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Jan Šaroch

Publications and source records attributed to Jan Šaroch.

17 recordsLinked to original sources
arXiv2025

Whitehead's problem and condensed mathematics

One of the better-known independence results in general mathematics is Shelah's solution to Whitehead's problem of whether $\mathrm{Ext}^1(A,\mathbb{Z})=0$ implies that an abelian group $A$ is free. The point of departure for the present work is Clausen and Scholze's proof that, in contrast, one natural interpretation of Whitehead's problem within their recently-developed framework of condensed mathematics has an affirmative answer in $\mathsf{ZFC}$. We record two alternative proofs of this result, as well as several original variations on it, both for their intrinsic interest and as a springboard for a broader study of the relations between condensed mathematics and set theoretic forcing. We show more particularly how the condensation $\underline{X}$ of any locally compact Hausdorff space $X$ may be viewed as an organized presentation of the forcing names for the points of canonical interpretations of $X$ in all possible set-forcing extensions of the universe, and we argue our main result by way of this fact. We show also that when interpreted within the category of light condensed abelian groups, Whitehead's problem is again independent of the $\mathsf{ZFC}$ axioms. In fact we show that it is consistent that Whitehead's problem has a negative solution within the category of $κ$-condensed abelian groups for every uncountable cardinal $κ$, but that this scenario, in turn, is inconsistent with the existence of a strongly compact cardinal.

math.LO↗
arXiv2024

Deconstructible abstract elementary classes of modules and categoricity

We prove a version of Shelah's Categoricity Conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $\mathcal{A}$ is a deconstructible class of modules that fits in an abstract elementary class $(\mathcal{A},\preceq)$ such that (1) $\mathcal{A}$ is closed under direct summands and (2) $\preceq$ refines direct summands, then $\mathcal{A}$ is closed under arbitrary direct limits. In an Appendix, we prove that the assumption (2) is not needed in some models of ZFC.

math.RT↗
arXiv2023

The cotorsion pair generated by the Gorenstein projective modules and $λ$-pure-injective modules

We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective modules over a ring $R$, then $\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair. Moreover, $\mathfrak{GP}$ is complete when all projective modules are $λ$-pure-injective for some infinite regular cardinal $λ$ (in particular, if $R$ is right $Σ$-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals. We also thoroughly study $λ$-pure-injective modules for an arbitrary infinite regular cardinal $λ$, proving along the way that: any cosyzygy module in an injective coresolution of a $λ$-pure-injective module is $λ$-pure-injective; the cotorsion pair cogenerated by a class of $λ$-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set. Finally, assuming the set-theoretic hypothesis that $0^\sharp$ does not exist, we prove that the category of right $R$-modules has enough $λ$-pure-injective objects if and only if the ring $R$ is right pure-semisimple. This, in turn, follows from a rather surprising result that $λ$-pure-injectivity amounts to pure-injectivity in the absence of $0^\sharp$.

math.RT↗
arXiv2023

Enochs Conjecture for cotorsion pairs and more

Enochs Conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes $\mathrm{Filt}(\mathcal S)$ where $\mathcal S$ consists of $\aleph_n$-presented modules for some fixed $n<ω$. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of $\aleph_n$-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs Conjecture holds for all classes of the form $\mathrm{Filt}(\mathcal{S})$ where $\mathcal{S}$ is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that the Enochs Conjecture holds (possibly under some additional set-theoretic assumption).

math.RA↗
arXiv2022

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

We show how to construct discretely ordered principal ideal subrings of $\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ of positive integers such that, for each prime integer $p$, the set $\{p\mathbb Z, d_1+p\mathbb Z,\dots, d_l+p\mathbb Z\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\dots, d_l)\in \mathcal D$, a cofinal set of progressions $(f, f+d_1, \dots, f+d_l)$ of prime elements in our principal ideal domain $R_τ$. Moreover, we can simultaneously guarantee that each positive prime $g\in R_τ\setminus\mathbb N$ is either in a prescribed progression as above or there is no other prime $h$ in $R_τ$ such that $g-h\in\mathbb Z$. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring $\hat{\mathbb{Z}}$ of profinite integers.

math.NT↗
arXiv2021

Enochs' conjecture for small precovering classes of modules

Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e. the classes of the form $\mathrm{Add}(M)$ where $M$ is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that $M$ has a perfect decomposition if $\mathrm{Add}(M)$ is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an $n<ω$ such that $M$ decomposes into a direct sum of $\aleph_n$-generated modules.

math.RA↗
arXiv2021

Pure semisimplicity conjecture and Artin problem for dimension sequences

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding $F\hookrightarrow M_n(G)$ exists provided that $n<5$. As a byproduct, we obtain a division ring extension $G\subseteq F$ such that the bimodule ${}_GF_F$ has the right dimension sequence $(1,2,2,2,1,4)$. Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.

math.RA↗
arXiv2020

Finitistic Dimension Conjectures via Gorenstein Projective Dimension

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander--Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin algebra, and the converse holds for Artin algebras over which the class $\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective modules) is contravariantly finite.

math.RT↗
arXiv2019

Test sets for factorization properties of modules

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property w.r.t. a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring $R$ and on additional set-theoretic hypotheses. For $R$ commutative noetherian of Krull dimension $0 < d < \infty$, we show that the assertion `projectivity is a factorization property w.r.t. a single epimorphism' is independent of ZFC + GCH. We also show that if $R$ is any ring and there exists a strongly compact cardinal $κ> |R|$, then the category of all projective modules is accessible.

math.RA↗
arXiv2018

Singular compactness and definability for $Σ$-cotorsion and Gorenstein modules

We introduce a general version of singular compactness theorem which makes it possible to show that being a $Σ$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

math.RT↗
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 tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [14], and it does not provide for approximations when $R$ has cardinality $\leq \aleph_0$, [6]. We remove the cardinality restriction on $R$ in the latter result. We also prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair $(\mathcal A,\mathcal B)$ is of countable type whenever the class $\mathcal B$ is closed under direct limits. In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the facts above to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs' problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [24]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [20].

math.RA↗
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↗
arXiv2015

On the non-existence of right almost split maps

We show that, over any ring, a module $C$ is a codomain of a right almost split map if and only if $C$ is a finitely presented module with local endomorphism ring; thus we give an answer to a 40 years old question by M. Auslander. Using the tools developed, we also provide a useful sufficient condition for a class of modules to be non-precovering. Finally, we show a non-trivial application in the general context of morphisms determined by object.

math.RT↗
arXiv2014

Quasi-Euclidean subrings of Q[x]

Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of Q[x] which are not k-stage Euclidean for any norm and positive integer k. These subrings can be either PID or non-UFD, depending on the choice of parameters in our construction. In both cases, there are 2^ω such domains up to ring isomorphism.

math.RA↗
arXiv2014

$Σ$-algebraically compact modules and $\mathbf L_{ω_1ω}$-compact cardinals

We prove that the property Add$(M)\subseteq$ Prod$(M)$ characterizes $Σ$-algebraically compact modules if $|M|$ is not $ω$-measurable. Moreover, under a large cardinal assumption, we show that over any ring $R$ where $|R|$ is not $ω$-measurable, any free module $M$ of $ω$-measurable rank satisfies Add$(M)\subseteq$ Prod$(M)$, hence the assumption on $|M|$ cannot be dropped in general (e.g. over small non-right perfect rings). In this way, we extend results from a recent paper by Simion Breaz.

math.LO↗