It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo‐resolutions. In this paper, we study the height 2 telescope conjecture at the prime 2 through the lens of tmf‐resolutions. To this end, we compute the structure of the tmf‐resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E1‐page of the tmf‐resolution possesses a decomposition into a v2‐periodic summand, and an Eilenberg–MacLane summand which consists of bounded v2‐torsion. However, unlike the height 1 case, the E2‐page of the tmf‐resolution exhibits unbounded v2‐torsion. We compare this to the work of Mahowald–Ravenel–Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v2‐torsion: either the unbounded v2‐torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2‐parabolas and the telescope conjecture is false. We also study how to use the tmf‐resolution to effectively give low‐dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)‐local Adams–Novikov spectral sequence for Z collapses.

We prove that the minimal v 2 -self-map of the 2-local spectrum A 1 has periodicity 32.

This paper is a contribution to the presentation of fractal sets in terms of final coal-gebras. The first result on this topic was Freyd's Theorem: the unit interval [0, 1] is the final coalgebra of a functor X → X ⊕ X on the category of bipointed sets. Leinster [L] offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called self-similar spaces using (a) bimodules (also called profunctors or distributors), (b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction of final coalgebras for the types of functors of interest using a notion of resolution. In addition to the characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces. Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions is not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces of interest as the Cauchy completion of an initial algebra, and this initial algebra is the set of points in a colimit of an ω-sequence of finite metric spaces. This generalizes Hutchinson's characterization of fractal attractors in [H] as closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is "computationally related" to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space of dyadic rational numbers in [0, 1]. Our second contribution is not completed at this time, but it is a set of results on metric space characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui [HJN], and our interest in quotient metrics comes from [HJN]. So in terms of (a)-(c) above, our work develops (a) and (b) in metric settings while dropping (c).

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