Separable Algebras

Proposition 1.5. The stack Agj is a separated algebraic stack of finite type over Spec(Z). 

The stack Agj is a separated algebraic stack of finite type over Spec(Z). This can be proved using the theory of Hilbert schemes as is done in [DM] 5 in the case of the stack classifying stable curves of genus g. 

Corollary 1.3. The stack S(g,p) is a separated algebraic stack of finite type over Spec(Z). Proof. We know that Ag is a separated algebraic stack of finite type over Spec(Z) (see for example [FC] I 4.11). However, it is easy to see that any stack which allows a representable morphism to an algebraic stack is an algebraic stack itself. Hence S(g,p) is an algebraic stack by Proposition 1.2. In the same manner we see that S(g>p) is separated and of finite type over Spec(Z). O... 

II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part... 

Corollary. Subalgebras, quotients, products, and tensor products of separable algebras are separable. 

Since separable algebras over k all look basically the same over ks, classifying them is a descent problem of the type we will study more generally in Chapter 17. But since usual Galois theory already classifies separable fields,... 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

Then in. .., Xr] we have fh1h2 = glg2.Asfis irreducible there, it divides either g2 or g2, and that factor therefore has at least as high a degree in Xj. Thus/is a minimal equation for xt over k(x2,..., xr). It involves X2 to some power not divisible by p, so it is separable. Thus L is separable algebraic over Ll = k(x2,. .., x ). By induction Lt is separable algebraic over some pure transcendental E, and L then is so also. ... 

Scheme 44 Schur s lemma 63 Semi-direct product 19 Semi-invariant element 34 Semisimple group 97 Separable algebra 47 Separable matrix 54 Sheaf 43...

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