By Emil Grosswald, Marvin Isadore Knopp, Mark Sheingorn
Emil Grosswald used to be a mathematician of significant accomplishment and memorable breadth of imaginative and prescient. This quantity will pay tribute to the span of his mathematical pursuits, that's mirrored within the wide variety of papers amassed right here. With contributions through best modern researchers in quantity concept, modular services, combinatorics, and comparable research, this publication will curiosity graduate scholars and experts in those fields. The prime quality of the articles and their shut connection to present examine traits make this quantity a needs to for any arithmetic library
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Extra resources for A Tribute to Emil Grosswald: Number Theory and Related Analysis
This proves the first assertion. The second assertion is proved by the argument of Proposition 14 (ii); note that any two-sided ideal of Maff is stable under conjugation by G, in view of (ii) above. t u Example 5. Let n be a positive integer, n the group scheme of nth roots of unity, and A an abelian variety containing n as a subgroup scheme (any ordinary elliptic curve will do). A2 ; /, and H the unit subgroup scheme of N ; then H Š n Gm . Next, let G WD A Gm ; this is a connected commutative algebraic group containing H as a subgroup scheme.
K/. V ˝ W / D W ˝ W as subspaces of V ˝ V . K/, this yields the desired equality. 32 M. Brion Since Y is associative on the dense subscheme K, it is associative everywhere; likewise, Y admits 1K as a neutral element. Thus, Y is an algebraic monoid law on Y . We may now form the induced monoid G K Y as in Sect. 2, to get the desired structure on X . 5 Algebraic Semigroups and Monoids over Perfect Fields In this subsection, we extend most of the above results to the setting of algebraic semigroups and monoids defined over a perfect field.
Moreover, by Lemma 2, the image of the morphism is exactly the kernel of S ; this is a simple algebraic semigroup in view of Proposition 5. One may thus deduce part of Theorem 6 from the structure of simple algebraic semigroups presented in Remark 3 (i). Yet we will provide a direct, selfcontained proof by adapting the arguments of Proposition 5. Proof of Theorem 6. One readily checks that the map (resp. ) as in the statement yields an algebraic semigroup structure on X G Y (resp. on S ); compare with Example 1 (ii).