Dictionary

group scheme en · NOUN

Meanings

1. (qualifier:scheme theory) A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.
   • Every finite flat group scheme of order p over S is of the form Gᴸ#95;#123;a,b#125; for some triple (L,a,b), and G#123;L#125;#95;#123;a,b#125; is isomorphic to G#123;L#95;1#125;#95;#123;a#95;1,b#95;1#125; if and only if there exists an isomorphism from L to L#95;1 carrying a to a#95;1 and b to b#95;1. 2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:
   • It is instructive to give a few explicit examples of such^([affine]) group schemes: 1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:
   • We will tend to use boldface characters, such as #92;mathbf#123;G#125;, for algebraic groups over k, as also for group schemes over #92;mathfrak#123;X#125; that are obtained from groups over k. A quintessential example is #92;mathbf#123;G#125;#95;#92;mathfrak#123;X#125;#61;#92;mathbf#123;G#125;#92;times#95;#123;k#125;#92;mathfrak#123;X#125;. For arbitrary group schemes, or more generally group functors, over #92;mathfrak#123;X#125; we shall tend to use german characters, such as #92;mathfrak#123;G#125;. 2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:

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