幺半群概形的形变理论 II：交换幺半群的预笛卡尔余扩张
Deformation Theory of Monoid Schemes II: Precartesian Coextensions of Commutative Monoids
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中文总结 AI 辅助
本文延续前期工作，将交换幺半群的预笛卡尔余扩张研究推广至交换幺半群系统，同时推广两类经典余扩张，明确了该框架的核心差异并提出需发展新上同调方法。
中文摘要 AI 辅助
本文是 arXiv:2606.17088 的续篇，在该文中，作者研究了阿贝尔群系统对交换幺半群的预笛卡尔余扩张。本文将其推广为研究交换幺半群系统的余扩张。其中，作者证明该版本能同时推广 Leech 的余扩张版本和 Rédei 的版本（也称为 Schreier 余扩张）。作者表明，从阿贝尔群系统过渡到交换幺半群系统所付出的代价是 $\boldsymbol{\textsf{Pcoex}(M, \boldsymbol{\textsf{L}})}$ 中的拟逆；具体而言，它们不再是对称范畴群，而仅成为对称幺半群群胚。此外，尽管未在正文中明确说明，另一个核心差异在于：对于作为幺半群函子的幺半群概形 $X$，预笛卡尔余扩张不再必须是幺半群概形，因此 $\boldsymbol{\textsf{Pcoex}(X, \boldsymbol{\textsf{L}})}$ 实际上不适用于研究幺半群概形的余扩张。该理论的其余部分仍成立，但需要开发一种新的上同调方法。
英文摘要
This paper is a continuation of arXiv:2606.17088, where I studied precartesian coextensions of commutative monoids by systems of abelian groups. In this paper, we generalise it to study coextensions by systems of commutative monoids. Among other things, we showcase that this version is able to simultaneously generalise Leech's version of coextensions and Redei's version, also called Schreier coextensions. We show that what going from systems of abelian groups to systems of commutative monoids costs us is quasi-inverses in $\mathsf{Pcoex}(M, \mathcal{L})$. Specifically, instead of a symmetric categorical group, they now only become symmetric monoidal groupoids. Moreover, though not explicitly stated in the body of the paper, another core difference is that for a monoid scheme $X$, regarded as a monoid functor, a precartesian coextension no longer need to be a monoid scheme. Thus, $\mathsf{Pcoex}(X, \mathcal{L})$ is, as stated, ineffective at studying monoid scheme coextensions. The rest of the theory goes through, but requires developing a new cohomological approach.
