Invariant Primes and the Chromatic Filtration
This presentation reveals how Morava stabilizer groups carve out a surprisingly simple structure inside Lubin-Tate deformation space. By combining Cartier theory, renormalized density arguments, and the Fargues-Fontaine period map, the authors prove that invariant prime ideals are exactly the height-stratified ideals that define chromatic filtration. The result settles the Hovey-Strickland classification of thick tensor ideals in K(n)-local spectra at every height and every prime, without relying on large-prime hypotheses.Script
In algebraic topology's chromatic world, symmetry can trap geometry. The Morava stabilizer group acts on Lubin-Tate space by preserving formal group laws, but which prime ideals remain unchanged under open subgroups of this symmetry? The answer turns out to be unexpectedly sharp: exactly the height filtration ideals and nothing else.
Every stabilizer-invariant prime ideal must land in one of two places. In the special fiber modulo p, it collapses to one of the height ideals generated by the first j deformation parameters. In mixed characteristic, if it avoids p entirely, it must be the zero ideal.
The proof avoids wrestling with all n variables simultaneously by choosing a test curve that encodes a connected-étale degeneration. Along this curve, the formal group breaks into a height s connected part and a complementary étale quotient, converting a multivariable algebraic problem into a one-parameter valued extension.
Honda endomorphisms produce obstruction classes that live in the connected-étale extension space, which after base change becomes r copies of the height s Honda group. When these obstruction points are renormalized by powers of p to the s, they become formally dense. This density is the heart of the argument: no nonzero formal equation can vanish on all of them, because Chai rigidity and the Fargues-Fontaine period map rule out any proper subvariety containing them.
The obstruction group is auxiliary, so a fixed-jet comparison links obstruction coordinates to actual stabilizer displacements in Lubin-Tate coordinates. This uniform control over finite Taylor orders ensures that stabilizer orbits, when renormalized, inherit the formal density of obstruction points and rule out any hypothetical strict enlargement of the height ideals.
The invariant radical classification immediately determines the thick tensor ideals of dualizable K(n)-local spectra: they are exactly the ideals generated by localized finite spectra of chromatic type k, for each height from 0 to n. This settles the Hovey-Strickland problem uniformly at every height and every prime. To explore more results in homotopy theory and stable homotopy, visit EmergentMind.com and create your own videos from the latest research.
