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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.

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3 Answers 3

up vote 7 down vote accepted

The category $P(R,\mathbb G_m)$ is defined in such a way that a morphism $(R^m,A) \to (R^n,B)$ is a matrix C with CA=BC. The category (groupoid) $iP(R,\mathbb G_m)$ of isomorphisms consists of those morphisms where C is invertible, i.e., where C is a similarity between A and B, and for good rings, such as commutative nonzero rings, that will force m=n. Perhaps that will help with the original "headache".

The automorphisms are those isomorphisms where m=n and A=B.

To write the category (groupoid) $iP(R,\mathbb G_m)$ of automorphisms as equivalent to a disjoint union of groupoids, each with just one object, one chooses an object in each isomorphism class and discards the others, which is what is what prompted the original question about classification results.

Some papers of mine discuss the K-theory of that category, but without investigating the original question, so whether they are relevant depends on the unstated goal. In case they are relevant, here they are:

http://www.math.uiuc.edu/~dan/cv.html#endo

http://www.math.uiuc.edu/~dan/cv.html#witt

http://www.math.uiuc.edu/~dan/cv.html#auto

http://www.math.uiuc.edu/~dan/cv.html#wfilt2

The approach chosen by the original poster, using $S^{-1} S$, will address only the direct sum K-theory of the category, not the (usual) exact sequence K-theory of the category. The difference between the two is relevant to the last paper on the list, and is treated by Suslin in

http://www.math.uiuc.edu/K-theory/0588/

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Wow, thank you for your answer. I am aware of all these papers, apart from the second one. In fact I have been working with them for a while know. This might be pretty forward, but since you have shown interest in my question, would you be open to discuss this in more detail? It would be much appreciated. –  Simon Markett Jun 18 '12 at 9:36
    
To clarify, I am not asking you to extend your answer here in any way. I am asking whether it would be ok if I wrote you an e-mail with my thoughts. I don't know whether this is appropriate but I figgured I might as well ask. –  Simon Markett Jun 18 '12 at 16:13
    
Yes............ –  Dan Grayson Jun 18 '12 at 19:58
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I believe that the best way to think of it is in terms of modules. Say we have a matrix with characteristic polynomial $f$. Then it gives $R^n$ the structure of a $R[\alpha]/f(\alpha)$-module. This module is often locally free of rank $1$, with different conditions depending on different ring properties.

So we are asked to understand line bundles on $\textrm{Spec} R[\alpha]/f(\alpha)$, or at least those line bundles that pushforward to trivial vector bundles on $\textrm{Spec} R$. This problem is quite subtle - it includes, for instance, computing the class numbers of every algebraic number field.

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Minimal polynomials are not well-defined in general. –  user2035 Jun 18 '12 at 10:17
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According to a theorem of Almkvist the characteristic polynomial injects your Grothendieck group into the multiplicative group of formal power series $1+r_1T+r_2T^2+\dots$. There is an old paper by Grayson about this, and about what this has to do with Witt vectors.

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To avoid confusion: Almkvist's theorem is about the endomorphism category, while the question seems to be about the automorphism category (?). I recall that Grayson says something about the automorphism category somewhere, I think in "K-theory and endomorphisms", and the story is quite complicated in general. –  Fabian Lenhardt Jun 15 '12 at 17:41
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