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Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of vector bundles over $X$ (this definition is due to Nori, if I am not mistaken). For example, any trivial or torsion line bundle is finite in this sense.

Now, let $k$ be a field and $X= \mathbb A_k^n -\{0\}$ with $n\geq 3$. My question is:

Are there non-trivial finite vector bundles on $X$? If the answer is no, are there elementary ways to see this?

In the case $k=\mathbb C$, I think the answer is no, as follows: by results of Nori in his thesis, finite bundles gives rise to representations of the fundamental groups scheme, so it is enough to see this group is trivial. But over $\mathbb C$, such group coincides with the etale fundamental group, and $X$ is simply connected. This argument seems to break down over arbitrary fields.

Motivation: I would like to mention this in a talk next week!

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2 Answers 2

If I understand your definition of "finite", then any stably trivial bundle is finite. In particular, if R denotes the real numbers, the bundle over $Spec(R[X,Y,Z]/(X^2+Y^2+Z^2=1)$ defined by the unimodular row (X,Y,Z) is finite. Unless I'm missing something, it should be easy to extend this example to the punctured affine space. $A_R^3-\{0\}$.

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Hi Steven, thank you. I do not totally understand, are you suggesting we take the first syzygy of $(x,y,z)$? But then wouldn't it give a non-trivial stably trivial bundle over the punctured spectrum of the local ring at $0$? The category of v.b over punctured spectrum of a regular local ring is Krull-Schmidt (Horrocks), so what am I missing? –  Hailong Dao Dec 31 '10 at 17:09
Hailong: On the punctured affine space, take the first syzygy of (x,y,z). This restricts to a non-trivial bundle on $Sp(R[X,Y,Z]/(X^2+Y^2+Z^2-1))$, so it is non-trivial. We can also restrict this bundle in the other direction to the punctured spectrum of the ring at the origin. You are arguing that it becomes trivial there. But I'm not sure why you see this as a contradiction. –  Steven Landsburg Dec 31 '10 at 17:45
Steven: but it can not be trivial on the punctured spectrum, if it is the module of sections would be free on the local ring, violating Auslander-Buchsbaum formula. So it can't be stably trivial on the punctured spectrum. –  Hailong Dao Dec 31 '10 at 18:43
Hailong: Over the punctured affine space, write $$0\rightarrow V\rightarrow O^3 \rightarrow O\rightarrow 0$$ where $O$ is the structure sheaf and the map $O^3\rightarrow O$ is $(f,g,h)\mapsto Xf+Yg+Zh$. Surely this map is split by the map that takes 1 to $(X,Y,Z)/(X^2+Y^2+Z^2)$. So this would seem to show that $V$ is stably trivial. And $V$ certainly restricts to a non-trivial bundle on $Spec(R[X,Y,Z]/(X^2+Y^2+Z^2 - 1)$, which makes it non-trivial. Does this seem right to you? If so, there must be something wrong with your Auslander-Buchsbaum argument. –  Steven Landsburg Dec 31 '10 at 19:39
Ah. Maybe I'm being stupid. X^2+Y^2+Z^2 is, I guess, not actually a unit on the punctured affine space, though it's non-zero at every real point. –  Steven Landsburg Dec 31 '10 at 19:46

By Nori's theorem, the existence of non-trivial finite vector bundles on a reduced connected scheme $X$ of finite type over a perfect field $k$ with a rational point is equivalent to the fact that for every finite group scheme $G$ over $k$, every $G$-torsor is a pullback from $k$.

On $X = \mathbb A^n\smallsetminus \{0\}$ over an algebraically closed field of positive characteristic there are non-trivial finite bundles: in this case $X$ has lots of non-trivial connected étale covers (for example, by Artin-Schreier).

In characteristic 0, however, this does not happen; it follows from the equivalence between étale covers of $X$ and of $\mathop{\rm Spec}k$, and from the fact that every finite group scheme over $k$ is étale, that there is an equivalence of $G$-torsors on $X$ and on $\mathop{\rm Spec}k$.

[Edit]: Of course Torsten is right, Nori's theorem applies to projective varieties. The answer above is just wrong.

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Something seems fishy here: On $\mathbb A^n$ there are no non-trivial vector bundles by Suslin-Quillen but lots of étale covers (Artin-Schreier for instance) in positive characteristics. –  Torsten Ekedahl Dec 31 '10 at 13:35
Dear Angelo, does the argument above show that there are finite bundles on projective spaces over some (perhaps non-agebraically closed) fields of positive characteristics? –  Hailong Dao Jan 1 '11 at 9:46

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