Finite presentation

We put Z = G/S faithfully flat since G is identified with a principal homogeneous space under S(a)z over Z. Thus we can use EGA IV 2.7.1 to see that the morphism G Z is of finite presentation. This allows us to use SGA Expose V Proposition 9.1 to conclude that Z is of finite type over Spec(Z). Finally, we leave it to the reader to show that Z is separated. 

Par06] R. Parker, Short theorems with long proofs in finitely presented groups, Nikolaus Conference, 2006, www.math.rwth-aachen.de 8001/Nikolaus2006/prog.html... 

Pi OOF If E has a finite presentation, there exist finitely generated free A-modules L lnd F and an exact sequence... 

Finally, since A(l) has zero-dimensional and hence finite fibers and since A(l) > S is proper (being obtained by the base change S —> A from the map p A — A which is proper by [E.G. A. II 5.4. 3(i)]), it follows that A(l)—>S is finite [E. G. A. IV 8.11.1]. Since A(l) -> S is flat, finite and of finite presentation it is finite and locally free [E.G. A. IV 1.4.7], Finally the statement about the rank follows immediately from [22 6]. Later we shall have much more to say about Barsotti-Tate groups of this type which we denote by A or A(< ). 

Hence, since is of finite presentation it is flat, then it is locally-... 

V (G(n)VF ). The other case is of course handled in the same way. To prove c) we observe that Gfn] is certainly finite and of finite presentation over S. Therefore to conclude it is locally free it suffices to show it is flat. But this follows immediately from b) because we have a commutative... 

C. If R is a local ring and M is an /2-module, and either R is artinian or M is a finitely presented /2-module (i.e., of the type Rn/finitely generated submodule), then M flat over R implies M is a free /2-module. 

Lemma 2.2.8. Given f X — S of finite presentation such that fJ U is finite and Stale. The following conditions are equivalent ... 

It follows then from SGA 1 IX 5 3 applied to the morphism cp that n p s) is topologically of finite presentation as soon as the have this property. 

Finite generation (in fact finite presentation) of it f. By our assumption on the divisors and from the way the formal scheme is constructed (8.2.6), we have that =Spf B with B a complete local... 

Ti P s) the largest profinite quotient of itj (S) of order prime to p. Then is topologically of finite presentation. 

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