Jan Šaroch

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 also show 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 \kappa -condensed abelian groups for every uncountable cardinal \kappa , but that this scenario, in turn, is inconsistent with the existence of a strongly compact cardinal.

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 Filt ⁡ (

We show how to construct discretely ordered principal ideal subrings of Q [ x ] \mathbb {Q}[x] with various types of prime behaviour. Given any set D \mathcal D consisting of finite strictly increasing sequences ( d 1 , d 2 , … , d l ) (d_1,d_2,\dots , d_l) of positive integers such that, for each prime integer p p , the set { p Z , d 1 + p Z , … , d l + p Z } \{p\mathbb {Z}, d_1+p\mathbb {Z},\dots , d_l+p\mathbb {Z}\} does not contain all the cosets modulo p p , we can stipulate to have, for each ( d 1 , … , d l ) ∈ D (d_1,\dots , d_l)\in \mathcal D , a cofinal set of progressions ( f , f + d 1 , … , f + d l ) (f, f+d_1, \dots , f+d_l) of prime elements in our principal ideal domain R τ R_\tau . Moreover, we can simultaneously guarantee that each positive prime g ∈ R τ ∖ N g\in R_\tau \setminus \mathbb {N} is either in a prescribed progression as above or there is no other prime h h in R τ R_\tau such that g − h ∈ Z g-h\in \mathbb {Z} . Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring Z ^ \widehat {\mathbb {Z}} of profinite integers.