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Non-(affine line)s - Revision history
2024-03-29T10:56:55Z
Revision history for this page on the wiki
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https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=1440&oldid=prev
Anton: fix typo /* A smooth non-(affine line) DM stack with non-separated diagonal */
2021-01-12T00:19:00Z
<p>fix typo <span dir="auto"><span class="autocomment">A smooth non-(affine line) DM stack with non-separated diagonal</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:19, 11 January 2021</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Remark.''' Matsushima's theorem (Theorem 12.15 of Alper<ref>Alper, [http://arxiv.org/abs/0804.2242 Good moduli spaces for Artin stacks]</ref>) says that a subgroup of a linearly reductive group is linearly reductive if and only if the quotient space is affine. Since $G$ is not affine, this shows that $H$ is not linearly reductive. In particular, this shows that linear reductivity of a relative group cannot be checked on fibers.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Remark.''' Matsushima's theorem (Theorem 12.15 of Alper<ref>Alper, [http://arxiv.org/abs/0804.2242 Good moduli spaces for Artin stacks]</ref>) says that a subgroup of a linearly reductive group is linearly reductive if and only if the quotient space is affine. Since $G$ is not affine, this shows that $H$ is not linearly reductive. In particular, this shows that linear reductivity of a relative group cannot be checked on fibers.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Now $B_{\AA^1}G$ is a smooth DM stack with a single stacky point with residual gerbe $B(\ZZ/2)$. However, it has ''non-separated diagonal'', so it is isomorphic to the usual $[\AA^1/(\ZZ/2)]$, with the action given by negation of the coordinate.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Now $B_{\AA^1}G$ is a smooth DM stack with a single stacky point with residual gerbe $B(\ZZ/2)$. However, it has ''non-separated diagonal'', so it is <ins style="font-weight: bold; text-decoration: none;">not </ins>isomorphic to the usual $[\AA^1/(\ZZ/2)]$, with the action given by negation of the coordinate.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Some smooth non-(affine line) Artin stacks =</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Some smooth non-(affine line) Artin stacks =</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=1221&oldid=prev
Anton: /* $\AA^1_\RR$ with a complex origin, various bug eyes, and a complex eye */
2012-08-09T04:21:41Z
<p><span dir="auto"><span class="autocomment">$\AA^1_\RR$ with a complex origin, various bug eyes, and a complex eye</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:21, 8 August 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example''' (space or stack with a ''non-lci'' singularity at the origin)'''.''' Let $C$ be the union of the coordinate axes in $\AA^3$, and let $\ZZ/3$ act by cyclic permutation of the three coordinates. Then $[C/(\ZZ/3)]$ (resp. the algebraic space obtained by tripling the origin in $C$ before quotienting) is a non-(affine line) stack (resp. algebraic space). The interesting thing about this example is that the singularity is not a local complete intersection singularity.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example''' (space or stack with a ''non-lci'' singularity at the origin)'''.''' Let $C$ be the union of the coordinate axes in $\AA^3$, and let $\ZZ/3$ act by cyclic permutation of the three coordinates. Then $[C/(\ZZ/3)]$ (resp. the algebraic space obtained by tripling the origin in $C$ before quotienting) is a non-(affine line) stack (resp. algebraic space). The interesting thing about this example is that the singularity is not a local complete intersection singularity.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin<del style="font-weight: bold; text-decoration: none;">, </del>various bug eyes<del style="font-weight: bold; text-decoration: none;">, and a complex eye </del>=</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin <ins style="font-weight: bold; text-decoration: none;">and </ins>various bug eyes =</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knutson<ref name="knutson"/> explains the following procedure for doing an "étale extension along a closed subscheme." Suppose $X\to Y$ is an étale morphism, $Z\subseteq Y$ is a closed subscheme, and $R=X\times_Y X$. Then $Y=X/R$. Since $X\to Y$ is \'etale, $R$ is the disjoint union of the diagonal and some other stuff: $R=X\sqcup R_0$. We can then remove the part of the