Closed embedding

Proof Let U C X be an open subset over which E is trivial. Then with respect to suitable local trivialisations over U the restriction of is the closed embedding... 

So is a closed embedding. We can see immediately that the image of the open subvariety... 

This is the graph of a morphism of affine schemes, hence it is a closed embedding. 

Let G be a finite group scheme. Show there is a closed embedding of G into the group scheme of units of k[G]°. If G is of multiplicative type, show this embeds G in a torus. 

In any closed embedding of G in GL , some element of GL (fc) conjugates G to a closed subgroup of the strict upper triangular group U . 

Corollary. A homomorphism G- H induces a Lie algebra map, injective if G-> H is a closed embedding. 

Theorem. Let F- G be a homomorphism of affine group scheme over afield. If the kernel is trivial, the map is a closed embedding. 

Cartan subgroup 77 Cartier duality 17 Center of a group scheme 27 Central simple algebra 145 Character 14 Charater group 55 Clopen set 42 Closed embedding 13 Closed set, closure 156 Closed set in k 28 Closed set in Spec A 42... 

Recall that S is a Noetherian scheme of finite dimension. Let z Z —> S be a closed embedding and j U —> S be the complimentary open embedding. For any simplicial sheaf we have a canonical commutative square in the simplicial homotopy category of the form... 

Proposition 2.24. — Let i Z be a closed embedding of smooth schemes over S. Then the tuuo morphisms and ax,z aTe A -weak equivalences. 

To prove the general case we proceed as follows. Fisrt of all since Z — X is a closed embedding of smooth schemes there exists a finite Zariski open covering X = UU, such that for any i the embedding ZPlU, —> U, satisfies the condition of Lemma 2.28. Note also that if this condition holds for Z —> X it also holds for Z fl LT —> U where U is any open subset of X. In particular, it holds for all intersections of the form... 

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