Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.

Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where $f_k=\sum_{i=0}^{[k/2]}(-1)^{k-i}{{k-i}\choose i}c_1^{k-2i}c_2^i$.

I want to compute the cup-length of the first Chern class $c_1$, i.e., the smallest positive integer $l$ such that $c_1^l=0$. (For the real case, I found a paper *Cup products in Grassmannians, R.E. Stong, Topology and its Applications, Volume 13, Issue 1, 1982, Pages 103–113*).

By considering the dimension, I have $l\leq 2n-1$. Tested by computer, I obtain $l=2n-1$ for $n\leq 30$.

Are there any references or method to prove $l=2n-1$? This question is equivalent to the following question: a problem about ideals of polynomial rings. Thanks!

Complex manifolds without potential theory, p. 82. $\endgroup$ – Francois Ziegler May 11 '15 at 6:20