# Syzygies of determinantal varieties: Looking for English text

I would like to understand the syzygies of the determinantal ideal $I_r$, generated by the $r\times r$ minors of a matrix $(X_{ij})$ of indeterminantes in the polynomial ring over an algebraically closed field of characteristic zero. The original resource for this object of study is the paper "Syzygies des variétés déterminantales" by Alain Lascoux [L]. While I would like to read it at some point, my rusty French is making it a bit cumbersome, and hence I was wondering if there were any translations of this treatment in English, possibly in some textbook. Thanks a lot in advance already.

[L] A. Lascoux, Syzygies des variétés déterminantales, Adv. Math. 30 (1978), 202–237.
Mathematical Reviews (MathSciNet): MR520233
Digital Object Identifier: doi:10.1016/0001-8708(78)90037-3

• Is the Bruns-Vetter book ( home.uni-osnabrueck.de/wbruns/brunsw/detrings.pdf ) of any relevance? Apr 10, 2013 at 22:22
• Bruns-Vetter only treats special cases of the syzygies of determinantal ideals. At the end of Chapter 2 they mention Lascoux's work and ask whether a minimal free resolution exists in general over the integers. The answer is no: Hashimoto (in "Determinantal ideals without minimal free resolutions") showed that in characteristic 3, the Betti numbers change for the $2 \times 2$ minors of a $5 \times 5$ matrix (this example can also be seen in Macaulay2 now). Apr 11, 2013 at 13:12