By Daniel I.A. Cohen
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3]. But the morphism ˛ is generally nonaffine, and the finiteness of ˇ is an open question in this setting. We now characterize algebraic monoids among equivariant embeddings: Theorem 4. Let X be an equivariant embedding of a connected algebraic group G. Then X has a structure of algebraic monoid with unit group G if and only if the Albanese morphism ˛ W X ! X / is affine. Proof. In view of Proposition 16, it suffices to show that X is an algebraic monoid if ˛ is affine. X /. X / Š G=K equivariantly for the left (or right) action of G.
S; / (over k) is defined over F , or an algebraic F -semigroup, if S is an F -variety and the morphism is defined over F . F /. Note that an algebraic F -semigroup may well have no F -point; for example, an F -variety without F -point equipped with the trivial semigroup law l or r . But this is the only obstruction to the existence of F -idempotents, as shown by the following: Proposition 17. S; / be an algebraic F -semigroup. S / (viewed as closed subsets of S ) are defined over F . (ii) If S is commutative, then its smallest idempotent is defined over F .
C tNn 1 , where c 2 k. xy/ D 0 for all x; y 2 S . Thus, xy belongs to the fiber of n at 0, a finite set containing 0. , D 0 ; a contradiction. Thus, we must have n D 2, and we obtain a nonconstant morphism D 2 W S ! A1 , where the semigroup law on A1 is the multiplication. The image of contains 0 and a nonempty open subset U of the unit group Gm . Then U U D Gm and hence is surjective. e/ D 1. Then e is the desired nonzero idempotent. t u Remark 10. One may also deduce the above theorem from the description of algebraic semigroup structures on abelian varieties (Proposition 21), when the irreducible curve S is assumed to be nonsingular and nonrational.
Basic techniques of combinatorial theory by Daniel I.A. Cohen