relation responsible for gluing together points in the fiber over $Z$ by replacing $R_0$ by $R_0'=R_0\times_Y (Y\smallsetminus Z)$. Then $R'=X\sqcup R_0'$ is an étale relation on $X$. The algebraic space quotient $Y'=X/R'$ has a morphism to $Y$ which is an isomorphism over the complement of $Z$, but $Y'\times_Y Z\cong X\times_Y Z$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knutson<ref name="knutson"/> explains the following procedure for doing an "étale extension along a closed subscheme." Suppose $X\to Y$ is an étale morphism, $Z\subseteq Y$ is a closed subscheme, and $R=X\times_Y X$. Then $Y=X/R$. Since $X\to Y$ is \'etale, $R$ is the disjoint union of the diagonal and some other stuff: $R=X\sqcup R_0$. We can then remove the part of the relation responsible for gluing together points in the fiber over $Z$ by replacing $R_0$ by $R_0'=R_0\times_Y (Y\smallsetminus Z)$. Then $R'=X\sqcup R_0'$ is an étale relation on $X$. The algebraic space quotient $Y'=X/R'$ has a morphism to $Y$ which is an isomorphism over the complement of $Z$, but $Y'\times_Y Z\cong X\times_Y Z$.</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=1220&oldid=prev
Anton at 04:19, 9 August 2012
2012-08-09T04:19:31Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:19, 8 August 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example''' (space or stack with a ''non-lci'' singularity at the origin)'''.''' Let $C$ be the union of the coordinate axes in $\AA^3$, and let $\ZZ/3$ act by cyclic permutation of the three coordinates. Then $[C/(\ZZ/3)]$ (resp. the algebraic space obtained by tripling the origin in $C$ before quotienting) is a non-(affine line) stack (resp. algebraic space). The interesting thing about this example is that the singularity is not a local complete intersection singularity.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example''' (space or stack with a ''non-lci'' singularity at the origin)'''.''' Let $C$ be the union of the coordinate axes in $\AA^3$, and let $\ZZ/3$ act by cyclic permutation of the three coordinates. Then $[C/(\ZZ/3)]$ (resp. the algebraic space obtained by tripling the origin in $C$ before quotienting) is a non-(affine line) stack (resp. algebraic space). The interesting thing about this example is that the singularity is not a local complete intersection singularity.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin =</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin<ins style="font-weight: bold; text-decoration: none;">, various bug eyes, and a complex eye </ins>=</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knutson<ref name="knutson"/> explains the following procedure for doing an "étale extension along a closed subscheme." Suppose $X\to Y$ is an étale morphism, $Z\subseteq Y$ is a closed subscheme, and $R=X\times_Y X$. Then $Y=X/R$. Since $X\to Y$ is \'etale, $R$ is the disjoint union of the diagonal and some other stuff: $R=X\sqcup R_0$. We can then remove the part of the relation responsible for gluing together points in the fiber over $Z$ by replacing $R_0$ by $R_0'=R_0\times_Y (Y\smallsetminus Z)$. Then $R'=X\sqcup R_0'$ is an étale relation on $X$. The algebraic space quotient $Y'=X/R'$ has a morphism to $Y$ which is an isomorphism over the complement of $Z$, but $Y'\times_Y Z\cong X\times_Y Z$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Knutson<ref name="knutson"/> explains the following procedure for doing an "étale extension along a closed subscheme." Suppose $X\to Y$ is an étale morphism, $Z\subseteq Y$ is a closed subscheme, and $R=X\times_Y X$. Then $Y=X/R$. Since $X\to Y$ is \'etale, $R$ is the disjoint union of the diagonal and some other stuff: $R=X\sqcup R_0$. We can then remove the part of the relation responsible for gluing together points in the fiber over $Z$ by replacing $R_0$ by $R_0'=R_0\times_Y (Y\smallsetminus Z)$. Then $R'=X\sqcup R_0'$ is an étale relation on $X$. The algebraic space quotient $Y'=X/R'$ has a morphism to $Y$ which is an isomorphism over the complement of $Z$, but $Y'\times_Y Z\cong X\times_Y Z$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It's also possible to do non-étale extensions along closed subschemes. For example, the squaring map $\AA^1\to \AA^1$ is étale away from the origin. We can modify the induced relation on $\AA^1$ to make it étale: consider $\AA^1\sqcup (\AA^1\setminus 0)\rightrightarrows \AA^1$, where the first copy is the diagonal, and the second copy is $x\mapsto x$ and $x\mapsto -x$. The quotient is the "bug-eyed" cover of $\AA^1$. An alternative description: let $\ZZ/2$ act on the non-separated line by $x\mapsto -x$ and switching the two origins, and consider the quotient. The same trick can be done with any of the non-(affine line)s in the previous section.