Group-like element

Theorem. Characters of an affine group scheme G represented by A correspond to group-like elements in A. 

Lemma. If A is a Hopf algebra over a field k, the group-like elements in A are linearly independent. 

Ilieorein. Forming G° commutes with base change, and G°(R) group-like elements in A R = Hom(GK, (Gm)R). 

Let k be a ring with nontrivial idempotents. Show that group-like elements in a Hopf algebra over k need not be linearly independent. 

The construction of A, commutes with base extension, since n0(CD L) = n0(CD) L. Hence to prove G, is of multiplicative type and G unipotent we may assume k = k. Then each CD/Rad CD is a product of copies of k, and the homomorphisms to k are group-like elements spanning C,. Thus A, is spanned by group-likes, and G, is diagonalizable. Also, any group-like b in C defines a homomorphism CD - k such a homomorphism vanishes on the radical, so b is in C,. Thus the other tensor factor of A, representing G , has no nontrivial group-likes. Hence by the previous corollary G is unipotent. ... 

Also the set of "group-like" elements in Bv is identical with G (A). 

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