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Register a new userFiltration Games and Potentially Projective Modules
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Authors
Sean D. Cox
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The notion of a textbf{$boldsymbol{mathcal{C}}$-filtered} object, where $mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the textbf{$boldsymbol{mathcal{C}}$-Filtration Game of length $boldsymbol{omega_1}$} on a module, paying particular attention to the case where $mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martins Maximum implies the determinacy of many $mathcal{C}$-Filtration Games of length $omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraiss{e} games of length $omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen cite{MR1191613}. Also, Martins Maximum implies that if $R$ is a countable hereditary ring, the class of textbf{$boldsymbol{sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $sigma$-closed forcing extension of the universe---is closed under $<aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.
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We prove a number of results on the determinacy of $sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $sigma$-projective games of a given countable length and of games with payoff in the smallest $sigma$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).
Let $M^sharp_n(mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(mathbb{R})$ class-many times. We characterize the sets of reals which are $Sigma_1$-definable from $mathbb{R}$ over $M_n(mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Pi^1_{n+1}$; (2) for odd $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $mathbb{R}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists for all $ninmathbb{N}$, and that determinacy of all projective games of length $omega^2$ with moves in $mathbb{N}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists and satisfies $mathsf{AD}$ for all $ninmathbb{N}$.
Let B be a commutative Bezout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes of modules over the localizations of B by the maximal ideals of B, and on the other hand, of the constructible subsets of MSpec(B). When B has good factorization, it allows us to derive decidability results for the class B-modules, in particular when B is the ring of algebraic integers or its intersection with real numbers or p-adic numbers.
We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.
In 2011, Rideau and Winskel introduced concurrent games and strategies as event structures, generalizing prior work on causal formulations of games. In this paper we give a detailed, self-contained and slightly-updated account of the results of Rideau and Winskel: a notion of pre-strategy based on event structures; a characterisation of those pre-strategies (deemed strategies) which are preserved by composition with a copycat strategy; and the construction of a bicategory of these strategies. Furthermore, we prove that the corresponding category has a compact closed structure, and hence forms the basis for the semantics of concurrent higher-order computation.
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