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= A smooth non-(affine line) DM stack with non-separated diagonal =</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= A smooth non-(affine line) DM stack with non-separated diagonal =</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l33">Line 33:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{todo|}}add bug-eyed cover of $\AA^1$ (how did I forget that?)</del>. <del style="font-weight: bold; text-decoration: none;">Perhaps also include </del>the <del style="font-weight: bold; text-decoration: none;">version </del>of <del style="font-weight: bold; text-decoration: none;">the bug-eyed cover that includes </del>a <del style="font-weight: bold; text-decoration: none;">$B\GG_m$ in the bug's eye. Also Kresch's </del>stack: $X=[\AA^2/_{<del style="font-weight: bold; text-decoration: none;">(</del>1<del style="font-weight: bold; text-decoration: none;">\ </del>-1<del style="font-weight: bold; text-decoration: none;">)</del>}\GG_m]$ with the <del style="font-weight: bold; text-decoration: none;">two origins glued together; note that </del>this <del style="font-weight: bold; text-decoration: none;">isn't </del>a quotient stack<del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In Kresch's [http://www.math.uzh.ch/fileadmin/user/kresch/publikation/contr</ins>.<ins style="font-weight: bold; text-decoration: none;">pdf Flattening stratification and </ins>the <ins style="font-weight: bold; text-decoration: none;">stack of partial stabilisations </ins>of <ins style="font-weight: bold; text-decoration: none;">prestable curves], he builds </ins>a <ins style="font-weight: bold; text-decoration: none;">very interesting </ins>stack: <ins style="font-weight: bold; text-decoration: none;">the quotient of </ins>$<ins style="font-weight: bold; text-decoration: none;">\</ins>X=[<ins style="font-weight: bold; text-decoration: none;">(</ins>\AA^2<ins style="font-weight: bold; text-decoration: none;">\smallsetminus \{0\})</ins>/_{<ins style="font-weight: bold; text-decoration: none;">\begin{pmatrix}</ins>1<ins style="font-weight: bold; text-decoration: none;">& </ins>-1<ins style="font-weight: bold; text-decoration: none;">\end{pmatrix}</ins>}\GG_m]$ <ins style="font-weight: bold; text-decoration: none;">by the &eacute;tale relation $\AA^1\rightrightarrows \X$ given by the two open immersions. The quotient is a smooth stack </ins>with <ins style="font-weight: bold; text-decoration: none;">a dense open substack isomorphic to $\AA^1$, whose complement is a closed substack of codimension 2. Interestingly, this stack has no non-trivial vector bundles, since sections of a vector bundle extend across codimension 2 on a normal stack. When any vector bundle is restricted to </ins>the <ins style="font-weight: bold; text-decoration: none;">copy of $\AA^1$, it has trivializing sections which extend to trivialize the whole bundle. Therefore, </ins>this <ins style="font-weight: bold; text-decoration: none;">stack is not </ins>a quotient stack<ins style="font-weight: bold; text-decoration: none;">!</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=1217&oldid=prev
Anton: /* Some smooth non-(affine line) Artin stacks */
2012-08-04T03:00:19Z
<p><span dir="auto"><span class="autocomment">Some smooth non-(affine line) Artin stacks</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:00, 3 August 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l33">Line 33:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{todo|add bug-eyed cover of $\AA^1$ (how did I forget that?). Perhaps also include the version of the bug-eyed cover that includes a $B\GG_m$ in the bug's eye. Also Kresch's stack: $X=[\AA^2/_{(1\ -1)}\GG_m]$ with the two origins glued together; note that this isn't a quotient stack.<del style="font-weight: bold; text-decoration: none;">}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{todo|<ins style="font-weight: bold; text-decoration: none;">}}</ins>add bug-eyed cover of $\AA^1$ (how did I forget that?). Perhaps also include the version of the bug-eyed cover that includes a $B\GG_m$ in the bug's eye. Also Kresch's stack: $X=[\AA^2/_{(1\ -1)}\GG_m]$ with the two origins glued together; note that this isn't a quotient stack.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=1216&oldid=prev
Anton: /* Some smooth non-(affine line) Artin stacks */
2012-08-04T02:58:55Z
