group scheme

English

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

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

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