One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated projective modules over a commutative unital ring $R$.

One way to do this is to consider $iF(R)$ the category of finitely generated free modules instead and then use a cofinality theorem. We can write the category $iF(R)$ as $$iF(R)=\coprod Gl_n(R),$$ where $Gl_n(R)$ is the category with one object and the general linear group as morphisms.

So far for the classical theory. I am considering the category $P(R,\mathbb G_m)$, the category of pairs $(P,\theta)$, where $P$ is finitely generated and projective and $\theta$ is an automorphism of $P$, morphisms have to respect the automorphisms. It would be desirable if I could write the category $iF(R,\mathbb G_m)$ in a similar fashion as $iF(R)$ above.

There are more objects but that's ok, what really gives me a headache are the morphisms. As far as I understand this there is a morphism $C:(R^n,A)\to (R^n,B)$ (now, $A,B$,and $C$ are matrices) if and only if $A$ and $B$ are similar and for any two morphisms $C,D$ we have $CD^{-1}\in Z(B)$ the centraliser of $B$.

So my question: are there any classification results for invertible matrices with entries in arbitrary rings? Sure, we have the characteristic polynomial but maybe there's more. Also is there a relation between the similarity class of a matrix and it's centraliser?

The question turned out to be pretty vague and I am sorry for that. My guess is that there are connections to representation theory. Any suggestion of literature is welcome, really.

Edit: The answers I got so far suggest to use the characteristic polynomials to reformulate the question as a question of modules of projective dimension 1 over the polynomial ring localised at the ideal of monic polynomials with constant part a unit (This is essentially what Grayson does in the paper K-theory of Endomorphisms). I tried this before I asked this question and it lead to a dead end, since the third part in the localisation sequence is as intractable as the one I am interested in. Another paper by Grayson (Weight filtrations via commuting automorphisms) suggests that the (reduced) K-theory space is the delooping of the (truncated ?) space for Karoubi-Villamayor K-theory. I am trying to prove this directly by using the universal property of higher K-theory (cf. group completion). For that I need to show that a certain map is a localisation on homology. My hope was that mimicking the proof for the classical Theorem (see above) would be helpful.

One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated projective modules over a commutative unital ring $R$.

One way to do this is to consider $iF(R)$ the category of finitely generated free modules instead and then use a cofinality theorem. We can write the category $iF(R)$ as $$iF(R)=\coprod Gl_n(R),$$ where $Gl_n(R)$ is the category with one object and the general linear group as morphisms.

So far for the classical theory. I am considering the category $P(R,\mathbb G_m)$, the category of pairs $(P,\theta)$, where $P$ is finitely generated and projective and $\theta$ is an automorphism of $P$, morphisms have to respect the automorphisms. It would be desirable if I could write the category $iF(R,\mathbb G_m)$ in a similar fashion as $iF(R)$ above.

There are more objects but that's ok, what really gives me a headache are the morphisms. As far as I understand this there is a morphism $C:(R^n,A)\to (R^n,B)$ (now, $A,B$,and $C$ are matrices) if and only if $A$ and $B$ are similar and for any two morphisms $C,D$ we have $CD^{-1}\in Z(B)$ the centraliser of $B$.

So my question: are there any classification results for invertible matrices with entries in arbitrary rings? Sure, we have the characteristic polynomial but maybe there's more. Also is there a relation between the similarity class of a matrix and it's centraliser?

The question turned out to be pretty vague and I am sorry for that. My guess is that there are connections to representation theory. Any suggestion of literature is welcome, really.