<p><span dir="auto"><span class="autocomment">Some smooth non-(affine line) Artin stacks</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:58, 3 August 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32">Line 32:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $\begin{pmatrix}a& -b\end{pmatrix}$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $\begin{pmatrix}1& -1\end{pmatrix}$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{todo|add bug-eyed cover of $\AA^1$ (how did I forget that?). Perhaps also include the version of the bug-eyed cover that includes a $B\GG_m$ in the bug's eye. Also Kresch's stack: $X=[\AA^2/_{(1\ -1)}\GG_m]$ with the two origins glued together; note that this isn't a quotient stack.}}</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=909&oldid=prev
Anton: /* Some smooth non-(affine line) Artin stacks */
2012-02-03T19:29:18Z
<p><span dir="auto"><span class="autocomment">Some smooth non-(affine line) Artin stacks</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:29, 3 February 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l29">Line 29:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Some smooth non-(affine line) Artin stacks =</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Some smooth non-(affine line) Artin stacks =</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Consider the action of $\GG_m$ on $\AA^2$ given by $t\cdot (x,y)=(tx,t^{-1}y)$. Then $[\AA^2/\GG_m]$ (which I would denote $[\AA^2/_{<del style="font-weight: bold; text-decoration: none;">(</del>1<del style="font-weight: bold; text-decoration: none;">\ </del>-1<del style="font-weight: bold; text-decoration: none;">)</del>}\GG_m]$) has good moduli space $\AA^1$, but contains a dense open copy of the non-separated line (namely $[(\AA^2\smallsetminus \{0\})/_{<del style="font-weight: bold; text-decoration: none;">(</del>1<del style="font-weight: bold; text-decoration: none;">\ </del>-1<del style="font-weight: bold; text-decoration: none;">)</del>}\GG_m]$). The non-separatedness disappears in the coarse space because there is a $B\GG_m$ which both origins specialize to.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Consider the action of $\GG_m$ on $\AA^2$ given by $t\cdot (x,y)=(tx,t^{-1}y)$. Then $[\AA^2/\GG_m]$ (which I would denote $[\AA^2/_{<ins style="font-weight: bold; text-decoration: none;">\begin{pmatrix}</ins>1<ins style="font-weight: bold; text-decoration: none;">& </ins>-1<ins style="font-weight: bold; text-decoration: none;">\end{pmatrix}</ins>}\GG_m]$) has good moduli space $\AA^1$, but contains a dense open copy of the non-separated line (namely $[(\AA^2\smallsetminus \{0\})/_{<ins style="font-weight: bold; text-decoration: none;">\begin{pmatrix}</ins>1<ins style="font-weight: bold; text-decoration: none;">& </ins>-1<ins style="font-weight: bold; text-decoration: none;">\end{pmatrix}</ins>}\GG_m]$). The non-separatedness disappears in the coarse space because there is a $B\GG_m$ which both origins specialize to.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $<del style="font-weight: bold; text-decoration: none;">(</del>a<del style="font-weight: bold; text-decoration: none;">\ </del>-b<del style="font-weight: bold; text-decoration: none;">)</del>$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $<del style="font-weight: bold; text-decoration: none;">(</del>1<del style="font-weight: bold; text-decoration: none;">\ </del>-1<del style="font-weight: bold; text-decoration: none;">)</del>$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>By considering a the weight $<ins style="font-weight: bold; text-decoration: none;">\begin{pmatrix}</ins>a<ins style="font-weight: bold; text-decoration: none;">& </ins>-b<ins style="font-weight: bold; text-decoration: none;">\end{pmatrix}</ins>$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $<ins style="font-weight: bold; text-decoration: none;">\begin{pmatrix}</ins>1<ins style="font-weight: bold; text-decoration: none;">& </ins>-1<ins style="font-weight: bold; text-decoration: none;">\end{pmatrix}</ins>$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=904&oldid=prev
Anton: /* A strange smooth non-(affine line) stack */
2012-02-01T19:03:10Z
