Complements to Projective Almost Complete Tilting Modules

ABSTRACT

An equivalent version of the Generalized Nakayama Conjecture states that any projective almost complete tilting module admits a finite number of non-isomorphic indecomposable complements. Motivated by this connection, we investigate the number of possible complements of projective almost complete tilting modules for some particular classes of Artin algebras, namely monomial algebras and algebras with exactly two simple modules.     

On the volume of a tilting module

We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK ₀ of finitely generated A-modules. We prove that the set of cones associated to tilting modules of projective dimension at most one defines a, not necessarily finite, fan Σ(A). IfA is of finite global dimension, the fan Σ(A) is smooth. Moreover, using the cone of a tilting module, we can associate a volume to each tilting module. Using the fan and the volume, we obtain new proofs for several classical results; we obtain certain convergent sums naturally associated to the algebraA and obtain criteria for the completeness of a list of tilting modules. Finally, we consider several examples. Dedicated to O. Riemenschneider on the occasion of his 65th birthday     

One Dimensional Tilting Modules are of Finite Type

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one. Presented by Claus Michael Ringel.     

On Modules over Endomorphism Algebras of Maximal Rigid Objects in 2-Calabi-Yau Triangulated Categories

This note investigates the modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. We study the possible complements for almost complete tilting modules. Combining with Happel's theorem, we show that the possible exchange sequences for tilting modules over such algebras are induced by the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated categories. For the modules of infinite projective dimension, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.     

Strict Mittag-Leffler Conditions and Gorenstein Modules

In this paper, firstly, we characterize some rings by strict Mittag-Leffler conditions. Then, we investigate when Gorenstein projective modules are Gorenstein flat by employing tilting modules and cotorsion pairs. Finally, we study the direct limits of Gorenstein projective modules.     

A Generalization of Auslander's Last Theorem

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander's result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

The subring generated by the units in a compact ring

ABSTRACT

In this paper, the authors introduce the concept of integrally closed modules and characterize Dedekind modules and Dedekind domains. They also show that a given domain R is integrally closed if and only if a finitely generated torsion-free projective R-module is integrally closed. In addition, it is proved that any invertible submodule of a finitely generated projective module over a domain is finitely generated and projective. Also they give the equivalent conditions for Dedekind modules and Dedekind domains.

     

Generalizations of projectivity and supplements revisited for superfluous ideals

We (re)introduce four ideal-related generalizations of classic module-theoretic notions: the ideal-superfluity, projective ideal-covers, the ideal-projectivity, and ideal-supplements. For a superfluous ideal I, the main theorem asserts the equivalence between the conditions: “I-supplements are direct summands in finitely generated projective modules”; “finitely generated I-projective modules are projective”; “projective modules with finitely generated factors modulo I are finitely generated”; “finitely generated flat modules with projective factors modulo I are projective.” Moreover, we provide a property of the ideal I which is sufficient for the equivalence to hold true. The property is expressed in terms of idempotent-lifting in matrix rings.     

Non-finitely generated projective modules over generalized Weyl algebras

We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules.     

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective–injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.     

Gorenstein Cotilting and Tilting Modules

Auslander and Solberg introduced the concepts of finitely generated cotilting and tilting modules in relative homological algebra considering subfunctors of the Ext-functor. In this article we generalize Auslander–Solberg relative notions by giving the definitions of infinitely generated Gorenstein cotilting and tilting modules by means of Gorenstein exact sequences. Using the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we prove a characterization of relative Gorenstein cotilting and tilting modules, which is a generalization of the beautiful characterization of relative cotilting and tilting modules given by Bazzoni.     

Yetter-Drinfeld modules over weak bialgebras

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. IfH is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.     

Concealed-Canonical Algebras and Separating Tubular Families

We characterise those concealed-canonical algebras which ariseas endomorphism rings of tilting modules, all of whose indecomposablesummands have strictly positive rank, as those artin algebraswhose module categories have a separating exact subcategory(that is, a separating tubular family of standard tubes). This paper develops further the technique of shift automorphismswhich arises from the tubular structure. It is related to the characterisation of hereditary noetheriancategories with a tilting object as the categories of coherentsheaves on a weighted projective line. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

A note on tilting sequences

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.      

有常秩的投射模

本文定义更具一般性的模(未必是有限生成投射模)的常秩的概念,并证明了如果M有常秩n,∧~n M是有限生成的,则M是有限生成的,还证明了若M是有常秩n的投射模,则M一定是有限生成的。     

Generalized tilting functors and Ringel-Hall algebras

It is known from [M. Auslander, M.I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46] and [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] that the Bernstein-Gelfand-Ponomarev reflection functors are special cases of tilting functors and these reflection functors induce isomorphisms between certain subalgebras of Ringel-Hall algebras. In [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] the result from [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] is generalized to the tilting module case by giving an isomorphism between two Ringel-Hall subalgebras. In [J. Miyashita, Tilting Modules of Finite Projective Dimension, Math. Z. 193 (1986) 113-146] Miyashita generalized the tilting theory by introducing the tilting modules of finite projective dimension. In this paper the result in [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] is generalized to the tilting modules of finite projective dimension.