Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.