A simple proof of Tyurin's babylonian tower theorem

TL;DR

This paper presents a simplified proof of Tyurin's theorem on vector bundle extensions and extends the method to prove the Babylonian tower theorem for locally complete intersection subschemes in projective spaces.

Contribution

It offers a straightforward proof of Tyurin's theorem and generalizes the Babylonian tower theorem using the method of Coandă and Trautmann.

Findings

Simplified proof of Tyurin's theorem for vector bundles.

Extension of the Babylonian tower theorem to locally complete intersection subschemes.

Demonstrates the effectiveness of the method of Coandă and Trautmann.

Abstract

Using the method of Coand\u{a} and Trautmann (2006), we give a simple proof of the following theorem due to Tyurin (1976) in the smooth case: if a vector bundle $E$ on a $c$-codimensional locally Cohen-Macaulay closed subscheme $X$ of the projective space $P^n$ extends to a vector bundle $F$ on a similar closed subscheme $Y$ of $P^N$, for every $N > n$, then $E$ is the restriction to $X$ of a direct sum of line bundles on $P^n$. Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.