交换幺半群的同调代数
Homological Algebra for Commutative Monoids
摘要：
我们首先研究交换，指出幺半群提供基本的定义和结果的方式类似交换环理论。包括结果链条件，初级分解以及规范化的一类特殊的monoid导致研究monoid计划，因子，皮卡德集团和类组。它表明，正规化的幺半群不一定是一个幺半群，但可能是一个幺半群计划。在给出了$A$ -集的定义和基本结果之后，我们对射影$A$-集进行了分类，并证明了它们完全由秩决定。随后，对幺半群$A$，我们计算了$K_0$和$K_1$，并证明了$G_0$的设计定理.在定义了$A$ -集的短正合序列的基础上，利用Hochschild余单纯集，描述了$A$ -集$X,Y$的扩张集$Ext(X,Y)$，并将幺半群$A$的平方零扩张集分类为$A$ -集$X$.我们还研究了单纯$A$ -集上的投影模型结构，展示了计算同伦群以及确定幺半群的派生类别所涉及的困难。本文对一类非交换范畴$\mathcal{C}$定义了双箭头复形范畴$\operatorname{Da}(\mathcal{C})$，并在$A$ -集的情况下，给出了它与单纯$A$ -集范畴的一个伴随.
英文摘要：
We first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. After giving the definition of, and basic results for, $A$-sets, we classify projective $A$-sets and show they are completely determine by their rank. Subsequently, for a monoid $A$, we compute $K_0$ and $K_1$ and prove the Devissage Theorem for $G_0$. With the definition of short exact sequence for $A$-sets in hand, we describe the set $Ext(X,Y)$ of extensions for $A$-sets $X,Y$ and classify the set of square-zero extensions of a monoid $A$ by an $A$-set $X$ using the Hochschild cosimplicial set. We also examine the projective model structure on simplicial $A$-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category $\operatorname{Da}(\mathcal{C})$ of double-arrow complexes for a class of non-abelian categories $\mathcal{C}$ and, in the case of $A$-sets, shows an adjunction with the category of simplicial $A$-sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
