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Jan Trlifaj

Publications and source records attributed to Jan Trlifaj.

33 records · Page 2Linked to original sources
arXiv2017

Faith's problem on R-projectivity is undecidable

In \cite{F}, Faith asked for what rings $R$ does the Dual Baer Criterion hold in Mod-$R$, that is, when does $R$-projectivity imply projectivity for all right $R$-modules? Such rings $R$ were called right testing. Sandomierski proved that if $R$ is right perfect, then $R$ is right testing. Puninski et al.\ \cite{AIPY} have recently shown for a number of non-right perfect rings that they are not right testing, and noticed that \cite{T2} proved consistency with ZFC of the statement {\lq}each right testing ring is right perfect{\rq} (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of non-right perfect rings such that the Dual Baer Criterion holds for {\lq}small{\rq} modules (where {\lq}small{\rq} means countably generated, or $\leq 2^{\aleph_0}$-presented of projective dimension $\leq 1$).

math.RA↗
arXiv2017

Pure Projective Tilting Modules

Let $T$ be a $1$-tilting module whose tilting torsion pair $({\mathcal T}, {\mathcal F})$ has the property that the heart ${\mathcal H}_t$ of the induced $t$-structure (in the derived category ${\mathcal D}({\rm Mod} \mbox{-} R)$ is Grothendieck. It is proved that such tilting torsion pairs are characterized in several ways: (1) the $1$-tilting module $T$ is pure projective; (2) ${\mathcal T}$ is a definable subcategory of ${\rm Mod} \mbox{-} R$ with enough pure projectives, and (3) both classes ${\mathcal T}$ and ${\mathcal F}$ are finitely axiomatizable. This study addresses the question of Saorín that asks whether the heart is equivalent to a module category, i.e., whether the pure projective $1$-tilting module is tilting equivalent to a finitely presented module. The answer is positive for a Krull-Schmidt ring and for a commutative ring, every pure projective $1$-tilting module is projective. A criterion is found that yields a negative answer to Saorín's Question for a left and right noetherian ring. A negative answer is also obtained for a Dubrovin-Puninski ring, whose theory is covered in the Appendix. Dubrovin-Puninski rings also provide examples of (1) a pure projective $2$-tilting module that is not classical; (2) a finendo quasi-tilting module that is not silting; and (3) a noninjective module $A$ for which there exists a left almost split morphism $m: A \to B,$ but no almost split sequence beginning with $A.$

math.RT↗
arXiv2016

Approximations and Mittag-Leffler conditions --- the applications

A classic result by Bass says that the class of all projective modules is covering, if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules $\mathcal C$, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when $\mathcal C = \mathcal A$, or $\mathcal C$ is the class of all locally $\mathcal A ^{\leq ω}$-free modules, where $\mathcal A$ is any class of modules fitting in a cotorsion pair $(\mathcal A, \mathcal B)$ such that $\mathcal B$ is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and artin algebras of infinite representation type.

math.RA↗
arXiv2016

Very flat, locally very flat, and contraadjusted modules

Very flat and contradjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum Spec(R) of a noetherian domain R: (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin's criterion for locally very flat modules over Dedekind domains.

math.AC↗
arXiv2016

Generalized injectivity and approximations

Injective, pure-injective and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the $C_1$, $C_2$, $C_3$, quasi-continuous, continuous, and quasi-injective modules. We show that, except for the class of all $C_1$-modules, each of the latter classes provides for approximations only when it coincides with the injectives (for quasi-injective modules, this forces R to be a right noetherian V-ring, in the other cases, R even has to be semisimple artinian). The class of all $C_1$-modules over a right noetherian ring R is (pre)enveloping, iff R is a certain right artinian ring of Loewy length at most 2; in this case, however, R may have an arbitrary representation type.

math.RT↗
arXiv2013

Cotilting modules over commutative noetherian rings

Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.

math.AC↗
arXiv2013

Colocalization and cotilting for commutative noetherian rings

For a commutative noetherian ring R, we investigate relations between tilting and cotilting modules in Mod-R and Mod-R_m where m runs over the maximal spectrum of R. For each finite n, we construct a 1-1 correspondence between (equivalence classes of) n-cotilting R-modules C and (equivalence classes of) compatible families F of n-cotilting R_m-modules (m \in mSpec R). It is induced by the assignment C |-> (C^m ; m \in mSpec R) where C^m is the colocalization of C at m, and its inverse F |-> \prod_{M \in F} M. We construct a similar correspondence for n-tilting modules using compatible families of localizations; however, there is no explicit formula for the inverse.

math.AC↗
arXiv2012

Approximations and locally free modules

For any set of modules S, we prove the existence of precovers (right approximations) for all classes of modules of bounded C-resolution dimension, where C is the class of all S-filtered modules. In contrast, we use infinite dimensional tilting theory to show that the class of all locally free modules induced by a non-sum-pure-split tilting module is not precovering. Consequently, the class of all locally Baer modules is not precovering for any countable hereditary artin algebra of infinite representation type.

math.RT↗
arXiv2012

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 Hochster's conjecture on the existence of finitely generated maximal Cohen-Macaulay modules.

math.AC↗
arXiv2011

Descent of restricted flat Mittag-Leffler modules and generalized vector bundles

A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was proved by Kaplansky's technique of dévissage already in \cite[II.\S3]{RG}. Since vector bundles coincide with $\aleph_0$-restricted Drinfeld vector bundles, a question arose in \cite{EGPT} of whether locality holds for $κ$-restricted Drinfeld vector bundles for each infinite cardinal $κ$. We give a positive answer here by replacing the d\' evissage with its recent refinement involving $\mathcal C$-filtrations and the Hill Lemma.

math.AC↗
arXiv2010

Tilting via torsion pairs and almost hereditary noetherian rings

We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.

math.RA↗
arXiv2009

Almost free modules and Mittag--Leffler conditions

Drinfeld recently suggested to replace projective modules by the flat Mittag--Leffler ones in the definition of an infinite dimensional vector bundle on a scheme $X$. Two questions arise: (1) What is the structure of the class $\mathcal D$ of all flat Mittag--Leffler modules over a general ring? (2) Can flat Mittag--Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi--coherent sheaves on $X$? We answer (1) by showing that a module $M$ is flat Mittag--Leffler, if and only if $M$ is $\aleph_1$--projective in the sense of Eklof and Mekler. We use this to characterize the rings such that $\mathcal D$ is closed under products, and relate the classes of all Mittag--Leffler, strict Mittag--Leffler, and separable modules. Then we prove that the class $\mathcal D$ is not deconstructible for any non--right perfect ring. So unlike the classes of all projective and flat modules, the class $\mathcal D$ does not admit the homotopy theory tools developed recently by Hovey . This gives a negative answer to (2).

math.RA↗
arXiv2004

On the cogeneration of cotorsion pairs

Let R be a Dedekind domain. Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP^+ implies that C is not cogenerated by a set whenever C is a cotorsion pair generated by a set which contains a non-cotorsion module.

math.LO↗
arXiv2000

Spectra of the Gamma-invariant of uniform modules

For a ring R, denote by Spec^R_kappa(Gamma) the kappa-spectrum of the Gamma-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that Spec^R_{aleph_1}(Gamma) is full for suitable von Neumann regular algebras R, but the techniques do not extend to cardinals kappa>aleph_1. By a direct construction, we prove that for any field F and any regular uncountable cardinal kappa there is an F-algebra R such that Spec^R_kappa(Gamma) is full. We also derive some consequences for the complexity of Ziegler spectra of infinite dimensional algebras.

math.LO↗