Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the [Bass conjecture](http://en.wikipedia.org/wiki/Bass_conjecture) states that $K\_0(A)$ is a finitely generated abelian group. As far as I am aware, this is wide open in general. My question is about whether something weaker is known to hold.



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> Let $n \in \mathbb{Z}$. Then does $A$ have only finitely many finitely generated projective modules of rank $n$, up to isomorphism?
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If I understand it correctly, it seems like Bass' conjecture implies a positive answer to my question. However, it is not clear to me that a positive answer to my question would imply the finite generation of $K\_0(A)$, so my hope is that it is weaker and known.



A weak version of Bass' conjecture
The following two statements might clarify the relation between Bass conjecture (for $K\_0$) and its weak version in the question. I only consider the case of commutative rings.


1. Contrary to the formulation of the question, the weak Bass conjecture implies the Bass conjecture for $A$ finitely generated over $\mathbb{F}\_q$.
2. The weak version of the Bass conjecture for $A$ smooth of dimension $\leq 3$ over a finite field can be deduced (using $\mathbb{A}^1$-homotopy classification of projective modules) from a refined version of the Bass conjecture and the Parshin conjecture.


I give a more detailed sketch of arguments below:


Ad 1: Serre proved that if $X$ is an affine scheme of Krull dimension $d$, then all vector bundles of rank $>d$ split off a free direct summand, cf. Theorem 1 in J.-P. Serre: Modules projectifs et espaces fibrés à fibre vectorielle. Séminaire Dubreil, Dubreil-Jacotin et Pisot, 1957/1958, Fasc 2, Exposé 23, p. 18, Sécretariat mathématique, Paris, 1958. This result implies that there is a global upper bound for the number of projective modules of given rank $n$ on an affine scheme of dimension $d$, namely the sum over $1\leq i\leq d$ of the numbers of projective modules of rank $i$. The weak Bass conjecture would say that this is a finite number. But then, the group completion of the monoid of isomorphism classes of projective modules will be finitely generated, so we have the ($K\_0$-case of the) Bass conjecture. So the weak version of Bass' conjecture is in fact a strong version of the Bass conjecture.


The result of Serre also implies that if the weak version of the Bass conjecture holds for an affine variety of dimension $d$ over $\mathbb{F}\_q$ and all ranks $\leq d$, then it holds for all ranks. 


Ad 2:
Consider $A$ a smooth variety of dimension $\leq 3$ over $\mathbb{F}\_q$. The following is a sketch of an argument that the refined Bass conjecture and the Parshin conjecture imply the weak version of the Bass conjecture in the question. This is not entirely precise, as it uses some boundary
cases of results from $\mathbb{A}^1$-homotopy theory (such as finite
base fields...) and there might be considerable work turning this into a proof. I still think it conveys a good intuition for the strength of the "weak Bass conjecture". The basic idea is that projective modules over a smooth algebra of dimension $\leq 3$ are classified essentially by their Chern classes which lie in Chow groups $CH^i(X)$, the refined Bass conjecture and Parshin conjecture together imply that these Chow groups are finite. 


First of all, the homotopy classification of projective modules: 
Fabien Morel (in his book on $\mathbb{A}^1$-algebraic topology) has
proved that for $X$ a smooth affine variety, there is a bijection
between rank $n$ projective modules over $X$ and
$\mathbb{A}^1$-homotopy classes of maps $X\to BGL\_n$. Note in Morel's
book there is a restriction $n\neq 2$ but that can be lifted using
work of his student Moser. With this, it is possible to use classical
topology obstruction methods to classify projective modules over
smooth algebras. There is a recent series of amazing papers by Aravind
Asok and Jean Fasel in which they are pursuing this line of thought. I
would in particular like to point to arXiv:1204.0770, where they show
that for a smooth affine $3$-fold over an algebraically closed field
of characteristic $\neq 2$, the rank $2$ vector bundles can be
classified by their Chern classes. Although this result needs algebraically closed fields, the methods adapt to finite fields: the proof uses the Postnikov tower for $BGL\_2$, and the classification of projective modules then can be made precise in terms of lifting classes in group
$H^i\_{\operatorname{Nis}}(X,\pi\_i(BGL\_2))$. 


