X₀≠0, then there exists a unique integer j with 0≤j≤t - 1 such that pd_A^(m)X_j - pd_A^(m)X_j+1 = j + 1, pd_A^(m)X_i = i + 1 for 0≤i≤j and pd_A^(m) X_i = i for j + 1≤i≤n.Corollary 2.2.7 Let A be representation-infinite and T a faithful almost complete tilting A^(m)-module with pd_A^(m)T≤m. Then the number of non-isomorphic indecomposable complements to T is either 2m + 1 or 2m + 2.In the second section of chapter 3, we give the definition of almost complete C-tilting modules and that of C-complements, and then give an equivalent condition that an almost complete C-tilting module admits a C-complement, motivated by results of paper [WX1] and [WX2]. The main result is as follows:Theorem 3.2.13 Let C be selforthogonal such that C^χhas a relative injective congenerator, C^χis finitely filtered, and C^χis closed under the kernels of epimorphisms . Assume that M is an almost complete C-tilting module. Then M admits a C-complement if and only if M^⊥∩C^χis covariantly finite in C^χ.In the third section of chapter 3, for an A-module T and a selforthogonal A-module C, We give the definition of C-faithful dimension of T and that of C-cofaithful dimension of T. And then we discuss the relationship between them and the number of the non-isomorphic indecomposable C-complements to the almost complete C-tilting module T. The main results are as follows:Theorem 3.3.4 Let C be selforthogonal such that C^χhas a relative injective congenerator, C^χis finitely filtered, and C^χis closed under the kernels of epimorphisms . Assume that M is an almost complete C-tilting module and that M has a C-complement.(1) There exists a unique indecomposable C-complement X₀ to M such that M^⊥∩C^χ= (M⊕X₀)^⊥∩C^χ, i.e., X₀ is a Bongartz C-complement to M.(2) If M is not C-faithful, then M has a unique indecomposable C-complement i.e., X₀.(3) If M is C-faithful, there exists an exact sequencewhere X_i = Coker h_i-1 for i≥1 and X_i (?) M_i^" is a minimal left add M-approximation of X_i for i≥0. Moreover, {X_i} are a complete list of pairwise non-isomorphic indecomposable C-complements to M.Theorem 3.3.5 Let C be selforthogonal such that C^χhas a relative injective congenerator, C^χis finitely filtered, and C^χis closed under the kernels of epimorphisms . Assume that M is an almost complete C-tilting module and that M has a C-complement. Then M has exactly n+1 non-isomorphic indecomposable C-complements if and only if C-cofd(M) = n. Theorem 3.3.6 Let the assumptions of Theorem 3.3.5 hold. Assume that M has n + 1 non-isomorphic indecomposable C-complements. Then C-fd (M)≥n.In chapter 4, we give the notion of W-C-tilting modules and of∞-C-titling modules, and show relationship between them. Moreover, we extend the Auslander-Reiten correspondence for tilting modules to the context of W-C-tilting modules. The main results are as follows:Theorem 4.2.4 Assume that T is a∞-C-tilting module and that C^χhas a relative injective cogenerator I. Then T is a W-C-tilting module.Proposition, 4.2.5 Assume that T∈C^χand that∞-pres (T) (?) C^χ. Then∞-pres (T) (?) C^χ.Theorem 4.3.6 Let C be selforthogonal such that C^χhas a relative injective cogenerator and C^⊥₁ is closed under the kernels of epimorphisms. Then T→T^χ∩C^χgives a one-one correspondence between isomorphism classes of basic W-C-tilting modules and subcategories V which are relative coresolving subcategories in C^χwith a relative projective generator, and is maximal among those with the same relative projective generator.Theorem 4.3.7 Let C be selforthogonal such that C^χhas a relative injective cogenerator and C^⊥₁ is closed under the kernels of epimorphisms. Then T→χ_T∩C^χgives a one-one correspondence between isomorphism classes of basic W-C-tilting modules and subcategories V which are relative resolving subcategories in C^χwith a relative injective cogenerator, and is maximal among those with the same relative injective cogenerator.