13 - Tori Acting on Schemes

Summary

Schemes with an action of a torus arise frequently in the theory of algebraic groups. In this chapter, we prove the basic theorems concerning such actions. In particular, we prove the Białynicki-Birula decomposition (13.47), which will allow us to show that the Bruhat decomposition exists on the level of scheme

Throughout, all schemes are algebraic over the field k. Recall that all algebraic groups are affine.

The smoothness of the fixed subscheme

Recall that tori are linearly reductive.

THEOREM 13.1. Let G be a linearly reductive group variety acting on a smooth variety X over k. Then the fixed-point scheme X_Gis smooth.

We shall need to use some basic results on regular local rings.

Let A be a local ring with maximal ideal m and residue field. Let d denote the Krull dimension of A. Every set of generators for m has at least d elements. If there exists a set with d elements, then A is said to be regular, and a set of generators with d elements is called a regular system of parameters for A (Matsumura 1986, p. 105).

(a)A local ring A is regular if and only if the canonical map

is an isomorphism (Matsumura 1986, 14.4).

(b) Assume that A is regular. Let be a regular system of parameters for A, and let for some. Then is local of dimension its maximal ideal is generated by, and so is regular. Every regular quotient of A is of this form (Matsumura 1986, 14.2).

We require several lemmas.

LEMMA 13.3. Let A be a regular local ring of dimension d and m the maximal ideal in A. Let a be an ideal in A, and let. If, for every, there exists a regular system of parameters for A such that

then is regular (of dimension).

PROOF. Let, and let n denote the maximal ideal of B.