These are notes for two talks given by Mark Hovey at the Summer School on the Interactions between Homotopy Theory and Algebra at the University of Chicago, July 26 to August 6, 2004. Because they are notes, they are a bit more chatty and a bit more likely to contain errors than a paper would be, so caveat lector. They are based on the papers [Hov02], [Gil04b], and [Gil04a], and concern the rel...

We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa’s formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

We introduce a natural way to define Selmer groups and $p$-adic $L$-functions for modular forms of weight 1. The corresponding Galois representation $\rho$ $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is 2-dimensional Artin with odd determinant. Thus, the dimension $d^{+}$ (+1)-eigenspace complex conjugation Choose prime $p$ such that restriction local group $\mathr...