global dimension of a subring

Let S be a ring with identity and RS a subring, such that R is contained in the center of S. In this case S is a (left) R-module via multiplicationPlanetmathPlanetmath. Throughout by modules we will understand left modules and by global dimension we will understand left global dimension (we will denote it by gl dim(S)).

PropositionPlanetmathPlanetmath. Assume that gl dim(S)=n<. If S is free as a R-module, then gl dim(R)n+1.

Proof. Let M be a R-module. Then, there exists exact sequencePlanetmathPlanetmathPlanetmathPlanetmath


of R-modules, where each Pi is projective (module K is just a kernel of a map PnPn-1). We will show, that K is also projective (and since M is arbitrary, it will show that gl dim(R)n+1).

Since S is free as a R-module, then the extension of scalars (-RS) is an exact functorMathworldPlanetmathPlanetmath from the cateogry of R-modules to the categoryMathworldPlanetmath of S-modules. Furthermore for any projective R-module M, the S-module MRS is projective (in the category of S-modules). Thus we have following exact sequence of S-modules


where each PiRS is a projective S-module. But projective dimension of MRS is at most n (since gl dim(S)=n). Thus KRS is a projective S-module (please, see this entry ( for more details).

Note that the restriction of scalars functorMathworldPlanetmath also maps projective S-modules into projective R-modules. Thus KRS is a projective R-module. But S is free R-module, so


for some index setMathworldPlanetmathPlanetmath I. Finally we have


This shows, that K is a direct summand of a projective R-module KRR and therefore K is projective, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title global dimension of a subring
Canonical name GlobalDimensionOfASubring
Date of creation 2013-03-22 19:05:09
Last modified on 2013-03-22 19:05:09
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 13D05
Classification msc 16E10
Classification msc 18G20