Global dimension of valuation rings

Abstract

pd(M) = n t (Vg)[kernel dn_ is projective], (id(M) = n (V.2)[image An is injective]). The right global dimension of R, gl.d(R), is given by gl.d(R) = sup{pd(M) M e R}. The Global Dimension Theorem (Auslander [1]) enables one to look only at dimensions of right ideals of R or of cyclic right R-modules, for gl.d(R) = sup{pd(R/I) I a right ideal of R} = 0 or 1 + sup{pd(I) I a right ideal of R}. This global dimension, defined in terms of projective resolutions of modules, turns out to be definable also in terms of injective resolutions, for we have gl.d(R) = sup{id(M) M e R}. (See [4, p. 111].) In general, there is no analogy to the Global Dimension Theorem in terms of injective dimensions of cyclic modules or right ideals, although if R is