Jan Trlifaj

Approximations and endomorphism algebras of modules

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules. In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Tilting Preenvelopes and Cotilting Precovers

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.

Whitehead test modules

A (right R-) module N is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module M , ExtR(M,N) = 0 implies M is projective. Dually, i-test modules are defined. For example, Z is a p-test abelian group iff each Whitehead group is free. Our first main result says that if R is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring R, there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring R is said to be fully saturated (κ-saturated) provided that all non-projective (≤ κ-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, GT (1, n, p, S, T ). The four parameters involved here are skew-fields S and T , and natural numbers n, p. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of ∗-modules. In modern algebra, the structure of rings, R, is studied by means of properties of corresponding module categories, Mod-R. In most cases, it is not possible to characterize Mod-R fully. Nevertheless, there are important subclasses of Mod-R that can be treated in detail and that shed light on the whole of Mod-R. Among the prominent ones are the classes of all projective and all injective modules. Recall that a module M is said to be projective (injective) provided that the functorHomR(M,−) (HomR(−,M)) preserves short exact sequences. There is also a universal algebraic aspect: each module is a factor module of a projective module, and a submodule of an injective module. So a possible strategy to investigate ModR consists in describing all injective modules, and for each injective module, I, all its submodules. The first step is usually relatively easy, but the second may be quite hard. For example, using this strategy for abelian groups, one meets serious difficulties already for I = Q⊕Q (see e.g. [E, Theorem 2]). Received by the editors March 17, 1995. 1991 Mathematics Subject Classification. Primary 16E30; Secondary 03E35, 20K35.

Tilting modules and Gorenstein rings

Abstract We apply tilting theory to study modules of finite projective dimension. We introduce the notion of finite and cofinite type for tilting and cotilting classes of modules, respectively, showing that, for each dimension, there is a bijection between these classes and resolving classes of modules. We then focus on Iwanaga-Gorenstein rings. Using tilting theory, we prove the first finitistic dimension conjecture for these rings. Moreover, we characterize them among noetherian rings by the property that Gorenstein injective modules form a tilting class. Finally, we give an explicit construction of families of (co)tilting modules of (co)finite type for one-dimensional commutative Gorenstein rings.

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.

Tilting Cotorsion Pairs

Let R be a ring and T be a 1-tilting right R-module. Then T is of countable type. Moreover, T is of finite type in case R is a Prüfer domain.

Tilting theory and the finitistic dimension conjectures

Let R be a right noetherian ring and let P<∞ be the class of all finitely presented modules of finite projective dimension. We prove that findimR = n < ∞ iff there is an (infinitely generated) tilting module T such that pdT = n and T⊥ = (p<∞)⊥. If R is an artin algebra, then T can be taken to be finitely generated iff p<∞ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalo.

Dually slender modules and steady rings.

A module M over a ring R is called dually slender if Hom (M, —) commutes with direct sums of -modules. For example, any finitely generated module is dually slender. A ring R is called right steady if each dually slender right -module is finitely generated. We provide a model theoretic necessary and sufficient condition for a countable ring to be right steady. Also, we prove that any right semiartinian ring of countable Loewy length is right steady. For each uncountable ordinal σ, we construct examples of commutative semiartinian rings Γσ, and Qa, of Loewy length σ +1 such that Τσ is, but Q0 is not, steady. Finally, we study relations among dually slender, reducing, and almost free modules. 1991 Mathematics Subject Classification: 16D40, 16D90, 16E50, 03C60.

Tilting, cotilting, and spectra of commutative noetherian rings

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochsters conjecture on the existence of finitely generated maximal Cohen-Macaulay modules.

Tilting modules over small Dedekind domains

Abstract A Dedekind domain R is called small if card(R)⩽2ω and card(Spec(R))⩽ω. Assuming Godels Axiom of Constructibility (V=L), we characterize tilting modules over small Dedekind domains. In particular, we prove that under V=L, a class of modules, T , is a tilting torsion class iff there is a set P⊆Spec(R) such that T ={M∈ Mod− R | Ext R 1 (R/p,M)=0 for all p∈P} .