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<title>Group-stack</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Group-stack</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>group-stack</b> is an <a href="Algebraic_stack" title="Algebraic stack">algebraic stack</a> whose categories of points have group structures or even <a href="Groupoid" title="Groupoid">groupoid</a> structures in a compatible way.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It generalizes a <a href="Group_scheme" title="Group scheme">group scheme</a>, which is a scheme whose sets of points have group structures in a compatible way.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>A group scheme is a group-stack. More generally, a <b>group algebraic-space</b>, an algebraic-space analog of a group scheme, is a group-stack.</li>

<li>Over a field <i>k</i>, a <b>vector bundle stack</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation>

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</math></span><img src="./47d69f309b6deb2e5008f6130ee11e09bbabd7b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}" loading="lazy"></span> on a Deligne–Mumford stack <i>X</i> is a group-stack such that there is a vector bundle <i>V</i> over <i>k</i> on <i>X</i> and a presentation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to {\mathcal {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle V\to {\mathcal {V}}}</annotation>

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</math></span><img src="./7cfd8fee6d69ebe001ef10d2e3c5879e1c7bb1b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.931ex; height:2.176ex;" alt="{\displaystyle V\to {\mathcal {V}}}" loading="lazy"></span>. It has an action by the affine line <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}}</annotation>

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</math></span><img src="./67530a5bd8c23c0be226ac63ddf5f6b2619e682d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}}" loading="lazy"></span> corresponding to <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>.</li>

<li>A <a href="Picard_stack" class="mw-redirect" title="Picard stack">Picard stack</a> is an example of a group-stack (or groupoid-stack).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Actions_of_group-stacks">Actions of group-stacks</h2></div>

<p>The definition of a <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of a group-stack is a bit tricky. First, given an algebraic stack <i>X</i> and a group scheme <i>G</i> on a base scheme <i>S</i>, a right action of <i>G</i> on <i>X</i> consists of

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<ol><li>a <a href="Morphism_of_algebraic_stacks" title="Morphism of algebraic stacks">morphism</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma :X\times G\to X}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma :X\times G\to X}</annotation>

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</math></span><img src="./c72be058c35280295aba70a5352ae3cdb3e489d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.508ex; height:2.176ex;" alt="{\displaystyle \sigma :X\times G\to X}" loading="lazy"></span>,</li>

<li>(associativity) a natural isomorphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \circ (m\times 1_{X}){\overset {\sim }{\to }}\sigma \circ (1_{X}\times \sigma )}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma \circ (m\times 1_{X}){\overset {\sim }{\to }}\sigma \circ (1_{X}\times \sigma )}</annotation>

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</math></span><img src="./4888461971ad742c44b1469bdbecdf52bf5d5ccf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.632ex; height:3.343ex;" alt="{\displaystyle \sigma \circ (m\times 1_{X}){\overset {\sim }{\to }}\sigma \circ (1_{X}\times \sigma )}" loading="lazy"></span>, where <i>m</i> is the multiplication on <i>G</i>,</li>

<li>(identity) a natural isomorphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{X}{\overset {\sim }{\to }}\sigma \circ (1_{X}\times e)}">

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<annotation encoding="application/x-tex">{\displaystyle 1_{X}{\overset {\sim }{\to }}\sigma \circ (1_{X}\times e)}</annotation>

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</math></span><img src="./d39033bce11e9d043810b2983f5ee9e2285cb11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.171ex; height:3.343ex;" alt="{\displaystyle 1_{X}{\overset {\sim }{\to }}\sigma \circ (1_{X}\times e)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:S\to G}">

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<annotation encoding="application/x-tex">{\displaystyle e:S\to G}</annotation>

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</math></span><img src="./9fe5cef1328c0e6216eea1256ed01ac10c89e2ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.961ex; height:2.176ex;" alt="{\displaystyle e:S\to G}" loading="lazy"></span> is the identity section of <i>G</i>,</li></ol>

<p>that satisfy the typical compatibility conditions.

</p><p>If, more generally, <i>G</i> is a group-stack, one then extends the above using local presentations.

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/231313">"Ag.algebraic geometry - Are Picard stacks group objects in the category of algebraic stacks"</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBehrendFantechi1997" class="citation journal cs1">Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". <i>Inventiones Mathematicae</i>. <b>128</b> (1): <span class="nowrap">45–</span>88. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/alg-geom/9601010">alg-geom/9601010</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997InMat.128...45B">1997InMat.128...45B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002220050136">10.1007/s002220050136</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>.</cite></li></ul>

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