See group scheme on Wiktionary
{
"forms": [
{
"form": "group schemes",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "group scheme (plural group schemes)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Category theory",
"orig": "en:Category theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"ref": "1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:",
"text": "It is instructive to give a few explicit examples of such^([affine]) group schemes:",
"type": "quote"
},
{
"ref": "2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:",
"text": "We will tend to use boldface characters, such as #x5C;mathbf#x7B;G#x7D;, for algebraic groups over k, as also for group schemes over #x5C;mathfrak#x7B;X#x7D; that are obtained from groups over k. A quintessential example is #x5C;mathbf#x7B;G#x7D;#x5F;#x5C;mathfrak#x7B;X#x7D;#x3D;#x5C;mathbf#x7B;G#x7D;#x5C;times#x5F;#x7B;k#x7D;#x5C;mathfrak#x7B;X#x7D;. For arbitrary group schemes, or more generally group functors, over #x5C;mathfrak#x7B;X#x7D; we shall tend to use german characters, such as #x5C;mathfrak#x7B;G#x7D;.",
"type": "quote"
},
{
"ref": "2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:",
"text": "Every finite flat group scheme of order p over S is of the form Gᴸ#x5F;#x7B;a,b#x7D; for some triple (L,a,b), and G#x7B;L#x7D;#x5F;#x7B;a,b#x7D; is isomorphic to G#x7B;L#x5F;1#x7D;#x5F;#x7B;a#x5F;1,b#x5F;1#x7D; if and only if there exists an isomorphism from L to L#x5F;1 carrying a to a#x5F;1 and b to b#x5F;1.",
"type": "quote"
}
],
"glosses": [
"A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group."
],
"id": "en-group_scheme-en-noun-glWlGyXA",
"links": [
[
"category theory",
"category theory"
],
[
"group object",
"group object"
],
[
"object",
"object"
],
[
"category",
"category"
],
[
"scheme",
"scheme"
],
[
"group",
"group"
]
],
"qualifier": "scheme theory",
"raw_glosses": [
"(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."
],
"topics": [
"category-theory",
"computing",
"engineering",
"mathematics",
"natural-sciences",
"physical-sciences",
"sciences"
],
"wikipedia": [
"group scheme"
]
}
],
"word": "group scheme"
}
{
"forms": [
{
"form": "group schemes",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "group scheme (plural group schemes)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with quotations",
"Pages with 1 entry",
"Pages with entries",
"en:Category theory"
],
"examples": [
{
"ref": "1973, Jean A. Dieudonné, Introduction to the Theory of Formal Groups, Marcel Dekker, page 6:",
"text": "It is instructive to give a few explicit examples of such^([affine]) group schemes:",
"type": "quote"
},
{
"ref": "2013, Philippe Gille, Arturo Pianzola, Torsors, Reductive Group Schemes and Extended Affine Lie Algebras, American Mathematical Society, page 11:",
"text": "We will tend to use boldface characters, such as #x5C;mathbf#x7B;G#x7D;, for algebraic groups over k, as also for group schemes over #x5C;mathfrak#x7B;X#x7D; that are obtained from groups over k. A quintessential example is #x5C;mathbf#x7B;G#x7D;#x5F;#x5C;mathfrak#x7B;X#x7D;#x3D;#x5C;mathbf#x7B;G#x7D;#x5C;times#x5F;#x7B;k#x7D;#x5C;mathfrak#x7B;X#x7D;. For arbitrary group schemes, or more generally group functors, over #x5C;mathfrak#x7B;X#x7D; we shall tend to use german characters, such as #x5C;mathfrak#x7B;G#x7D;.",
"type": "quote"
},
{
"ref": "2014, J. S. Milne, “The Work of John Tate”, in Helge Holden, Ragni Piene, editors, The Abel Prize 2008-2012, Springer,, page 303:",
"text": "Every finite flat group scheme of order p over S is of the form Gᴸ#x5F;#x7B;a,b#x7D; for some triple (L,a,b), and G#x7B;L#x7D;#x5F;#x7B;a,b#x7D; is isomorphic to G#x7B;L#x5F;1#x7D;#x5F;#x7B;a#x5F;1,b#x5F;1#x7D; if and only if there exists an isomorphism from L to L#x5F;1 carrying a to a#x5F;1 and b to b#x5F;1.",
"type": "quote"
}
],
"glosses": [
"A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group."
],
"links": [
[
"category theory",
"category theory"
],
[
"group object",
"group object"
],
[
"object",
"object"
],
[
"category",
"category"
],
[
"scheme",
"scheme"
],
[
"group",
"group"
]
],
"qualifier": "scheme theory",
"raw_glosses": [
"(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."
],
"topics": [
"category-theory",
"computing",
"engineering",
"mathematics",
"natural-sciences",
"physical-sciences",
"sciences"
],
"wikipedia": [
"group scheme"
]
}
],
"word": "group scheme"
}
Download raw JSONL data for group scheme meaning in All languages combined (2.6kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.