Descent theory

1 If X, = Nerve(gi) (A) for some faithfully flat morphism g of S-schemes, then A, is a simplicial groupoid. 

S-schemes and Z, is a simplicial groupoid, then we have A, is also a simplicial groupoid. 

We prove the only if part. As X, is a simplicial groupoid, there is a faithfully flat S -morphism g Z and a faithfully flat cartesian morphism f, Z, X, of simplicial S -schemes, where Z, = Nerve(3) (A). It is easy to see that (do) i (A) is nothing but the base change by g, and it 

Corollary 10.9. Let the notation he as in the lemma. Then there is a func-torial isomorphism 

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This refined version of condition (1) in the previous theorem is not needed now but will be crucial in the descent theory of Part V. 

Let G be a fixed affine group scheme, A = k[G]. The structure of principal homogeneous space X for G is one to which descent theory applies. If kpf] = N, the structure on N is given by a multiplication N N - N and an action map N-> A N, the axioms are that certain diagrams commute, that N has a map to some faithfully flat k-algebra, and that a certain map... 

Quadratic forms are another type of structure uncomplicated over algebraically closed fields, and descent theory can be applied to them. Even in the simplest case we can see some interesting results. On k2 let Q be the quadratic form Q(xet + ye2) = xy. It is a fact that over k every nondegenerate rank 2 quadratic form looks like Q, even in characteristic 2. Such forms are therefore classified by Aut(Q)). 

The following well-known theorem in descent theory contained in [33] is now easy to prove. 

Proof. — This follows immediately from the standard etale descent theory for vector bundles cind our definitions. 

The last part of this chapter shows that in the public sphere, the periodic system was connected to questions concerning the understanding of nature, its internal connections, and the descent theory. After 1910 the situation regarding the periodic system was changing rapidly as a result of the new understanding of atomic theory and the law of Henry Moseley (1887-1915). This time period, however, is not discussed in this chapter. 

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