By Michael Spivak

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**Extra resources for A Comprehensive Introduction to Differential Geometry, Vol. 5, Third Edition**

**Sample text**

E. F ∈ AY [t]. ), so that R(f ) = 0 and hence R = 0 by minimality of deg(F ). It follows that AX is a free AY -module of rank d = deg(f ). e. for f ∈ AX the image of the basic open set D(f ) by ϕ is open. Let Φ = td + fd−1 td−1 + · · · + f0 ∈ AY [t] be the characteristic polynomial of multiplication by f on the free AY -module AX . We ∪ show that ϕ(D(f )) = D(fi ). On the one hand, if P is a maximal ideal of AX not containing f (this corresponds to a point of D(f )), then P does not contain all the fi , for otherwise the equation Φ(f ) = 0 (Cayley-Hamilton theorem) would imply f d ∈ P and hence f ∈ P by primeness of P , a contradiction.

Closed orbit lemma) If Y is aﬃne or projective1, an orbit of minimal dimension is closed. Proof. Let OP be such an orbit, Z its closure. Then Z is the union of orbits of G, because if Q ∈ Z has an open neighbourhood UQ containing P ′ ∈ OP , then the open neighbourhood gUQ of gQ contains gP ′ . By the lemma Z \OP is a closed subset. It does not contain any irreducible component of Z, because Z is the union of the closures of the irreducible components of OP which are themselves irreducible. 4 applied to each irreducible component of Z we thus get that Z \OP is a union of orbits of smaller dimension, and hence must be empty.

5. If P is a smooth point on a variety X, then the local ring OX,P is a unique factorisation domain. Proof. Recall that we have identiﬁed TP (X) with the dual k-vector space of MP /MP2 , where MP is the maximal ideal of OX,P . It follows that P is a smooth point if and only if dim k MP /MP2 = dim X = dim OX,P . g. 3). There is also a direct proof of the special case we need which goes back to Zariski. It proceeds by comparing OX,P with its completion which is a power series ring, hence a UFD; one shows that the UFD property ‘descends’ from the completion to OX,P .