{"paper":{"title":"On the limit of Frobenius in the Grothendieck group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Kazuhiko Kurano, Kosuke Ohta","submitted_at":"2014-07-15T21:56:15Z","abstract_excerpt":"Considering the Grothendieck group modulo numerical equivalence, we obtain the finitely generated lattice $\\overline{G_0(R)}$ for a Noetherian local ring $R$. Let $C_{CM}(R)$ be the cone in $\\overline{G_0(R)}_{\\Bbb R}$ spanned by cycles of maximal Cohen-Macaulay $R$-modules. We shall define the fundamental class $\\overline{\\mu_R}$ of $R$ in $\\overline{G_0(R)}_{\\Bbb R}$, which is the limit of the Frobenius direct images (divided by their rank) $[{}^e R]/p^{de}$ in the case ${ch}(R) = p > 0$. The homological conjectures are deeply related to the problems whether $\\overline{\\mu_R}$ is in the Cohe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4159","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}