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Let $A$ be an algebra over some field $k$. Let $K_P(A)$ be the Grothendieck group of the category of projective $A$-modules and $K_F(A)$ the category of finite dimensional $A$-modules. I've been told there are examples where $K_P(A)$ and $K_F(A)$ have different rank, but I've never seen an example.

Does anyone have such an example?

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What does PIMS stand for? – Theo Buehler Feb 19 '11 at 1:25
projective indecomposable modules – David Hill Feb 19 '11 at 1:45
I guess the first use of PIM was as an abbreviation for "principal indecomposable module", before the notion of projective module became widespread in this kind of algebra. Fortunately "principal" and "projective" both start with the same letter, though it's actually a little more natural to write "indecomposable projective module". – Jim Humphreys Feb 20 '11 at 20:16
up vote 4 down vote accepted

Let $A=k[x]$. Then $K_P(A)$ has rank 1 (I assume that you consider only finitely generated projective modules) and $K_F(A)$ has infinite rank.

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Do you know one where both are finite dimensional? – David Hill Feb 19 '11 at 1:43

If k is an algebraically closed field, then A = { f(x)/g(x) : g(0)⋅g(1) ≠ 0; f,g in k[x] } ≤ k(x) is a commutative ring with exactly two maximal ideals, (x) and (x−1). It has two simple modules and both happen to be one dimensional as k-vector spaces. It is a PID, so it has only one projective indecomposable module, which is of course free and cyclic as an A-module, but infinite dimensional as a k-vector space.

Hence the rank of KP(A) is 1, and the rank of KF(A) is 2.

This is just a modification of Victor Ostrik's example to cut down on the number of simple modules.

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An example where both groups are of finite rank but different is the Weyl algebra. There are no finite dimensional modules, so one of the groups is trivial, and $K_0(A)=\mathbb Z$.

(I'm considering the $K_0$ if f.g. projectives...)

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Is there anything in between? In some sense, this is the same example with the reciprocal of the rank. – David Hill Feb 19 '11 at 2:10
«this is the same example with the reciprocal of the rank»? I don't know what you mean by that. – Mariano Suárez-Alvarez Feb 19 '11 at 2:13
In Victor's example, there were infinitely many simples but one projective. In your example, there is 1 projective and no simples. The algebras are related by x↔\partial/\partial x so it's like you're taking reciprocals – David Hill Feb 19 '11 at 2:19
The Weyl algebra is generated by $x$ and $d/dx$. Also, it has many, many simple modules, but all of them infinite-dimensional. – Mariano Suárez-Alvarez Feb 19 '11 at 2:21
Also, there are many non-isomorphic projectives, but they collapse in $K_0$. – Mariano Suárez-Alvarez Feb 19 '11 at 2:22

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