On homology of the $MSU$ spectrum

We give a complete proof the Novikov isomorphism $\varOmega^{SU}\otimes \mathbb Z[\textstyle\frac12]\cong\mathbb Z[{\textstyle\frac12}][y_2,y_3,\ldots],\quad\mathrm{deg} y_i=2i$, where $\varOmega^{SU}$ is the $SU$-bordism ring. The proof uses the Adams spectral sequence and a description of the comodule structure of $H_*(MSU; \mathbb F_p)$ over the dual Steenrod algebra $\mathfrak A_p^*$ with odd prime $p$, which was also missing in the literature.