Edit (I edited my question without refreshing my browser first. Hence, even though the edit is newer, it doesn't refer to the last answer): The answers I got so far suggest to use the characteristic polynomials to reformulate the question as a question of modules of projective dimension 1 over the polynomial ring localised at the ideal of monic polynomials with constant part a unit (This is essentially what Grayson does in the paper K-theory of Endomorphisms). I tried this before I asked this question and it lead to a dead end, since the third part in the localisation sequence is as intractable as the one I am interested in. Another paper by Grayson (Weight filtrations via commuting automorphisms) suggests that the (reduced) K-theory space is the delooping of the (truncated ?) space for Karoubi-Villamayor K-theory. I am trying to prove this directly by using the universal property of higher K-theory (cf. group completion). For that I need to show that a certain map is a localisation on homology. My hope was that mimicking the proof for the classical Theorem (see above) would be helpful.

One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated projective modules over a commutative unital ring $R$.

One way to do this is to consider $iF(R)$ the category of finitely generated free modules instead and then use a cofinality theorem. We can write the category $iF(R)$ as $$iF(R)=\coprod Gl_n(R),$$ where $Gl_n(R)$ is the category with one object and the general linear group as morphisms.

So far for the classical theory. I am considering the category $P(R,\mathbb G_m)$, the category of pairs $(P,\theta)$, where $P$ is finitely generated and projective and $\theta$ is an automorphism of $P$, morphisms have to respect the automorphisms. It would be desirable if I could write the category $iF(R,\mathbb G_m)$ in a similar fashion as $iF(R)$ above.

There are more objects but that's ok, what really gives me a headache are the morphisms. As far as I understand this there is a morphism $C:(R^n,A)\to (R^n,B)$ (now, $A,B$,and $C$ are matrices) if and only if $A$ and $B$ are similar and for any two morphisms $C,D$ we have $CD^{-1}\in Z(B)$ the centraliser of $B$.

So my question: are there any classification results for invertible matrices with entries in arbitrary rings? Sure, we have the characteristic polynomial but maybe there's more. Also is there a relation between the similarity class of a matrix and it's centraliser?

The question turned out to be pretty vague and I am sorry for that. My guess is that there are connections to representation theory. Any suggestion of literature is welcome, really.

One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated projective modules over a commutative unital ring $R$.

One way to do this is to consider $iF(R)$ the category of finitely generated free modules instead and then use a cofinality theorem. We can write the category $iF(R)$ as $$iF(R)=\coprod Gl_n(R),$$ where $Gl_n(R)$ is the category with one object and the general linear group as morphisms.

So far for the classical theory. I am considering the category $P(R,\mathbb G_m)$, the category of pairs $(P,\theta)$, where $P$ is finitely generated and projective and $\theta$ is an automorphism of $P$, morphisms have to respect the automorphisms. It would be desirable if I could write the category $iF(R,\mathbb G_m)$ in a similar fashion as $iF(R)$ above.

There are more objects but that's ok, what really gives me a headache are the morphisms. As far as I understand this there is a morphism $C:(R^n,A)\to (R^n,B)$ (now, $A,B$,and $C$ are matrices) if and only if $A$ and $B$ are similar and for any two morphisms $C,D$ we have $CD^{-1}\in Z(B)$ the centraliser of $B$.

So my question: are there any classification results for invertible matrices with entries in arbitrary rings? Sure, we have the characteristic polynomial but maybe there's more. Also is there a relation between the similarity class of a matrix and it's centraliser?

The question turned out to be pretty vague and I am sorry for that. My guess is that there are connections to representation theory. Any suggestion of literature is welcome, really.

Edit: The answers I got so far suggest to use the characteristic polynomials to reformulate the question as a question of modules of projective dimension 1 over the polynomial ring localised at the ideal of monic polynomials with constant part a unit (This is essentially what Grayson does in the paper K-theory of Endomorphisms). I tried this before I asked this question and it lead to a dead end, since the third part in the localisation sequence is as intractable as the one I am interested in. Another paper by Grayson (Weight filtrations via commuting automorphisms) suggests that the (reduced) K-theory space is the delooping of the (truncated ?) space for Karoubi-Villamayor K-theory. I am trying to prove this directly by using the universal property of higher K-theory (cf. group completion). For that I need to show that a certain map is a localisation on homology. My hope was that mimicking the proof for the classical Theorem (see above) would be helpful.