<p><span dir="auto"><span class="autocomment">A strange smooth non-(affine line) stack</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:03, 1 February 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= A <del style="font-weight: bold; text-decoration: none;">strange </del>smooth non-(affine line) stack =</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= A smooth non-(affine line) <ins style="font-weight: bold; text-decoration: none;">DM </ins>stack <ins style="font-weight: bold; text-decoration: none;">with non-separated diagonal </ins>=</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the relative group $(\ZZ/2)\times \AA^1$ over $\AA^1$. Then $H=\AA^1\sqcup (\AA^1\smallsetminus \{0\})$ is an open subgroup. The quotient $G=(\ZZ/2\times \AA^1)/H$ is the affine line with a doubled origin, regarded as a group over $\AA^1$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the relative group $(\ZZ/2)\times \AA^1$ over $\AA^1$. Then $H=\AA^1\sqcup (\AA^1\smallsetminus \{0\})$ is an open subgroup. The quotient $G=(\ZZ/2\times \AA^1)/H$ is the affine line with a doubled origin, regarded as a group over $\AA^1$.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l26">Line 26:</td>
<td colspan="2" class="diff-lineno">Line 26:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now $B_{\AA^1}G$ is a smooth DM stack with a single stacky point with residual gerbe $B(\ZZ/2)$. However, it has ''non-separated diagonal'', so it is isomorphic to the usual $[\AA^1/(\ZZ/2)]$, with the action given by negation of the coordinate.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now $B_{\AA^1}G$ is a smooth DM stack with a single stacky point with residual gerbe $B(\ZZ/2)$. However, it has ''non-separated diagonal'', so it is isomorphic to the usual $[\AA^1/(\ZZ/2)]$, with the action given by negation of the coordinate.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">= Some smooth non-(affine line) Artin stacks =</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Consider the action of $\GG_m$ on $\AA^2$ given by $t\cdot (x,y)=(tx,t^{-1}y)$. Then $[\AA^2/\GG_m]$ (which I would denote $[\AA^2/_{(1\ -1)}\GG_m]$) has good moduli space $\AA^1$, but contains a dense open copy of the non-separated line (namely $[(\AA^2\smallsetminus \{0\})/_{(1\ -1)}\GG_m]$). The non-separatedness disappears in the coarse space because there is a $B\GG_m$ which both origins specialize to.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">By considering a the weight $(a\ -b)$ action of $\GG_m$ on $\AA^2$ for positive integers $a$ and $b$ (instead of the weight $(1\ -1)$ action), we get a similar stack, but where the two origins have stabilizers $\mu_a$ and $\mu_b$.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{todo|maybe throw in some Artin stack examples like $[\AA^2/_{(1\ -1)}\GG_m]$}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Blog]] [[Category:Note]]</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=903&oldid=prev
Anton: /* Non-(affine line) spaces with various singularities at the origin */
2012-02-01T18:55:18Z
<p><span dir="auto"><span class="autocomment">Non-(affine line) spaces with various singularities at the origin</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:55, 1 February 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{todo|using </del>axes in 3<del style="font-weight: bold; text-decoration: none;">-space mod action </del>of $\ZZ/3$<del style="font-weight: bold; text-decoration: none;">, can even get </del>non-<del style="font-weight: bold; text-decoration: none;">lci </del>singularity <del style="font-weight: bold; text-decoration: none;">at the origin}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Example''' (space or stack with a ''non-lci'' singularity at the origin)'''.''' Let $C$ be the union of the coordinate </ins>axes in <ins style="font-weight: bold; text-decoration: none;">$\AA^3$, and let $\ZZ/</ins>3<ins style="font-weight: bold; text-decoration: none;">$ act by cyclic permutation </ins>of <ins style="font-weight: bold; text-decoration: none;">the three coordinates. Then </ins>$<ins style="font-weight: bold; text-decoration: none;">[C/(</ins>\ZZ/3<ins style="font-weight: bold; text-decoration: none;">)]</ins>$ <ins style="font-weight: bold; text-decoration: none;">(resp. the algebraic space obtained by tripling the origin in $C$ before quotienting) is a </ins>non-<ins style="font-weight: bold; text-decoration: none;">(affine line) stack (resp. algebraic space). The interesting thing about this example is that the singularity is not a local complete intersection </ins>singularity<ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin =</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin =</div></td></tr>
</table>
Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=902&oldid=prev
Anton: /* Non-(affine line) spaces with various singularities at the origin */
2012-02-01T18:48:35Z
