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We determine the integral Chow and cohomology rings of the moduli stack$\mathcal{B}_{r,d}$ of rank $r$, degree $d$ vector bundles on $\mathbb{P}^1$bundles. We first show that the rational Chow ring$A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$ is a free $\mathbb{Q}$-algebra on $2r+1$generators. The isomorphism class of this ring happens to be independent of$d$. Then, we prove that the integral Chow ring $A^*(\mathcal{B}_{r,d})$ istorsion-free and provide multiplicative generators for $A^*(\mathcal{B}_{r,d})$as a subring of $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$. From this description,we see that $A^*(\mathcal{B}_{r,d})$ is not finitely generated as a$\mathbb{Z}$-algebra. Finally, the cohomology ring of $\mathcal{B}_{r,d}$ isisomorphic to its Chow ring.

The intersection theory of the moduli stack of vector bundles on $\mathbb{P}^1$