论文标题
在相对背景下的完整合并对的明确自偶型结构
An explicit self-dual construction of complete cotorsion pairs in the relative context
论文作者
论文摘要
令$ r \ to $为联想环的同态，让$（\ Mathcal f，\ Mathcal c）$是$ \ r \ mathsf {-mod} $中的遗传完整的cotorsion对。令$（\ Mathcal f_a，\ Mathcal C_A）$为$ a \ Mathsf {-mod} $中的CotorSion对，其中$ \ Mathcal f_a $是所有左$ a $ a $ a $ r $ -modules属于$ \ mathcal f $的$ a $ -mmodules的类。假设每个左$ r $ -MODULE的$ \ MATHCAL F $分辨率是有限的，并且$ \ MATHCAL F $由Coinduction Founctor $ \ opperatorname {hom} _r（a， - ）$保留，我们显示了$ \ \ \ \ \ \ \ nathcal c_a $ by the triptire tirect tirect tirect yect y y Mathcal co_a co）co）$ -OD $ -OD $ -OD $ -MOD（ $ a $ - 模块从$ r $ - 模块造成的$ \ Mathcal c $。假设类$ \ Mathcal f $在可数产品下关闭，并由函数$ \ permatatOrname {hom} _r（a， - - ）$保留，我们证明$ \ nathcal c_a $是$ a $ a $ cofile cofter $ a $ cofile cof $ cof $ cof $ coful cof $ coful cof $ cof的$ coful coful cof的$ rys $ ry的所有直接总和。自然数索引的降低过滤。一个组合结果，基于以下假设：从$ \ Mathcal F $中的模块的可数产品具有有限的$ \ Mathcal F $分辨率尺寸，该尺寸由$ k $界定，涉及由序数$ω+k $索引的共滤。双重结果也可以得出，可以通过相同的技术回到作者关于半无限同源代数Arxiv的专着：0708.3398。此外，我们讨论了$ n $ cotilting和$ n $ tISTING COTORSION对，为此，我们使用合适的Bongartz-Ringel Lemma版本获得了更好的结果。作为本文主要结果的例证，我们考虑了与弯曲的DG模型相互矛盾和编码类别相关的某些合并对。
Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let $(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in which $\mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal F$. Assuming that the $\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\mathcal F$ is preserved by the coinduction functor $\operatorname{Hom}_R(A,-)$, we show that $\mathcal C_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal C$. Assuming that the class $\mathcal F$ is closed under countable products and preserved by the functor $\operatorname{Hom}_R(A,-)$, we prove that $\mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal F$ have finite $\mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $ω+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.
