I am trying to understand Horrocks's  [construction of vector bundles](http://dl.dropbox.com/u/3849644/Construction%20of%20Bundles.pdf). However I have been stuck on the proof the first theorem in the paper.


In the paper, a trivial bundle is a direct sum of Hopf bundles $\mathcal{O}(p)$. 


Theorem: Let $E$ be a vector bundle without a trivial direct summand. Then there exist a trivial bundle $T$ such that $E\oplus T$ has a filtration
$$E\oplus T=F^0 \supseteq F^1\supseteq F^2\supseteq\cdots\supseteq F^N=0$$
with $F^i/F^{i+1}$ a twisted exterior power of the tangent bundle $T\_{\mathbb{P}^n}$.


Here is his proof:


Take a resolution $L$ of the dual of $E$ by trivial sheaves which is exact as a resolution of graded modules. The dual $L^\*$ can be dismantled into Koszul complexes. 


Here are my questions.
How to break up $L^\*$ into Koszul complexes?
What are the Koszul complexes? 
Where does the $T$ come from?


Thanks in advance!



Proof of a Theorem in the paper "Construction of bundles on P^n" by Horrocks
That is a pretty terse proof! Let me give an outline of a proof that I know. First, one could deduce the statement from a more general:


**Theorem 1**: Let $R$ be a regular local ring, $E$ be a reflexive $R$-module locally free on $U\_R$, the punctured spectrum such that $E$ has no free direct summand. Then one can find a free module $T$ and a filtration:


$$E\oplus T = F\_0 \supseteq F\_1 \supseteq \cdots F\_N =0$$


with $F\_i/F\_{i+1}$ a syzygy of $k=R/m$. 


Why is this local statement implies what you want? 


Let $A=k[x\_0,\cdots, x\_n],m=(x\_0,\cdots,x\_n), X=Proj(A)=\mathbb P^n, R=A\_m$. There is natural functor from the category of vector bundles on $X$ to that of vector bundles on $U\_R$, which is the same as the category of reflexive $R$-modules which are locally free on $U\_R$. This is used by Horrocks all the time and is explained in Section 9 of his [paper](http://plms.oxfordjournals.org/cgi/pdf_extract/s3-14/4/689): "Vector bundles on punctured spectrum of a regular local ring". 


A proof of Theorem 1 can be found in Chapter 5 (theorem 5.2) of the book "Syzygy" by Evans-Griffith. A brief outline in case you can't find the book:


As suggested in the paper you quoted, one starts with a minimal resolution of $E^\*$. Then dualizing gives a complex (remember that $E^{\*\*} \cong E$ as $E$ is reflexive):


$0 \to E \to L\_0 \to L\_1 \cdots $ 


whose cohomologies are $Ext^i(E^\*,R)$. Let $i>0$ be the smallest number such that $X=Ext^i(E^\*,R) \neq 0$ Break the l.e.s in to the exact sequences:


$0\to E \to L\_0 \to L\_1 \cdots \to L\_i \to N \to 0 (\*)$ 


and $0 \to X \to N \to N/X \to 0$. Now build free resolutions for $X$ and $N/X$ and map them onto $(\*)$ as in [Horseshoe Lemma](http://en.wikipedia.org/wiki/Horseshoe_lemma), stopping at the spot $E$, one gets a s.e.s: 


$0 \to B \to E\oplus T \to C \to 0$
here $T$ is free and $C$ is a syzygy of $X = Ext^i(E^\*,R)$. Repeat if necessary and you have a filtration whose quotient are syzygies of various $ Ext^i(E^\*,R)$. But each of this $Ext$ modules has finite length (as $E$ is locally free on $U\_R$), so they can be filtered by copies of $k$. Now use the same trick to build a finer filtration whose quotients are syzygies of $k$. Since $R$ is regular, the resolution of $k$ is the Koszul complex, answering your second question. 


