Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let $E^{(a)}$ denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then $X^{(2)}$ computes $A^{(\omega )}$, we show that there is a ceer R such that $R\ge {\mathrm{Id}}^{(n)}$, for every finite ordinal n, but, for all k, $R^{(k)}?{\mathrm{Id}}^{(\omega )}$ (here Id is the identity equivalence relation). We show that if $a,b$ are notations of the same ordinal less than ${\omega }^2$, then $E^{(a)}\equiv E^{(b)}$, but there are notations $a,b$ of ${\omega }^2$ such that ${\mathrm{Id}}^{(a)}$ and ${\mathrm{Id}}^{(b)}$ are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form ${\mathrm{Id}}^{(a)}$ where a is a notation for ${\omega }^2$.     

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 $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.