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We describe the Chow ring with rational coefficients of $\overline{M}_{0,1}(\mathbb{P}^n,d)$ as the subring of invariants of a ring $B^*(\overline{M}_{0,1}(\mathbb{P}^n,d);\mathbb{Q})$ , relative to the action of the group of symmetries Sd. We compute $B^*(\overline{M}_{0,1}(\mathbb{P}^n,d);\mathbb{Q})$ by following a sequence of intermediate spaces for $\overline{M}_{0,1}(\mathbb{P}^n,d)$ .

Let R be a ring. We prove that the homotopy category K(R-Proj) is always $\aleph_1$ -compactly generated, and, depending on the ring R, it may or may not be compactly generated. We use this to give a description of K(R-Proj) as a quotient of K(R-Flat). The remarkable fact is that this new description of K(R-Proj) generalizes to non-affine schemes; this will appear in Murfet’s thesis.