AI 中文总结
本文在阿贝尔范畴带乘积的复形范畴中提出同伦极限概念，证明其可计算导出极限复形、与极限拟同构，且是同伦上极限的对偶，余局部化子范畴对其稳定。
AI 中文摘要
我们提出了阿贝尔范畴上带乘积的复形范畴中同伦极限的概念，该概念通过将Roos复形的经典构造整体化得到，而Roos复形用于计算导出逆极限。对于非交换环上的模复形，我们证明该同伦极限构造可计算导出极限复形；在逆系统满足无环性假设时，它与极限拟同构。我们进一步证明，该构造一般而言是[Alonso、Jeremías和Souto：《复形范畴的局部化与无界预解》，载于《加拿大数学杂志》（2000年）]中复形同伦上极限构造的恰当对偶，且模的导出范畴中导出极限与上极限也存在对偶行为。最后，我们证明余局部化子范畴对同伦极限是稳定的。
英文摘要
We propose a notion of homotopy limit in the category of complexes over an abelian category with products by totalizing the classic construction of the Roos' complex that computes derived inverse limits. For complexes of modules over a non-necessarily commutative ring, we show that our construction of homotopy limits computes the derived limit complex, and under an acyclicity hypothesis on the inverse system, we prove that it is quasi-isomorphic to the limit. We further show that, in general, the construction is appropriately dual of the previous construction of homotopy colimits of complexes from [Alonso, Jeremías and Souto: Localization in categories of complexes and unbounded resolutions. \textit{Canad. J. Math.} (2000)], and that there also is a dual behavior between derived limits and colimits in derived categories of modules. Finally, we show that colocalizing subcategories are stable for homotopy limits.
Comments31 pages. Comments welcome
