Question

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\mathcal{E}$ be a vector bundle of rank $r$, with $\mathcal{O}_X$ the usual sheaf of functions on $X = \mathbb{P}^1$, then we can write our line bundles as the invertible sheaves $\mathcal{O}_{X}(n)$ with $n \in \mathbb{Z}$. Thus, the decomposition can be stated as $$\mathcal{E} \cong \oplus_{i=1}^n \mathcal{O}(n_i) \quad n_1 \leq ... \leq n_r.$$

If we use the usual open cover of $\mathbb{P}^1$ with two affine lines $U_0 = \mathbb{P}^1 - \{\infty\}$ and $U_1 = \mathbb{P}^1 - \{0\}$, note that $\mathcal{O}_{U_0 \cap U_1} = k[x,x^{-1}]$ (with $\mathcal{O}_{U_0} = k[x]$ and $\mathcal{O}_{U_1} = k[x^{-1}]$). A vector bundle (up to isomorphism) $\mathcal{E}$ of rank $n$ is then a linear automorphism on $\mathcal{O}_{U_0 \cap U_1}^r$ modulo automorphisms of each $\mathcal{O}_{U_i}^r$ for $i = 0,1$. (I am looking at the definition given in Hartshorne II.5.18 where $A = k[x,x^{-1}]$, the linear automorphisms are $\psi_1^{-1} \circ \psi_0$ where $\psi_i: \mathcal{O}_{U_i}^r \rightarrow \left.\mathcal{E}\right|_{U_i}$ are isomorphisms, and the definition of isomorphism of vector bundles allows us to change bases of $\mathcal{O}_{U_i}^r$.

Thinking of this in linear algebra terms, these linear automorphisms on $\mathcal{O}_{U_0 \cap U_1}^r$ are elements of $GL_r(k[x,x^{-1}])$, and changing coordinates in $\mathcal{O}_{U_i}^r$ are elements of $GL_r(k[x])$ for $i = 0$ and $GL_r(k[x^{-1}])$ for $i = 1$. Thus up to isomorphism, the vector bundles of rank $r$ on $\mathbb{P}^1$ are elements of the double quotient $$ GL_r(k[x^{-1}]) \left\backslash \large{GL_r(k[x,x^{-1}])} \right/ GL_r(k[x].$$ The decomposition of vector bundles into line bundles SHOULD mean that these double cosets can be represented by matrices of the form $$\left(\begin{array}{cccc} x^{n_1} & & & 0 \\ & x^{n_2}& & \\ & & \ddots & \\ 0 & & & x^{n_r}\end{array}\right) \quad n_1 \leq n_2 \leq ... \leq n_r.$$ I want to know whether there is a way to prove this fact purely via linear algebra (equivalently, if the geometric proof [cf. Lemma 4.4.1 in Le Potier's "Lectures on Vector Bundles"] has a linear algebraic interpretation).

[Note: For the affine case, taking the double quotient $$GL_n(k[x]) \left \backslash M_{n,m}(k[x]) \right/ GL_m(k[x])$$ gives the classification of vector bundles over $\mathbb{A}^1_k$ (and of course, when replacing $k[x]$ with an arbitrary PID, gives the usual structure theorem of finitely generated modules over PID).]

Answer 1

I must admit I have never read this reference, but I remember it from a similar discussion on a German forum, according to which there is a simple proof in

Michiel Hazewinkel, A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line, Journal of pure and applied algebra 25 (1982), pp. 207 - 211.

Answer 2

Dear Ila, the linear algebra result you mention is due to Dedekind-Weber and was published in Crelle's Journal dated 1882, in their article "Theorie der algebraischen Funktionen einer Veränderlichen". Their motivation was proving Riemann-Roch on an arbitrary smooth projective curve $X$ by presenting the curve as a ramified covering of $\mathbb P^1$ and pushing down the line bundle associated to a divisor on $X$ (they used the language of function fields).

Of course Grothendieck was not aware of this result, which was also rediscoverd by Birkhoff in an analytic setting in 1913, by Plemelj in1908, by Hilbert in 1905...

You will find some details on the metamorphosis of this linear algebra theorem into Grothendieck's result in Scharlau's interesting paper

http://wwwmath.uni-muenster.de/u/scharlau/scharlau/grothendieck/Grothendieck.pdf

Answer 3

When I was a graduate student, my advisor Phillip Griffiths told me that the Grothendieck splitting theorem was equivalent to the Kronecker pencil lemma, which gives a normal form for a 2-dimensional space of rectangular matrices. I recall working out the equivalence, and I actually used it in my Ph.D. thesis (which was not on algebraic geometry). The statement and proof of the Kronecker pencil lemma can be found in Gantmacher's book, "The Theory of Matrices" and relies only on linear algebra. I don't know anything about the Dedekind-Weber result cited by Georges Elencwajg. I recall that I found the book "Vector Bundles on Complex Projective Spaces" by Okonek et al to be very helpful.