K0 of a semilocal ring

Abstract

Let R be a semilocal ring, that is, R modulo its Jacobson radical J R is artinian. Then K 0 R/J R is a partially ordered abelian group with order-unit, isomorphic to Z n ≤ u , where ≤ denotes the componentwise order on Z n and u is an order-unit in Z n ≤ . Moreover, the canonical projection π: R → R/J R induces an embedding of partially ordered abelian groups with order-unit K 0 π : K 0 R → K 0 R/J R . In this paper we prove that every embedding of partially ordered abelian groups with order-unit G → Z n can be realized as the mapping K 0 π : K 0 R → K 0 R/J R for a suitable hereditary semilocal ring R.

Direct-sum decompositions of modules with semilocal endomorphism rings

“…Moreover, if M R is the direct sum of the modules in V(C), E is the endomorphism ring End(M R ), S E denotes the full subcategory of Mod-E consisting of finitely generated projective right E-modules with semilocal endomorphism ring, and C is viewed as a full subcategory of Mod-R, then the categories C and S E turn out to be equivalent via the functors Hom R (M R , −) : C → S E and − ⊗ E M : S E → C. In particular, the monoids V(C) and V(S E ) are isomorphic. The main theorem of this paper states that every reduced Krull monoid arises in this fashion, as V(S R ) for a suitable ring R. Thus reduced Krull monoids coincide both with the monoids that can be realized as V(C) for some class C of modules satisfying the three properties above, and with the monoids that can be realized as V(S R ) for some ring R. To compare our result with the earlier realization theorems, we note that the classes C in [13,16] that represent a given positive normal monoid do indeed satisfy the three properties above, and that the positive normal monoids are exactly the finitely generated reduced Krull monoids (see Proposition 1.4).…”

“…(These monoids have also been called Diophantine monoids in the literature.) The first author and Herbera [13] showed, conversely, that given a positive normal monoid M, there is semilocal ring R such that M ∼ = V(proj-R). Thus the direct-sum behavior of finitely generated projective modules over a semilocal ring is precisely delineated by the structure of positive normal monoids.…”

Classifying Projective Modules Over Some Semilocal Rings

“…If P is finitely generated projective, then P is projective cover of P/ J(P ), hence P is uniquely determined by its dimension vector. The set of all tuples dim(P ), where P runs over finitely generated projective R-modules, is a subsemigroup of N n called a finite projective spectrum of R. Facchini and Herbera [6] proved that finite projective spectra of semilocal rings can be described as semigroups G ∩ N n with order units, where G is a subgroup of Z n . It may be difficult to calculate the finite projective spectrum of a particular semilocal ring.…”

Subdirect Products of Preadditive Categories and Weak Equivalences

“…Thus V(−) ("taking the isomorphism classes of objects of the category") is a metafunctor from the metacategory of all additive categories to the metacategory of all large commutative monoids [11, p. 7-14] (or, to avoid set theoretic difficulties, a functor from the category of all small additive categories to the category of all commutative monoids). The first author has been studying for the last years the monoid V(A) for an additive category A; see, e.g., [1,3,4,5,6,7,8]. In particular, he studied properties of the monoid V(A) that can be lifted to properties of the corresponding category A ( [5] and [6]).…”