Mixing
Is every commutative ring a limit of noetherian rings?
$begingroup$
Let $mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $mathsf{CRing}$ of commutative rings with one.
Let $A$ be in $mathsf{CRing}$.
Question 1. Is there a functor from a small category to $mathsf{Noeth}$ whose limit in $mathsf{CRing}$ is $A$?
(I know that there is a functor from a small category to $mathsf{Noeth}$ whose colimit is $A$.)
Let $f:Ato B$ be a morphism in $mathsf{CRing}$ such that the map
$$
circ f:text{Hom}_{mathsf{CRing}}(B,C)totext{Hom}_{mathsf{CRing}}(A,C)
$$
sending $g$ to $gcirc f$ is bijective for all $C$ in $mathsf{Noeth}$.
Question 2. Does this imply that $f$ is an isomorphism?
Yes to Question 1 would imply yes to Question 2.
Question 3. Does the inclusion functor $iota:mathsf{Noeth}tomathsf{CRing}$ commute with colimits? That is, if $Ainmathsf{Noeth}$ is the colimit of a functor $alpha$ from a small category to $mathsf{Noeth}$, is $A$ naturally isomorphic to the colimit of $iotacircalpha$?
Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do not exist in $mathsf{Noeth}$: see this answer of Martin Brandenburg.
One may try to attack the first question as follows:
Let $A$ be in $mathsf{CRing}$ and $I$ the set of those ideals $mathfrak a$ of $A$ such that $A/mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/mathfrak a$ with $mathfrak ain I$, and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective.
commutative-algebra category-theory noetherian limits-colimits
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
$endgroup$
|
show 3 more comments
$begingroup$
Let $mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $mathsf{CRing}$ of commutative rings with one.
Let $A$ be in $mathsf{CRing}$.
Question 1. Is there a functor from a small category to $mathsf{Noeth}$ whose limit in $mathsf{CRing}$ is $A$?
(I know that there is a functor from a small category to $mathsf{Noeth}$ whose colimit is $A$.)
Let $f:Ato B$ be a morphism in $mathsf{CRing}$ such that the map
$$
circ f:text{Hom}_{mathsf{CRing}}(B,C)totext{Hom}_{mathsf{CRing}}(A,C)
$$
sending $g$ to $gcirc f$ is bijective for all $C$ in $mathsf{Noeth}$.
Question 2. Does this imply that $f$ is an isomorphism?
Yes to Question 1 would imply yes to Question 2.
Question 3. Does the inclusion functor $iota:mathsf{Noeth}tomathsf{CRing}$ commute with colimits? That is, if $Ainmathsf{Noeth}$ is the colimit of a functor $alpha$ from a small category to $mathsf{Noeth}$, is $A$ naturally isomorphic to the colimit of $iotacircalpha$?
Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do not exist in $mathsf{Noeth}$: see this answer of Martin Brandenburg.
One may try to attack the first question as follows:
Let $A$ be in $mathsf{CRing}$ and $I$ the set of those ideals $mathfrak a$ of $A$ such that $A/mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/mathfrak a$ with $mathfrak ain I$, and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective.
commutative-algebra category-theory noetherian limits-colimits
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
$endgroup$
$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
1
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
1
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52
|
show 3 more comments
$begingroup$
Let $mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $mathsf{CRing}$ of commutative rings with one.
Let $A$ be in $mathsf{CRing}$.
Question 1. Is there a functor from a small category to $mathsf{Noeth}$ whose limit in $mathsf{CRing}$ is $A$?
(I know that there is a functor from a small category to $mathsf{Noeth}$ whose colimit is $A$.)
Let $f:Ato B$ be a morphism in $mathsf{CRing}$ such that the map
$$
circ f:text{Hom}_{mathsf{CRing}}(B,C)totext{Hom}_{mathsf{CRing}}(A,C)
$$
sending $g$ to $gcirc f$ is bijective for all $C$ in $mathsf{Noeth}$.
Question 2. Does this imply that $f$ is an isomorphism?
Yes to Question 1 would imply yes to Question 2.
