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Direct limits of modules of nite projective dimension Lidia Angeleri Hugel and Jan Trlifaj
 

Summary: Direct limits of modules of nite projective dimension
Lidia Angeleri Hugel and Jan Trlifaj 
Dedicated to Paul Eklof on his 60th birthday
Abstract
We describe in homological terms the direct limit closure of a class C of modules over
a ring R. We also determine the closure of the cotorsion pair C = (A; B) cogenerated by
C. As an application, we solve a problem of Fuchs and Salce on the structure of direct
limits of modules of projective dimension at most one over commutative domains. Then
we consider the case when R is a right coherent ring and C = P <1 , the class of all
nitely presented modules of nite projective dimension. If ndim R < 1 then C is a
tilting cotorsion pair induced by a tilting module T . We characterize closure properties
of A in terms of properties of T . Finally, we discuss an example where A is not closed
under direct limits.
Let R be a ring. Denote by P the class of all modules of nite projective dimension,
and by P <1 the class of all nitely presented modules in P. For n < ! let P n be
the class of all modules of projective dimension at most n, and let P <1
n
be the
corresponding subclass of P <1 .
In this paper, we study the categories lim ! P n and lim ! P <1

  

Source: Angeleri Hügel, Lidia - Dipartimento di Informatica, Universita` di Verona

 

Collections: Mathematics