So I was concerned with one of Hatcher's remarks. When he is computing the cohomology ring of the Grasmannian he makes a short aside by saying that $H^\ast(F_n(\mathbb{C}^k);\mathbb{Z})$ is equal to the free module generated by $x_1^{j_1}\cdots x_n^{j_n}$ where $j_i\leq k-i$. In other words this can be realized as just $\mathbb{Z}[z_1]/(z_1^{k})\otimes\cdots\otimes\mathbb{Z}[z_n]/(z_n^{k-n+1})$. But Hatcher also claims that the cohomology ring can be rewritten as $\mathbb{Z}[z_1,...,z_n]/(\sigma_1,...,\sigma_n)$, where $\sigma_i$ are the symmetric polynomials. Are these two spaces equivalent or have I misunderstood something. I would really appreciate some feedback. Thanks.
EDIT: Note that the first half can be seen from page 436.