https://stacky.net/wiki/api.php?action=feedcontributions&user=170.140.151.70&feedformat=atomstacky wiki - User contributions [en]2022-08-19T17:37:32ZUser contributionsMediaWiki 1.24.4https://stacky.net/wiki/index.php?title=Non-(affine_line)s&diff=898Non-(affine line)s2012-01-26T23:38:34Z<p>170.140.151.70: /* Non-(affine line)s with various singularities at the origin */</p>
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<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.<br />
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= Non-(affine line)s with various singularities at the origin =<br />
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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.<br />
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An alternative description of this example is to let $C$ be $V(y^2-x^2)$ ''with a doubled origin'', and let $\ZZ/2$ act by negating $x$ and swapping the two origins. Then the affine line with a doubled tangent direction is the algebraic space $C/(\ZZ/2)$.<br />
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'''Example.''' Generalizing the above example, we can consider the action of $\ZZ/2$ on $V(y^n-x^2)$ with a doubled origin. The space quotient is an affine line with a higher (tac)node (if $n$ is even) or higher cusp (if $n$ is odd) at the origin. More generally, any singularity of the form $f(y)-x^k$ can appear on a tweaked copy of the affine line by "$k$-folding up" the points where $x=0$ and acting by $\ZZ/k$. (Assuming we're working over a base where $\ZZ/k$ is isomorphic to $\mu_k$)<br />
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'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.<br />
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= $\AA^1_\RR$ with a complex origin =<br />
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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 $Y'\times_Y Z\cong X\times_Y Z$.<br />
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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 $\CC$.<br />
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= A ''smooth'' non-(affine line) stack =<br />
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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 $\AA^1$.<br />
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'''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.<br />
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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.<br />
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{{todo|maybe throw in some Artin stack examples like $[\AA^2/_{(1\ -1)}\GG_m]$}}<br />
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[[Category:Blog]] [[Category:Note]]</div>170.140.151.70