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It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freeness of finitely generated $k$-submodules of the coordinate ring of $G$. The same argument (appropriately formulated) then works when $k$ is a PID. (I don't know offhand if the result is true over a non-PID Dedekind domain, but I won't ask that here since it could come down to a little trick.)

The question is this: is the above result true for all artin local rings $k$, or even just the ring of dual numbers over a field? Or can one give a counterexample? Since monic homomorphisms between finite type groups over an artin ring are closed immersions, an equivalent formulation which may be more vivid is: does $G$ admit a (functorially) faithful linear representation on a finite free $k$-module?

(I originally thought I needed an affirmative such result over artin local rings to prove a certain general fact for smooth affine group schemes over noetherian rings, but eventually that motivation got settled in another way. So for me it is now an idle question, though I think a very natural one from the viewpoint of deformation theory of smooth linear algebraic groups.)

It sounds like the sort of thing which must have been thought about back in the 1960's when SGA3 was being written, so I mentioned the question to a couple of the SGA3 collaborators as well as some other experts in these matters. Unfortunately nobody whom I have asked knew one way or the other, even for the dual numbers. One of them suggested a couple of days ago that I should "advertise this problem; it is very provocative." Fair enough; I suppose this kind of advertising on MO is OK.

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freeness of finitely generated $k$-submodules of the coordinate ring of $G$. The same argument (appropriately formulated) then works when $k$ is a PID. (edit: I originally mentioned that I didn't know if this is also true over any Dedekind domain, and wasn't asking about it; nonetheless, comments from George and Kevin below give proof for Dedekind base case.)

The question is this: is the above result true for all artin local rings $k$, or even just the ring of dual numbers over a field? Or can one give a counterexample? Since monic homomorphisms between finite type groups over an artin ring are closed immersions, an equivalent formulation which may be more vivid is: does $G$ admit a (functorially) faithful linear representation on a finite free $k$-module?

(I originally thought I needed an affirmative such result over artin local rings to prove a certain general fact for smooth affine group schemes over noetherian rings, but eventually that motivation got settled in another way. So for me it is now an idle question, though I think a very natural one from the viewpoint of deformation theory of smooth linear algebraic groups.)

It sounds like the sort of thing which must have been thought about back in the 1960's when SGA3 was being written, so I mentioned the question to a couple of the SGA3 collaborators as well as some other experts in these matters. Unfortunately nobody whom I have asked knew one way or the other, even for the dual numbers. One of them suggested a couple of days ago that I should "advertise this problem; it is very provocative." Fair enough; I suppose this kind of advertising on MO is OK.

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freeness of finitely generated $k$-submodules of the coordinate ring of $G$. The same argument (appropriately formulated) then works when $k$ is a PID. (I don't know offhand if the result is true over a non-PID Dedekind domain, but I won't ask that here since it could come down to a little trick.)

The question is this: is the above result true for all artin local rings $k$, or even just the ring of dual numbers over a field? Or can one give a counterexample? Since monomorphisms between finite type groups over an artin ring are closed immersions, an equivalent formulation which may be more vivid is: does $G$ admit a (functorially) faithful linear representation on a finite free $k$-module?

(I originally thought I needed an affirmative such result over artin local rings to prove a certain general fact for smooth affine group schemes over noetherian rings, but eventually that motivation got settled in another way. So for me it is now an idle question, though I think a very natural one from the viewpoint of deformation theory of smooth linear algebraic groups.)

It sounds like the sort of thing which must have been thought about back in the 1960's when SGA3 was being written, so I mentioned the question to a couple of the SGA3 collaborators as well as some other experts in these matters. Unfortunately nobody whom I have asked knew one way or the other, even for the dual numbers. One of them suggested a couple of days ago that I should "advertise this problem; it is very provocative." Fair enough; I suppose this kind of advertising on MO is OK.

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freeness of finitely generated $k$-submodules of the coordinate ring of $G$. The same argument (appropriately formulated) then works when $k$ is a PID. (I don't know offhand if the result is true over a non-PID Dedekind domain, but I won't ask that here since it could come down to a little trick.)

The question is this: is the above result true for all artin local rings $k$, or even just the ring of dual numbers over a field? Or can one give a counterexample? Since monic homomorphisms between finite type groups over an artin ring are closed immersions, an equivalent formulation which may be more vivid is: does $G$ admit a (functorially) faithful linear representation on a finite free $k$-module?

(I originally thought I needed an affirmative such result over artin local rings to prove a certain general fact for smooth affine group schemes over noetherian rings, but eventually that motivation got settled in another way. So for me it is now an idle question, though I think a very natural one from the viewpoint of deformation theory of smooth linear algebraic groups.)

It sounds like the sort of thing which must have been thought about back in the 1960's when SGA3 was being written, so I mentioned the question to a couple of the SGA3 collaborators as well as some other experts in these matters. Unfortunately nobody whom I have asked knew one way or the other, even for the dual numbers. One of them suggested a couple of days ago that I should "advertise this problem; it is very provocative." Fair enough; I suppose this kind of advertising on MO is OK.