By Hardy G.H., Wright E.M.

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This is the second notion of dimension we referred to earlier - the geometric one. 9 Theorem. If V is the germ of a holomorphic variety and the dimension of V is equal to the Krull dimension of V H. V H is its local ring, then Proof. Note that it follows from the previous theorem that if W ⊂ W are germs of irreducible subvarieties of V and dim W − dim W ≥ 2 then there is another irreducible subvariety W which is properly contained in W and properly contains W . Indeed, there must be a germ f ∈ V H which belongs to id W but not to id W .

Proof. We first show that C is actually a finitely generated B-module. The fact that C is finitely generated as an A-algebra means that it is also finitely generated as a B-algebra and, hence, that every element of C is a polynomial in a finite set of generators c1 , . . , ck with coeficients in B. However, the fact that C is integral over B means that for each c ∈ C there is an integer nc such that every polynomial in c is equal to one of degree less than or equal to nc . Thus, the algebra generated over B by c1 is a finitely generated B-module.

If there is a n − k-dimensional linear subspace L with V ∩ L = 0 then choose coordinates for which L = {z ∈ Cn : z1 = · · · = zk = 0}. 21 there are elements fj ∈ j H ∩ id V for j = k + 1, . . n such that fj is regular in zj . But then after some linear change of variables in the first k coordinates we may assume that id V is regular in zm+1 , . . , zn for some m ≤ k. 2 it follows that m is necessarily dim V . Hence, dim V ≤ k. On the other hand, if k = dim V let V = ∪i Vi be the decomposition of V into irreducibles and.