Meaning of group scheme | Babel Free

Definitions

A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.

Examples

“It is instructive to give a few explicit examples of such^([affine]) group schemes:”

“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;.”

“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.”

CEFR level