group scheme

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group scheme

Wikipedia

Noun

group scheme (plural group schemes)

1. (category theory, 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.
   • 1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:

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

   • 2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:

     We will tend to use boldface characters, such as ${\displaystyle \mathbf {G} }$, for algebraic groups over ${\displaystyle k}$, as also for group schemes over ${\displaystyle {\mathfrak {X}}}$ that are obtained from groups over ${\displaystyle k}$. A quintessential example is ${\displaystyle \mathbf {G} _{\mathfrak {X}}=\mathbf {G} \times _{k}{\mathfrak {X}}}$. For arbitrary group schemes, or more generally group functors, over ${\displaystyle {\mathfrak {X}}}$ we shall tend to use german characters, such as ${\displaystyle {\mathfrak {G}}}$.

   • 2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:

     Every finite flat group scheme of order ${\displaystyle p}$ over ${\displaystyle S}$ is of the form ${\displaystyle G_{a,b}^{L}}$ for some triple ${\displaystyle (L,a,b)}$, and ${\displaystyle G_{a,b}^{L}}$ is isomorphic to ${\displaystyle G_{a_{1},b_{1}}^{L_{1}}}$ if and only if there exists an isomorphism from ${\displaystyle L}$ to ${\displaystyle L_{1}}$ carrying ${\displaystyle a}$ to ${\displaystyle a_{1}}$ and ${\displaystyle b}$ to ${\displaystyle b_{1}}$.

Further reading

• Group-scheme action on Wikipedia.Wikipedia
• Groupoid scheme on Wikipedia.Wikipedia
• Group scheme on Encyclopedia of Mathematics on Wikipedia.Wikipedia
• group scheme on nLab on Wikipedia.Wikipedia