The lifting groups and projective module classification: the relevant homotopy group sheaves for $BGL\_2$ for classification of projective modules over varieties of dimension $\leq 3$ are
$\pi\_1^{\mathbb{A}^1}BGL\_2\cong \mathbb{G}\_m$ (Morel-Voevodsky),
$\pi\_2^{\mathbb{A}^1}BGL\_2\cong K^M\_2$ (Morel) and
$\pi\_3^{\mathbb{A}^1}BGL\_2$ is an extension of $KSp\_3$ by some
quotient of Milnor $K\_4$ and the fifth power of the fundamental ideal
in the Witt ring (Asok-Fasel, in the paper cited above). 
Now the relevant Nisnevich cohomology groups are
$H^1(X,\mathbb{G}\_m)$, $H^2(X,K^M\_2)\cong CH^2(X)$ and $H^3(X,KSp\_3)$
is the cokernel of $Sq^2:Ch^2(X)\to Ch^3(X)$. This is not entirely honest, there is some finite amount from orientability issues because $\mathbb{F}\_q$ is not quadratically closed. Anyway, over a finite field, the major information for projective module classification lies in 
$Pic(X)$, $CH^2(X)$ and $Ch^3(X)$. 


Finiteness conjectures: Now what do the above groups have to do with the Bass conjecture? The relation is not that direct, Chow groups are not
K-theory. Nevertheless, the conjecture is that Chow groups are
finitely generated - this is called ``refined Bass conjecture'' in [these talk notes](http://www.math.sci.hokudai.ac.jp/sympo/100809/pdf/geisser.pdf) of Thomas Geisser. In those same talk notes, you can also find Parshin's conjecture, which would imply that $CH^2(X)$ and $CH^3(X)$ are not just finitely generated but also torsion (if the base field is finite). 
Provided you believe the homotopy classification of projective modules above, the refined Bass conjecture and Parshin conjecture imply the weak Bass conjecture for $A$ finitely generated of Krull dimension $\leq 3$ over $\mathbb{F}\_q$. 




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Some further remarks: 


1. The counterexample proposed in Daniel Litt's comment would not be a counterexample if the refined Bass conjecture and Parshin conjecture are true. For a smooth affine surface over $\mathbb{F}\_q$ with trivial Picard group, all the information would be in $CH^2(X)$ (maybe an oriented Chow group would be needed). So, although there are many $0$-dimensional local complete intersections in the affine surface, the isomorphism type of the vector bundle only depends on the class in $CH^2(X)$ - and there would only be finitely many such classes.
2. Generally, the classification of projective modules will be impossible to do, much like it is impossible to classify vector bundles on compact manifolds in general. Nevertheless, the expectation is that there is a range (called
metastable range by Suslin) in which the projective module
classification can be done in terms of Chow groups, K-theory,
hermitian K-theory and operations connecting them. In this range,
finiteness of projective modules (over function rings of smooth
affine schemes over $\mathbb{F}\_q$) would follow from finite generation of Chow groups (refined Bass conjecture) and Parshin's conjecture.
3. The extension of the above argument to finitely generated $\mathbb{Z}$-algebras would be a lot more subtle. It would need the full Bass-Quillen conjecture on projective modules over polynomial rings. Moreover, the $\mathbb{A}^1$-homotopy classification results for projective modules have not been done (though that obstacle seems a lot smaller than the general Bass-Quillen conjecture in the non-geometric case). It is also not clear to me if the resulting motivic cohomology groups in arithmetic situations would be expected to be finite, or just finitely generated. A search for counterexamples in such settings might be more promising.


