Dimension

Abstract

As we know, any hypersurface of An or Pn has transcendent dimension n- 1 (see Exercise 10.1.20 ). As a first result of this chapter, we invert this result. We start with the following: Let V, W be quasi-projective varieties with W⊆ V. Then dim tr(W) ≤ dim tr(V). If, in addition, W is closed in V and dim tr(W) = dim tr(V), then V= W. It suffices to reduce to the case in which V and W are affine. Then we may assume W⊆ V⊆ An, so that A(V) and A(W) are generated, as K -algebras, by x1, …, xn. Let m= dim tr(V). Then any (m+ 1 ) -tuple (xi1,…,xim+1) of elements of { x1, …, xn} is algebraically dependent. This implies that there is a non-zero polynomial F∈K[Ti1,…,Tim+1], such that F(xi1,…,xim+1)=0 in A(V).