Classifying subcategories of modules
Mark Hovey
Wesleyan University
Middletown, CT 06459
mhovey@wesleyan.edu
AMS classification: 13C05, 13D30, 18G35, 55U35
In this paper, we classify certain subcategories of modules over a ring
R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod
that is closed under extensions. We claim that these wide subcategories
are analogous to thick subcategories of the derived category D(R).
Indeed, let C_0 denote the wide subcategory generated by R; C_0 is the
collection of all finitely presented modules precisely when R is
coherent. When R is a quotient of a regular commutative coherent ring
by a finitely generated ideal, we classify wide subcategories of C_0.
In fact, they are on 1-1 correspondence with thick subcategories of
small objects of D(R). The proof relies heavily on Thomason's thick
subcategory theorem for D(R).
We also classify wide subcategories closed under arbitrary coproducts;
these are analogous to localizing subcategories of D(R). In this case,
we must assume that R is Noetherian, where we use Neeman's
classification of localizing subcategories of D(R).