Non-(affine line)s - Revision history

Anton: fix typo /* A smooth non-(affine line) DM stack with non-separated diagonal */

2021-01-12T00:19:00Z

fix typo A smooth non-(affine line) DM stack with non-separated diagonal

← Older revision		Revision as of 16:19, 11 January 2021
Line 27:		Line 27:
	'''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.		'''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.

	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.		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 not isomorphic to the usual $[\AA^1/(\ZZ/2)]$, with the action given by negation of the coordinate.

	= Some smooth non-(affine line) Artin stacks =		= Some smooth non-(affine line) Artin stacks =

Anton: /* $\AA^1_\RR$ with a complex origin, various bug eyes, and a complex eye */

2012-08-09T04:21:41Z

$\AA^1_\RR$ with a complex origin, various bug eyes, and a complex eye

← Older revision		Revision as of 20:21, 8 August 2012
Line 13:		Line 13:
	'''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.		'''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.

	= $\AA^1_\RR$ with a complex origin, various bug eyes~~, and a complex eye~~ =		= $\AA^1_\RR$ with a complex origin and various bug eyes =

	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 $Y'\times_Y Z\cong X\times_Y Z$.		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 $Y'\times_Y Z\cong X\times_Y Z$.

Anton at 04:19, 9 August 2012

2012-08-09T04:19:31Z

← Older revision		Revision as of 20:19, 8 August 2012
Line 13:		Line 13:
	'''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.		'''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.

	= $\AA^1_\RR$ with a complex origin =		= $\AA^1_\RR$ with a complex origin, various bug eyes, and a complex eye =

	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 $Y'\times_Y Z\cong X\times_Y Z$.		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 $Y'\times_Y Z\cong X\times_Y Z$.

	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 $\CC$.		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 $\CC$.

			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.

	= A smooth non-(affine line) DM stack with non-separated diagonal =		= A smooth non-(affine line) DM stack with non-separated diagonal =
Line 33:		Line 35:
	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 $\mu_b$.		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 $\mu_b$.

	~~{{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.		In Kresch's [http://www.math.uzh.ch/fileadmin/user/kresch/publikation/contr.pdf Flattening stratification and the stack of partial stabilisations of prestable curves], he builds a very interesting stack: the quotient of $\X=[(\AA^2\smallsetminus \{0\})/_{\begin{pmatrix}1& -1\end{pmatrix}}\GG_m]$ by the étale relation $\AA^1\rightrightarrows \X$ given by the two open immersions. The quotient is a smooth stack with 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 the copy of $\AA^1$, it has trivializing sections which extend to trivialize the whole bundle. Therefore, this stack is not a quotient stack!

	<references/>		<references/>

	[[Category:Blog]] [[Category:Note]]		[[Category:Blog]] [[Category:Note]]

Anton: /* Some smooth non-(affine line) Artin stacks */

2012-08-04T03:00:19Z

Some smooth non-(affine line) Artin stacks

← Older revision		Revision as of 19:00, 3 August 2012
Line 33:		Line 33:
	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 $\mu_b$.		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 $\mu_b$.

	{{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.}}		{{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.

	<references/>		<references/>

	[[Category:Blog]] [[Category:Note]]		[[Category:Blog]] [[Category:Note]]

Anton: /* Some smooth non-(affine line) Artin stacks */

2012-08-04T02:58:55Z

Some smooth non-(affine line) Artin stacks

← Older revision		Revision as of 18:58, 3 August 2012
Line 32:		Line 32:

	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 $\mu_b$.		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 $\mu_b$.

			{{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.}}

	<references/>		<references/>

	[[Category:Blog]] [[Category:Note]]		[[Category:Blog]] [[Category:Note]]

Anton: /* Some smooth non-(affine line) Artin stacks */

2012-02-03T19:29:18Z

Some smooth non-(affine line) Artin stacks

← Older revision		Revision as of 11:29, 3 February 2012
Line 29:		Line 29:
	= Some smooth non-(affine line) Artin stacks =		= Some smooth non-(affine line) Artin stacks =

	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.		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 $[\AA^2/_{\begin{pmatrix}1& -1\end{pmatrix}}\GG_m]$) has good moduli space $\AA^1$, but contains a dense open copy of the non-separated line (namely $[(\AA^2\smallsetminus \{0\})/_{\begin{pmatrix}1& -1\end{pmatrix}}\GG_m]$). The non-separatedness disappears in the coarse space because there is a $B\GG_m$ which both origins specialize to.

	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 $\mu_b$.		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 $\mu_b$.

	<references/>		<references/>

	[[Category:Blog]] [[Category:Note]]		[[Category:Blog]] [[Category:Note]]

Anton: /* A strange smooth non-(affine line) stack */

2012-02-01T19:03:10Z

A strange smooth non-(affine line) stack

← Older revision		Revision as of 11:03, 1 February 2012
Line 19:		Line 19:
	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 $\CC$.		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 $\CC$.

	= A ~~strange~~ smooth non-(affine line) stack =		= A smooth non-(affine line) DM stack with non-separated diagonal =

	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 $\AA^1$.		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 $\AA^1$.
Line 26:		Line 26:

	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.		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.

			= Some smooth non-(affine line) Artin stacks =

			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.

			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 $\mu_b$.

	<references/>		<references/>

	~~{{todo\|maybe throw in some Artin stack examples like $[\AA^2/_{(1\ -1)}\GG_m]$}}~~

	[[Category:Blog]] [[Category:Note]]		[[Category:Blog]] [[Category:Note]]

Anton: /* Non-(affine line) spaces with various singularities at the origin */

2012-02-01T18:55:18Z

Non-(affine line) spaces with various singularities at the origin

← Older revision		Revision as of 10:55, 1 February 2012
Line 11:		Line 11:
	'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.		'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.

	~~{{todo\|using~~ axes in 3~~-space mod action~~ of $\ZZ/3$~~, can even get~~ non-~~lci~~ singularity ~~at the origin}}~~		'''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.

	= $\AA^1_\RR$ with a complex origin =		= $\AA^1_\RR$ with a complex origin =

Anton: /* Non-(affine line) spaces with various singularities at the origin */

2012-02-01T18:48:35Z

Non-(affine line) spaces with various singularities at the origin

← Older revision		Revision as of 10:48, 1 February 2012
Line 10:		Line 10:

	'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.		'''Example.''' All the above singularities can appear on Deligne-Mumford stacks with coarse space $\AA^1$. Just don't introduce nonseparatedness: $[V(f(y)-x^k)/(\ZZ/k)]$.

			{{todo\|using axes in 3-space mod action of $\ZZ/3$, can even get non-lci singularity at the origin}}

	= $\AA^1_\RR$ with a complex origin =		= $\AA^1_\RR$ with a complex origin =

Anton at 17:39, 27 January 2012

2012-01-27T17:39:49Z

← Older revision		Revision as of 09:39, 27 January 2012
Line 1:		Line 1:
	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.		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.

	= Non-(affine line)s with various singularities at the origin =		= Non-(affine line) spaces with various singularities at the origin =

	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.		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.
Line 17:		Line 17:
	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 $\CC$.		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 $\CC$.

	= A ''smooth'' non-(affine line) stack =		= A strange smooth non-(affine line) stack =

	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 $\AA^1$.		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 $\AA^1$.