Picard groups and composition nilpotence for finite cellular isotropic spectra

Abstract

Let $k=k_0(t_1,t_2,\ldots)$ 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 $$Pic\bigl(SH(k/k)^c_{cell}\bigr)\cong\mathbb Z^2; $$ thus every tensor-invertible finite cellular isotropic spectrum is a unique bigraded suspension of $\mathbb X$. More generally, $E_{**}$ 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 $E_{**}$. For every nonzero finite cellular $M$, the kernel of $$End(M)\longrightarrow End(E\wedge_{\mathbb X}M)$$ 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.