Projective modules over semilocal rings

On Semilocal Modules and Rings

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

On Projective Modules over Semi-hereditary Rings

This theorem, already known for finitely generated projective modules[l, I, Proposition 6.1], has been recently proved for arbitrary projective modules over commutative semi-hereditary rings by I. Kaplansky [2], who raised the problem of extending it to the noncommutative case. We recall two results due to Kaplansky: Any projective module (over an arbitrary ring) is a direct sum of countably ge...

Tits Indices over Semilocal Rings

We present a simplified version of Tits’ proof of the classification of semisimple algebraic groups, which remains valid over semilocal rings. We also provide explicit conditions on anisotropic groups to appear as anisotropic kernels of semisimple groups of a given index.

Projective modules over polynomial rings and dynamical Gröbner bases

Modules over Projective Schemes

Definition 1. Let S be a graded ring, set X = ProjS and letM a graded S-module. We define a sheaf of modulesM ̃ on X as follows. For each p ∈ ProjS we have the local ring S(p) and the S(p)module M(p) (GRM,Definition 4). Let Γ(U,M ̃) be the set of all functions s : U −→ ∐p∈U M(p) with s(p) ∈M(p) for each p, which are locally fractions. That is, for every p ∈ U there is an open neighborhood p ∈ V ⊆...