<p><span dir="auto"><span class="autocomment">Non-(affine line) spaces with various singularities at the origin</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:48, 1 February 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{todo|using axes in 3-space mod action of $\ZZ/3$, can even get non-lci singularity at the origin}}</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin =</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= $\AA^1_\RR$ with a complex origin =</div></td></tr>
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Anton
https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=899&oldid=prev
Anton at 17:39, 27 January 2012
2012-01-27T17:39:49Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:39, 27 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This is my collection of examples of algebraic spaces and stacks that look something like $\AA^1$. If you like the affine line with a doubled origin, and the stack $[\AA^1/\mu_2]$, you've found the right place.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This is my collection of examples of algebraic spaces and stacks that look something like $\AA^1$. If you like the affine line with a doubled origin, and the stack $[\AA^1/\mu_2]$, you've found the right place.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= Non-(affine line)<del style="font-weight: bold; text-decoration: none;">s </del>with various singularities at the origin =</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= Non-(affine line) <ins style="font-weight: bold; text-decoration: none;">spaces </ins>with various singularities at the origin =</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Gluing two copies of $\AA^1$ along the complement of the origin gives the usual line with a doubled origin. An example in the introduction of Knutson<ref name="knutson">Knutson, [http://books.google.com/books/about/Algebraic_spaces.html?id=eqIZAQAAIAAJ Algebraic Spaces]</ref> is the "affine line with a doubled tangent direction", obtained by taking two intersecting lines $C=V(y^2-x^2)$, considering the $\ZZ/2$-action which negates $y$, "removing the action at the origin," and taking the algebraic space quotient. More precisely, take the algebraic space quotient by the relation $R=C\sqcup C'\rightrightarrows C$, where $C'$ is the complement of the origin in $C$, and the two maps $C'\to C$ are given by the inclusion and the inclusion followed by negating $x$. The result looks like a line, but with a "doubled tangent direction" at the origin since it has an étale cover by two intersecting lines.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Gluing two copies of $\AA^1$ along the complement of the origin gives the usual line with a doubled origin. An example in the introduction of Knutson<ref name="knutson">Knutson, [http://books.google.com/books/about/Algebraic_spaces.html?id=eqIZAQAAIAAJ Algebraic Spaces]</ref> is the "affine line with a doubled tangent direction", obtained by taking two intersecting lines $C=V(y^2-x^2)$, considering the $\ZZ/2$-action which negates $y$, "removing the action at the origin," and taking the algebraic space quotient. More precisely, take the algebraic space quotient by the relation $R=C\sqcup C'\rightrightarrows C$, where $C'$ is the complement of the origin in $C$, and the two maps $C'\to C$ are given by the inclusion and the inclusion followed by negating $x$. The result looks like a line, but with a "doubled tangent direction" at the origin since it has an étale cover by two intersecting lines.</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Applying this to the case $Y=\AA^1_\RR$, $X=\AA^1_\CC$, and $Z=\{0\}$, we get an algebraic space $Y'$ which looks like $\AA^1_\RR$, except the residue field at the origin is $\CC$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= A <del style="font-weight: bold; text-decoration: none;">''</del>smooth<del style="font-weight: bold; text-decoration: none;">'' </del>non-(affine line) stack =</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= A <ins style="font-weight: bold; text-decoration: none;">strange </ins>smooth non-(affine line) stack =</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the relative group $(\ZZ/2)\times \AA^1$ over $\AA^1$. Then $H=\AA^1\sqcup (\AA^1\smallsetminus \{0\})$ is an open subgroup. The quotient $G=(\ZZ/2\times \AA^1)/H$ is the affine line with a doubled origin, regarded as a group over $\AA^1$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the relative group $(\ZZ/2)\times \AA^1$ over $\AA^1$. Then $H=\AA^1\sqcup (\AA^1\smallsetminus \{0\})$ is an open subgroup. The quotient $G=(\ZZ/2\times \AA^1)/H$ is the affine line with a doubled origin, regarded as a group over $\AA^1$.</div></td></tr>
</table>
Anton