Cohomology ring of infinite complex projective space

Cell structure: by [F1] with $n=1$ the finite Grassmannians ${\mathbb{C}\mathbb{P}}^N={\mathrm{Gr}}_1({\mathbb{C}}^{N+1})$ have exactly one cell in each real dimension $0,2,\dots ,2N$, and by [F2] their union ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ is a CW complex with exactly one cell in each even dimension and none in odd dimensions.

The cellular complex: by [F3] the cellular cochain complex of ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ with $\mathbb{Z}$ coefficients has $C^{2k}=\mathbb{Z}$ and $C^{2k+1}=0$ for all $k\ge 0$; in particular every differential of the complex is zero, so $H^{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}$ and $H^{2k+1}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=0$. By [F4] the cellular chain complex likewise gives $H_{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}$, $H_{2k+1}=0$, hence freeness and finite generation in each degree.

Generators: by [F6], naturality of the Euler class identifies the restriction of $u=e({\gamma }_{\mathbb{R}})$ along ${\mathbb{C}\mathbb{P}}^k\hookrightarrow {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ with the finite-stage class $x$. By [F5], $H^{\ast }({\mathbb{C}\mathbb{P}}^k;\mathbb{Z})=\mathbb{Z}[x]\mathrm{/}(x^{k+1})$ and $x^k$ generates $H^{2k}$. Hence the restriction of $u^k$ is $x^k\mathrm{\ne }0$, so $u^k\mathrm{\ne }0$ in the infinite cyclic group $H^{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})$ of step 2.1. Moreover the cellular restriction to the $2k$-skeleton sends the degree-$2k$ cellular coordinate isomorphically to the sole degree-$2k$ cell, so this nonzero restriction has coefficient $\pm 1$; thus $u^k$ is a generator.

Ring structure: the multiplication is generated by $u$ in degree two, and by step 3.1 each power $u^k$ is a generator of the infinite cyclic group $H^{2k}$; therefore $H^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}[u]$ as a graded ring, which with step 2.1 gives the full assertion.

Boundary cases. For $k=0$ the statement reads $H^0=\mathbb{Z}$, the class $u^0=1$ being a generator; the empty space does not occur, and the coefficient ring $\mathbb{Z}$ is nonzero. The degrees are unbounded above, but each degree is a single cyclic group, so no finiteness in dimension is asserted. The trivial line $u=0$ occurs only over the empty base, which is excluded. AC enters only through [A1].