Nakayama’s lemma

Fix a point s i 5, let e e(s) and let a. ..,2 3 be such that their images in ys8k(s) form a basis. From Nakayama s lemma il follows that the homomorphism f.Q - defined by av...a, is suriective therefore there is an open neighborhood U of s to which f extends defining a surjective morphism f Oy8 —> Djj. With a similar argument applied to keKf) we may find an affine open neighborhood IXs) of s contained in U and an exact sequence... 

The last morphism is an isomorphism, and Tor,(6,It) (0) because 6 is A-flat this follows immediately from the flatness of A — B and of B — 6, and from the faithful flatness of A — A. Hence k - (0). Applying Nakayama s lemma as before we deduce that I = (0), equivalently that... 

Note that we have been able to apply Nakayama s lemma to the a priori not finitely generated A-moduie cokertp) because m is nilpotent... 

Since H1(Koa) = (0) because Fov. ... Fq is a regular sequence, t is surjective from Nakayama s lemma it follows that H K ) = (0), whence the assertion follows. 

Since the sheaf k V (u) is locally free, from Nakayama s lemma it follows that i and 3Z are isomorphisms this proves the claim. 

The last assertion follows from the fact that, by Nakayama s lemma, J/(XXMO) iff J - (0). 

A finite group scheme G over arbitrary fc is called etale if Q gi = 0. Show that G is etale if the base-change Gt/n is etale for all maximal ideals Af of k. [See (13.2) and Nakayama s lemma.]... 

Induction now shows that we can take any polynomial in the f with coefficient in (p0(A) and reduce it to have all exponents less than q. Hence A is a finitely generated module over B = cp0 A). This implies first of all that under A - B the dimension cannot go down. But since G is connected, A modulo its nilradical is a domain (6.6), and from (12.4) we see then that the kernel of k. Let M be the kernel, a maximal ideal of B. As B injects into A, we know BM injects into AM, and thus Am is a nontrivial finitely generated BM-module. By Nakayama s lemma then MAm Am, and so MA A. Any homomorphism x A- A/MA - fc then satisfies q>(x) = y. ... 

Nakayama s Lemma. Let R be a local ring with maximal ideal /, and M a finitely generated R-module. If I M = M, then M = 0. 

A basic tool in dealing with coherent modules is Nakayama s lemma which we want to recall in several forms here ... 

This corollary shows a very significant and far-reaching thing that the vector spaces mxfm2x, which a priori axe a collection of unrelated vector spaces, one for each closed point x, can all be derived from one coherent sheaf QX/k on X. This enables us a) to use the machinery of Nakayama s lemma ( 2), and b) to handle non-closed points. 

Note that d is upper-semi-continuous by version II of Nakayama s lemma. Now assume that X is a variety and n = dimX. Then... 

If h G A generates A/m- A over k, then by Nakayama s lemma, h generates A. Therefore A is a principal ideal. By Lemma 1, this shows that O is a UFD. ... 

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