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Do closed immersions preserve stalks?

Stalks of the graph of a morphismClosed subsets and closed subschemesShowing closed immersions are stable under base extension without using that they are affine.$f:Xrightarrow Y$ is a closed immersion iff $f:f^-1(U_i)rightarrow U_i$ is a closed immersion.A category of closed immersionsSchemes as closed subschemes in étale schemesWhat does Liu mean by “topological open/closed immersion” in his book “Algebraic Geometry and Arithmetic Curves”?Closed subschemes as subsetsHelp understanding closed subschemes and closed immersionsDefinition of Immersion

1

$begingroup$

This questions stems from an earlier confusion about the distinction between open and closed immersions between schemes.

I understand that an open immersion $Uto S$ can simply be read as an open subset $Usubseteq S$, as the only scheme structure you can put on $U$ which is in a sense "compatible" with the inclusion into $S$ is that given by restricting the structure sheaf of $S$ to $U$. Consequently the stalk $O_U, u$ for any $uin U$ should be isomorphic to that same stalk $O_S, u$ for $S$.

I used to hold the false belief that closed immersions could similarly be identified with closed subsets. I now understand that a closed subset of a scheme can of course admit many different scheme structures - in fact, every closed subset can be considered a closed immersion from a reduced scheme! However, I wonder if at least some of that intuition is correct. Do closed immersions still preserve stalks?

I suppose this can be reduced to the affine case, so that we can simply ask "if $f:Rto T$ is a surjective morphism of rings, and $pin Spec(T)$, then is it the case that $T_p cong R_f^-1(p)$?" This does not appear obviously to be true.

A further question if this is false is ... what is the point of closed immersions? What is the intuition behind them? The course I am studying never really addressed such questions.

schemes

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asked Mar 28 at 14:22

NethesisNethesis

1,9221823

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1

$begingroup$

This questions stems from an earlier confusion about the distinction between open and closed immersions between schemes.

I understand that an open immersion $Uto S$ can simply be read as an open subset $Usubseteq S$, as the only scheme structure you can put on $U$ which is in a sense "compatible" with the inclusion into $S$ is that given by restricting the structure sheaf of $S$ to $U$. Consequently the stalk $O_U, u$ for any $uin U$ should be isomorphic to that same stalk $O_S, u$ for $S$.

I used to hold the false belief that closed immersions could similarly be identified with closed subsets. I now understand that a closed subset of a scheme can of course admit many different scheme structures - in fact, every closed subset can be considered a closed immersion from a reduced scheme! However, I wonder if at least some of that intuition is correct. Do closed immersions still preserve stalks?

I suppose this can be reduced to the affine case, so that we can simply ask "if $f:Rto T$ is a surjective morphism of rings, and $pin Spec(T)$, then is it the case that $T_p cong R_f^-1(p)$?" This does not appear obviously to be true.

A further question if this is false is ... what is the point of closed immersions? What is the intuition behind them? The course I am studying never really addressed such questions.

schemes

share|cite|improve this question

asked Mar 28 at 14:22

NethesisNethesis

1,9221823

$endgroup$

add a comment | 

$begingroup$

This questions stems from an earlier confusion about the distinction between open and closed immersions between schemes.

I understand that an open immersion $Uto S$ can simply be read as an open subset $Usubseteq S$, as the only scheme structure you can put on $U$ which is in a sense "compatible" with the inclusion into $S$ is that given by restricting the structure sheaf of $S$ to $U$. Consequently the stalk $O_U, u$ for any $uin U$ should be isomorphic to that same stalk $O_S, u$ for $S$.

I used to hold the false belief that closed immersions could similarly be identified with closed subsets. I now understand that a closed subset of a scheme can of course admit many different scheme structures - in fact, every closed subset can be considered a closed immersion from a reduced scheme! However, I wonder if at least some of that intuition is correct. Do closed immersions still preserve stalks?

I suppose this can be reduced to the affine case, so that we can simply ask "if $f:Rto T$ is a surjective morphism of rings, and $pin Spec(T)$, then is it the case that $T_p cong R_f^-1(p)$?" This does not appear obviously to be true.

A further question if this is false is ... what is the point of closed immersions? What is the intuition behind them? The course I am studying never really addressed such questions.

schemes

share|cite|improve this question

asked Mar 28 at 14:22

NethesisNethesis

1,9221823

$endgroup$

share|cite|improve this question

asked Mar 28 at 14:22

NethesisNethesis

1,9221823

share|cite|improve this question

asked Mar 28 at 14:22

NethesisNethesis

1,9221823

asked Mar 28 at 14:22

NethesisNethesis

1,9221823

asked Mar 28 at 14:22

NethesisNethesis

1,9221823