What does it mean to say "the cohomology of $\mathbb{C}P^{n-1}$ is of rank $n$"? What does it mean to say "the cohomology of $\mathbb{C}P^{n-1}$ is of rank $n$"? From my knowledge, when considering the rank you would consider the individual ranks of each $H^{k}(\mathbb{C}P^{n-1}; G)$ component, and so I am confused what it means by simply saying cohomology.

The integral cohomology ring of $\mathbb{CP}^{n-1}$ is $H^*(\mathbb{CP}^{n-1}; \mathbb{Z}) \cong \mathbb{Z}[\alpha]/\langle\alpha^n\rangle$ where $|\alpha| = 2$. Note that $\{1, \alpha, \dots, \alpha^{n-1}\}$ is a basis for this $\mathbb{Z}$-module, so it is a free $\mathbb{Z}$-module of rank $n$.
