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Group functor
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<h2 id="group-functor-as-a-generalization-of-a-group-scheme">Group functor as a generalization of a group scheme</h2> <p>A scheme may be thought of as a contravariant functor from the category S c h S {\displaystyle {\mathsf {Sch}}_{S}} of <i>S</i>-schemes to the category of sets satisfying the <a href="/facts/Gluing_axiom/V3MnSb2f">gluing axiom</a>; the perspective known as the <a href="/facts/Rational_points/j5v4cxuE">functor of points</a>. Under this perspective, a group scheme is a contravariant functor from S c h S {\displaystyle {\mathsf {Sch}}_{S}} to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology). </p><p>For example, if Γ is a finite group, then consider the functor that sends Spec(<i>R</i>) to the set of locally constant functions on it. For example, the group scheme </p> S L 2 = Spec ( Z [ a , b , c , d ] ( a d − b c − 1 ) ) {\displaystyle SL_{2}=\operatorname {Spec} \left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}}\right)} <p>can be described as the functor </p> Hom CRing ( Z [ a , b , c , d ] ( a d − b c − 1 ) , − ) {\displaystyle \operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},-\right)} <p>If we take a ring, for example, C {\displaystyle \mathbb {C} } , then </p> S L 2 ( C ) = Hom CRing ( Z [ a , b , c , d ] ( a d − b c − 1 ) , C ) ≅ { [ a b c d ] ∈ M 2 ( C ) : a d − b c = 1 } {\displaystyle {\begin{aligned}SL_{2}(\mathbb {C} )&=\operatorname {Hom} _{\textbf {CRing}}\left({\frac {\mathbb {Z} [a,b,c,d]}{(ad-bc-1)}},\mathbb {C} \right)\\&\cong \left\{{\begin{bmatrix}a&b\\c&d\end{bmatrix}}\in M_{2}(\mathbb {C} ):ad-bc=1\right\}\end{aligned}}} <h2 id="group-sheaf">Group sheaf</h2> <p>It is useful to consider a group functor that respects a topology (if any) of the underlying category; namely, one that is a sheaf and a group functor that is a sheaf is called a group sheaf. The notion appears in particular in the discussion of a <a href="/facts/Torsor_(algebraic_geometry)/nMKyR4eF">torsor</a> (where a choice of topology is an important matter). </p><p>For example, a <a href="/facts/P-divisible_group/6d1VLkWb"><i>p</i>-divisible group</a> is an example of a fppf group sheaf (a group sheaf with respect to the fppf topology).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Automorphism_group_functor/65vImKwe">automorphism group functor</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/William_C._Waterhouse/PsgRvukt">Waterhouse, William</a> (1979), <i>Introduction to affine group schemes</i>, Graduate Texts in Mathematics, vol. 66, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-6217-6">10.1007/978-1-4612-6217-6</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90421-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0547117">0547117</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Course Notes -- J.S. Milne". <a href="http://www.jmilne.org/math/CourseNotes/" target="_blank">http://www.jmilne.org/math/CourseNotes/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Archived copy" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2018-03-26.{{cite web}}: CS1 maint: archived copy as title (link) <a href="https://web.archive.org/web/20161020141125/http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf" target="_blank">https://web.archive.org/web/20161020141125/http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>