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Failure of Artin Approximation for non excellent schemes
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One of the results that can be deduced from the Artin Approximation Theorem (in its modern formulation) is the following:
https://stacks.math.columbia.edu/tag/0CAV. It says that, if $S$ is a scheme such that $O_S,s$ is a G-ring, and $X,Y$ are schemes over $S$ locally of finite type, that satisfy:
$$widehatO_X,xcong widehatO_Y,y $$
for points $x,y$, then there exists a common étale neighborhood of $X,Y$ in the points $x,y$.
My questions are:
which counterexample do we have of this criterion? Are there noetherian schemes that have completed local rings isomorphic as above, but there exist no common étale neighborhood?
Are there at least examples of noetherian non G-rings? By now I have found just this one on wikipedia https://en.wikipedia.org/wiki/Excellent_ring#A_J-2_ring_that_is_not_a_G-ring.
algebraic-geometry commutative-algebra
asked Jul 30 at 20:21
Serser
799
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up vote
1
down vote
favorite
One of the results that can be deduced from the Artin Approximation Theorem (in its modern formulation) is the following:
https://stacks.math.columbia.edu/tag/0CAV. It says that, if $S$ is a scheme such that $O_S,s$ is a G-ring, and $X,Y$ are schemes over $S$ locally of finite type, that satisfy:
$$widehatO_X,xcong widehatO_Y,y $$
for points $x,y$, then there exists a common étale neighborhood of $X,Y$ in the points $x,y$.
My questions are:
which counterexample do we have of this criterion? Are there noetherian schemes that have completed local rings isomorphic as above, but there exist no common étale neighborhood?
Are there at least examples of noetherian non G-rings? By now I have found just this one on wikipedia https://en.wikipedia.org/wiki/Excellent_ring#A_J-2_ring_that_is_not_a_G-ring.
algebraic-geometry commutative-algebra
asked Jul 30 at 20:21
Serser
799
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
One of the results that can be deduced from the Artin Approximation Theorem (in its modern formulation) is the following:
https://stacks.math.columbia.edu/tag/0CAV. It says that, if $S$ is a scheme such that $O_S,s$ is a G-ring, and $X,Y$ are schemes over $S$ locally of finite type, that satisfy:
$$widehatO_X,xcong widehatO_Y,y $$
for points $x,y$, then there exists a common étale neighborhood of $X,Y$ in the points $x,y$.
My questions are:
which counterexample do we have of this criterion? Are there noetherian schemes that have completed local rings isomorphic as above, but there exist no common étale neighborhood?
Are there at least examples of noetherian non G-rings? By now I have found just this one on wikipedia https://en.wikipedia.org/wiki/Excellent_ring#A_J-2_ring_that_is_not_a_G-ring.
algebraic-geometry commutative-algebra
asked Jul 30 at 20:21
Serser
799
One of the results that can be deduced from the Artin Approximation Theorem (in its modern formulation) is the following:
https://stacks.math.columbia.edu/tag/0CAV. It says that, if $S$ is a scheme such that $O_S,s$ is a G-ring, and $X,Y$ are schemes over $S$ locally of finite type, that satisfy:
$$widehatO_X,xcong widehatO_Y,y $$
for points $x,y$, then there exists a common étale neighborhood of $X,Y$ in the points $x,y$.
My questions are:
which counterexample do we have of this criterion? Are there noetherian schemes that have completed local rings isomorphic as above, but there exist no common étale neighborhood?
Are there at least examples of noetherian non G-rings? By now I have found just this one on wikipedia https://en.wikipedia.org/wiki/Excellent_ring#A_J-2_ring_that_is_not_a_G-ring.
algebraic-geometry commutative-algebra
asked Jul 30 at 20:21
Serser
799
asked Jul 30 at 20:21
Serser
799
asked Jul 30 at 20:21
Serser
799
asked Jul 30 at 20:21
Serser
799
799
add a comment |Â
add a comment |Â
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