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Tuesday, October 9, 2018

**Abstract:** It is generally accepted that the structure of the $E(n)$-local homotopy theory, where $E(n)$ is the p-local Johnson-Wilson spectrum at height $n$, becomes increasingly algebraic when $p$ is large with respect to $n$. To give two examples of this phenomena, when the prime is large, the $E(n)$-based Adams spectral sequence collapses on the second page for degree reasons, giving an algebraic description of the homotopy of the $E(n)$-local sphere. Moreover, it is a result of Hovey and Sadofsky that the only invertible spectra in this range are the spheres, showing that the $E(n)$-local Hopkins' Picard group is isomorphic to the integers at large primes, in stark contrast to what happens when $p$ is small. In this talk, we show that when $p > n^2+n+1$, the homotopy category of $E(n)$-local spectra is equivalent to the homotopy category of differential $E(n)_*E(n)$-comodules, giving a precise sense in which chromatic homotopy is "algebraic" at large primes. This extends the work of Bousfield at $n = 1$ to all heights, and affirms a conjecture of Franke.