Lemma 4

Let k be a field, K an algebraic extension of k , B a k-algebra whose spectrum is connected. For every canonical factor C of B_{(K)} , one has \dim(C) = \dim(B) .

Let us first treat the case where the extension K is of finite degree over k . The B-module C is a direct factor of a free module, hence finitely generated projective. Since the function p \mapsto \mathrm{rg}_p C is constant on Spec(B) (II, § 5, No. 3), the support of the B-module C is equal to Spec(B) , so that the B-module C is faithfully flat. This implies \dim(C) \geq \dim(B) (VIII, § 2, No. 1, prop. 2), hence the required equality since one has \dim(C) \leq \dim(B_{(K)}) = \dim(B) (VIII, § 2, No. 4, cor. to prop. 5).

Let us pass to the general case. Let e be the idempotent element of B_{(K)} such that C = B_{(K)}/B_{(K)}e . There exists a subextension K' of K of finite degree such that e is the image of an idempotent element e' of B_{(K')} . Put C' = B_{(K')}/B_{(K')}e' . The ring C' \otimes_{K'} K is identified with C , and C' is a canonical factor of B_{(K')} . One has \dim(C') = \dim(B) by the case already treated, and \dim(C') = \dim(C) by loc. cit. , whence the lemma.