Chromatic homotopy is algebraic when p>n²+n+1

In chromatic homotopy theory, one stratifies the stable homotopy category by fixing a prime and looking at the \$E(n)\$-local parts, which informally see "information up to height \$n\$". As the height grows, these categories become increasingly intricate and converge to the p-local homotopy theory in a precise sense. 

On the other hand, it has been observed that when the prime is large relative to the height, then the \$E(n)\$-local category simplifies considerably - for example, the \$E(n)\$-local homotopy groups of spheres admit a purely algebraic description.  

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 theory is algebraic at large primes. This extends the work of Bousfield at \$n = 1\$ to all heights, and affirms a conjecture of Franke.