标题： Borel可定义同调代数在局部紧群中的应用 显示英文标题

标题： Applications of Borel-definable homological algebra to locally compact groups

摘要： 我们证明从局部紧致波兰阿贝尔群范畴$\mathbf{LCPAb}$到波兰阿贝尔群范畴$\mathbf{PAb}$的$\mathrm{Hom}$函子有一个全右导函子，改进了Hoffmann和Spitzweck对其上同调右导函子的构造。 我们还将子范畴$\mathbf{PAb}$的左心脏的描述应用于具有波兰覆盖和可测定义的群同态的群，以完全表征$\mathbf{LCPAb}$的左心脏中的内射对象和投射对象，以及由以下群生成的其全子范畴的左心脏中的内射对象和投射对象：有限生成群、李群、完全不连通群、拓扑扭群、拓扑$p$-群、有限秩的局部紧群、有限秩的拓扑扭群、有限秩的拓扑$p$-群以及类型$\mathbb{A}$群。 显示英文摘要

摘要： We show that the $\mathrm{Hom}$ functor from the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups to the category $\mathbf{PAb}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck's construction of its cohomological right derived functor. We also apply the description of the left heart of subcategories of $\mathbf{PAb}$ in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological $p$-groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological $p$-groups of finite ranks, and type $\mathbb{A}$ groups.