Reconsidering Resolutions

In Willing, Wanting, Waiting, Richard Holton lays out a detailed account of resolutions, arguing that they enable agents to resist temptation. Holton claims that temptation often leads to inappropriate shifts in judgment, and that resolutions are a special kind of first- and second-order intention pair that blocks such judgment shift. In this paper, I elaborate upon an intuitive but underdeveloped objection to Holton’s view – namely, that his view does not enable agents to successfully block the transmission of temptation in the way that he claims, because the second-order intention is as equally susceptible to temptation as the first-order intention alone would be. I appeal to independently compelling principles – principles that Holton should accept, because they help fill an important explanatory gap in his account – to demonstrate why this objection succeeds. This argument both shows us where Holton’s view goes wrong and points us to the kind of solu-tion we need. In conclusion, I sketch an alternative account of resolutions as a first-order intention paired with a second-order desire. I argue that my account is not susceptible to the same objection because a temptation that cannot be blocked by an intention can be blocked by a desire

Rational acyclic resolutions

Let X be a compactum such that dim_Q X 1. We prove that there is a Q-acyclic resolution r: Z-->X from a compactum Z of dim < n+1. This allows us to give a complete description of all the cases when for a compactum X and an abelian group G such that dim_G X 1 there is a G-acyclic resolution r: Z-->X from a compactum Z of dim < n+1.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-12.abs.htm

Filtering free resolutions

A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest "wild" quiver.Comment: We correct a mistake in the proof of Corollary 4.2 in the published version of this paper. The mistake involves an incorrect definition for when two degree sequences are "sufficiently separated". The new definition weakens Theorem 1.3 somewhat, but the examples survive. We thank Amin Nematbakhsh and to Gunnar Floystad for bringing this mistake to our attention. We also correct some minor typo