AI 中文总结
研究有限胞腔各向同性谱，证明\(Pic(SH(k/k)^c_{cell})\cong\mathbb{Z}^2\)，指出\(E_{**}\)性质及权重结构作用，还表明特定核为合成幂零理想，可检测合成幂零性并给出典范分解。
AI 中文摘要
设\(k = k_0(t_1,t_2,\ldots)\)是特征不为\(2\)的灵活域，\(\mathbb{X}\)是模\(2\)各向同性球面，\(E=\mathbb{X}\wedge MBP\)。证明了\(Pic(SH(k/k)^c_{cell})\cong\mathbb{Z}^2\)，即每个张量可逆的有限胞腔各向同性谱是\(\mathbb{X}\)的唯一双分次悬架。更一般地，\(E_{**}\)在有限胞腔对象上是保守的，对角线上的集中导致悬浮各向同性球面的有限直和。有界对角权重结构从\(E_{**}\)恢复精确权重和最小复项。对于每个非零有限胞腔\(M\)，\(End(M)\to End(E\wedge_{\mathbb{X}}M)\)的核是一个合成幂零理想，指数至多为\(d(M)(2L(M)-1)\)，这产生了合成幂零性的检测和典范菲廷分解。
英文摘要
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.
Comments25 pages
