Nilpotent approximation and completion of $\mathbb{E}_\infty$-algebra objects of stable symmetric monoidal model categories

Mathematics > Category Theory [Submitted on 27 Sep 2023 (v1), last revised 18 Jun 2026 (this version, v2)] Title:Nilpotent approximation and completion of $\mathbb{E}_\infty$-algebra objects of stable symmetric monoidal model categories View PDF HTML (experimental)Abstract:We develop a nilpotent approximation theory for Smith ideals, extending adic completion for commutative rings to monoid objects in locally presentable symmetric monoidal abelian categories and to $\mathbb{E}_\infty$-algebra objects in stable symmetric monoidal model categories. The main result is a formal completeness theorem: finite generation of a Smith ideal forces completeness of its nilpotent approximation. This gives a categorical analogue of the finite generation completeness phenomenon in classical adic completion, while remaining distinct from ordinary adic completion of quotient rings. As applications, we construct an almost mathematics version of nilpotent approximation and prove a homotopical completeness theorem for weakly compact Smith ideals. We then apply the general theory to motivic spectra. For the canonical morphism from algebraic cobordism to algebraic K-theory, we construct the corresponding K-theoretic nilpotent approximation of algebraic cobordism, prove its homotopical completeness and Bott periodicity, and establish a mod-$\ell$ Gabber rigidity theorem for the analogous approximation of $\mathbf{MGL}/\ell$ by $\mathbb{K}/l$. Submission history From: Yuki Kato [view email][v1] Wed, 27 Sep 2023 11:26:37 UTC (19 KB) [v2] Thu, 18 Jun 2026 09:07:50 UTC (131 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.