17 research outputs found
A spectral sequence for the Hochschild cohomology of a coconnective dga
A spectral sequence for the computation of the Hochschild cohomology of a
coconnective dga over a field is presented. This spectral sequence has a
similar flavour to the spectral sequence constructed by Cohen, Jones and Yan
for the computation of the loop homology of a closed orientable manifold. Using
this spectral sequence we identify a class of spaces for which the Hochschild
cohomology of their mod-p cochain algebra is Noetherian. This implies, among
other things, that for such a space the derived category of mod-p chains on its
loop-space carries a theory of support varieties.Comment: Final version. The new version adds an application of the results to
the construction of support varieties for modules over the chains algebra of
certain loop-spaces. See Corollary 1.7 and Proposition 2.
Cellular approximations and the Eilenberg-Moore spectral-sequence
We set up machinery for recognizing k-cellular modules and k-cellular
approximations, where k is an R-module and R is either a ring or a
ring-spectrum. Using this machinery we can identify the target of the
Eilenberg-Moore cohomology spectral sequence for a fibration in various cases.
In this manner we get new proofs for known results concerning the
Eilenberg-Moore spectral sequence and generalize another result.Comment: 35 page
Complete intersections and mod p cochains
We give homotopy invariant definitions corresponding to three well known
properties of complete intersections, for the ring, the module theory and the
endomorphisms of the residue field, and we investigate them for the mod p
cochains on a space, showing that suitable versions of the second and third are
equivalent and that the first is stronger. We are particularly interested in
classifying spaces of groups, and we give a number of examples.
This paper follows on from arXiv:0906.4025 which considered the classical
case of a commutative ring and arXiv:0906.3247 which considered the case of
rational homotopy theory.Comment: To appear in AG
Stratifying derived categories of cochains on certain spaces
In recent years, Benson, Iyengar and Krause have developed a theory of
stratification for compactly generated triangulated categories with an action
of a graded commutative Noetherian ring. Stratification implies a
classification of localizing and thick subcategories in terms of subsets of the
prime ideal spectrum of the given ring. In this paper two stratification
results are presented: one for the derived category of a commutative
ring-spectrum with polynomial homotopy and another for the derived category of
cochains on certain spaces. We also give the stratification of cochains on a
space a topological content.Comment: 27 page
Rational Secret Sharing, Revisited
We consider the problem of secret sharing among n rational players. This problem was introduced by Halpern and Teague (STOC 2004), who claim that a solution is impossible for n = 2 but show a solution for the case n >= 3. Contrary to their claim, we show a protocol for rational secret sharing among n = 2 players; our protocol extends to the case n 3, where it is simpler than the Halpern-Teague solution and also o#ers a number of other advantages. We also show how to avoid the continual involvement of the dealer, in either our own protocol or that of Halpern and Teague. Ou
Rationality and adversarial behavior in multi-party computation
Abstract. We study multi-party computation in the model where none of n participating parties are honest: they are either rational, acting in their selfish interest to maximize their utility, or adversarial, acting arbitrarily. In this new model, which we call the mixed-behavior model, we define a class of functions that can be computed in the presence of an adversary using a trusted mediator. We then give a protocol that allows the rational parties to emulate the mediator and jointly compute the function such that (1) assuming that each rational party prefers that it learns the output while others do not, no rational party has an incentive to deviate from the protocol; and (2) the rational parties are protected from a malicious adversary controlling ⌈ n ⌉ − 2 of the participants: the 2 adversary can only either cause all rational participants to abort (so no one learns the function they are trying to compute), or can only learn whatever information is implied by the output of the function.