Question 3. Does the inclusion functor $iota:mathsf{Noeth}tomathsf{CRing}$ commute with colimits? That is, if $Ainmathsf{Noeth}$ is the colimit of a functor $alpha$ from a small category to $mathsf{Noeth}$, is $A$ naturally isomorphic to the colimit of $iotacircalpha$?
Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do not exist in $mathsf{Noeth}$: see this answer of Martin Brandenburg.
One may try to attack the first question as follows:
Let $A$ be in $mathsf{CRing}$ and $I$ the set of those ideals $mathfrak a$ of $A$ such that $A/mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/mathfrak a$ with $mathfrak ain I$, and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective.
commutative-algebra category-theory noetherian limits-colimits
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
$endgroup$
Let $mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $mathsf{CRing}$ of commutative rings with one.
Let $A$ be in $mathsf{CRing}$.
Question 1. Is there a functor from a small category to $mathsf{Noeth}$ whose limit in $mathsf{CRing}$ is $A$?
(I know that there is a functor from a small category to $mathsf{Noeth}$ whose colimit is $A$.)
Let $f:Ato B$ be a morphism in $mathsf{CRing}$ such that the map
$$
circ f:text{Hom}_{mathsf{CRing}}(B,C)totext{Hom}_{mathsf{CRing}}(A,C)
$$
sending $g$ to $gcirc f$ is bijective for all $C$ in $mathsf{Noeth}$.
Question 2. Does this imply that $f$ is an isomorphism?
Yes to Question 1 would imply yes to Question 2.
Question 3. Does the inclusion functor $iota:mathsf{Noeth}tomathsf{CRing}$ commute with colimits? That is, if $Ainmathsf{Noeth}$ is the colimit of a functor $alpha$ from a small category to $mathsf{Noeth}$, is $A$ naturally isomorphic to the colimit of $iotacircalpha$?
Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do not exist in $mathsf{Noeth}$: see this answer of Martin Brandenburg.
One may try to attack the first question as follows:
Let $A$ be in $mathsf{CRing}$ and $I$ the set of those ideals $mathfrak a$ of $A$ such that $A/mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/mathfrak a$ with $mathfrak ain I$, and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective.
commutative-algebra category-theory noetherian limits-colimits
commutative-algebra category-theory noetherian limits-colimits
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
edited Feb 1 at 13:39
Pierre-Yves Gaillard
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
asked Feb 1 at 12:11
Pierre-Yves GaillardPierre-Yves Gaillard
13.5k23184
13.5k23184
$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
1
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
1
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52
|
show 3 more comments
$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
1
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
1
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52
$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
1
1
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
1
1
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52
|
show 3 more comments
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$begingroup$
I guess by question 1 you mean if there's a functor $F$ from a small category to ${bf Noeth}$ such that $A$ is the limit of $EF$. ($E:{bf Noeth} rightarrow {bf CRing}$ is the inclusion functor.)
$endgroup$
– sqtrat
Feb 1 at 13:24
$begingroup$
@sqtrat - Thanks! Yes. I've added "in $mathsf{CRing}$". I hope it's clear enough now.
$endgroup$
– Pierre-Yves Gaillard
Feb 1 at 13:41
$begingroup$
Doesn't question 2 follow from the fact that a ring is colimit of noetherian rings ?(plug in $C_i$, where $varinjlim C_i =A$ to get $id_A$ has an antecedent, then plug in $D_i$ where $varinjlim D_i = B$ to get that the inverse we get was a $2$-sided inverse) (I used $varinjlim$ to denote the colimit, because in fact we can choose the colimit to be a "direct limit")
$endgroup$
– Max
Feb 1 at 14:12
1
$begingroup$
@Pierre-YvesGaillard : sorry I actually wrote it down and it didn't work
$endgroup$
– Max
Feb 1 at 16:53
1
$begingroup$
Crossposted on MathOverflow: mathoverflow.net/q/323136/461
$endgroup$
– Pierre-Yves Gaillard
Feb 13 at 12:52