Quotient Maps

We call a homomorphism F - G a quotient map ii k[G] - fc[F] is injective. Clearly this property is unaffected by extension of the base field. For matrix groups it is easy to see what it means ... 

Finally, we already know several properties preserved under passage to quotient. If for instance F is connected and F - G is a quotient map, then G is connected, since n0 k[G] n0 k[F], If k[F] has enough homomorphisms to... 

In this case we see that quotients have the meaning one would naively expect. But this is a substantial theorem, and definitely fails for kfrk. The squaring homomorphism G - Gm, for instance, is a quotient map, but not every element in k need be a square. We will later investigate the way in which a quotient map can fail to be surjective. First, however, we fill in another gap in our earlier material. 

Theorem. Let F - G be a quotient map with kernel N. Then any homomorphism F - H vanishing on N factors through G. 

Corollary. If F -+G and F - G are quotient maps with the same kernel, then G G. ... 

The last result confirms that we have the right concept of quotient, but its functor meaning is still obscure, since a quotient map F G need not map F(k) onto G(lt). By (15.2) however we do know that each element of G(k) is the image of some point in F(fc) in other words, it does appear in the image, but only after we have made some reasonable extension of k. We now show that a similar statement holds for the functor as a whole. 

Theorem. Let F G be a homomorphism of affine group schemes over a field k. It is a quotient map iff it has the following property ... 

A quotient map with finite kernel is called an isogeny it is a separable (reap. purely inseparable) isogeny if the kernel is etale (resp. connected). Prove ... 

Theorem. Let G be an affine group scheme over a field. Let N be a closed normal subgroup. Then there is a quotient map G - H with kernel precisely N. 

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