group scheme

English

Noun

group scheme (plural group schemes)

(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, 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 ${\\displaystyle b_{1}}$ .

单词	group scheme
释义	group scheme English Noun group scheme (plural group schemes) (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, 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
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group scheme

English

Noun

Further reading