Picard groups and composition nilpotence for finite cellular isotropic spectra

Abstract

Let $k=k_0(t_1,t_2,\dots )$ be a flexible field of characteristic different from $2$, let $\mathbb{X}$ be the mod-$2$ isotropic sphere, and set $E=\mathbb{X}\wedge MBP$. We prove $$Pic(SH(k/k{)}_{cell}^c)\cong {\mathbb{Z}}^2;$$ thus every tensor-invertible finite cellular isotropic spectrum is a unique bigraded suspension of $\mathbb{X}$. More generally, $E_{\ast \ast }$ is conservative on finite cellular objects, and concentration on one diagonal forces a finite direct sum of suspended isotropic spheres. A bounded diagonal weight structure recovers the exact weights and minimal-complex terms from $E_{\ast \ast }$. For every nonzero finite cellular $M$, the kernel of $$End(M)⟶End(E{\wedge }_{\mathbb{X}}M)$$ is a composition-nilpotent ideal, with exponent at most $d(M)(2L(M)-1)$, where $L(M)$ is the diagonal width and $d(M)$ the maximal number of distinct Tate degrees on one diagonal. This yields detection of composition nilpotence and canonical Fitting decompositions.

25 pages