Kaplansky classes, finite character and ℵ1-projectivity
-
Jan Šaroch
and
Jan Trlifaj
Abstract.
Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class 
of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair 
is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case 
for a class 
of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open).
If is an AEC of roots of Ext, then is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class 
of all ℵ1-projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class 
equals the class of all flat modules and consists of modules of projective dimension 
. Assuming the Singular Cardinal Hypothesis, we prove that is not precovering for each countable non-right perfect ring R.
© 2012 by Walter de Gruyter Berlin Boston
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- Masthead
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- Higher integrability in parabolic obstacle problems
- Fundamental solutions for sum of squares of vector fields operators with C1,α coefficients
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Articles in the same Issue
- Masthead
- Conjugacy in normal subgroups of hyperbolic groups
- Surgery obstructions on closed manifolds and the inertia subgroup
- Higher integrability in parabolic obstacle problems
- Fundamental solutions for sum of squares of vector fields operators with C1,α coefficients
- Kuykian fields
- Multivariable manifold calculus of functors
- Weighted geometric means
- Kaplansky classes, finite character and ℵ1-projectivity
