"group scheme" meaning in English

{
  "forms": [
    {
      "form": "group schemes",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "group scheme (plural group schemes)",
      "name": "en-noun"
    }
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  "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": "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 group schemes:",
          "type": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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"
      ]
    }
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  "head_templates": [
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      "expansion": "group scheme (plural group schemes)",
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  "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",
        "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 group schemes:",
          "type": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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"
}