A Simple Proof of Tyurin's Babylonian Tower Theorem

Abstract

Using the method of Coandă and Trautmann [4], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of 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.

Key Words:

• Babylonian tower
• Projective scheme
• Vector bundle

2010 Mathematics Subject Classification:

• Primary: 14F05
• Secondary: 14J60
• 14B10

Notes

Communicated by L. Ein.