论文标题
$ t $ - 带有Grothendieck Hearts的结构通过函数类别
$t$-Structures with Grothendieck hearts via functor categories
论文作者
论文摘要
我们研究三角形类别中的T结构的核心$ \ Mathcal {d} $带有cocroducts的核心是AB5或Grothendieck类别。如果$ \ MATHCAL {D} $满足棕色的可表示性，则T结构具有带有含量的cogogenerator和coproduct Proveserving相关的同源函数的AB5心脏，并且只有Coaisle具有纯粹的注射式T-cogenerative对象。如果$ \ Mathcal {d} $是标准生成良好的，那么这样的心自动是Grothendieck类别。对于紧凑的T结构（在任何与共同生物的环境三角类别中），我们证明心脏是局部有限的Grothendieck类别。 我们使用函子类别，证明依赖于两种主要成分。首先，我们表达了任何三角类别中任何T结构的核心，作为有限呈现的加性函数类别的Serre serre，以选择过道或我们分别称为T-Generation或T-eneration t-eneration或t-cogeneration的子类别的适当选择。其次，我们从$ \ Mathcal {d} $中研究了合并赋予的同源函数，以用贡献的cogogogenerator来完成AB5 Abelian类别，并根据$ \ Mathcal {d} $中的纯注射对象对它们进行分类，并将其分类至所谓的计算对象。这使我们能够证明，任何标准生成的三角形类别$ \ MATHCAL {D} $都具有这种通用的这种coproduct Product Proserving同源函数，以开发纯度理论，并证明在此类三角形类别中始终纯粹的注射对象始终在此类三角类别中进行T结构。
We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $\mathcal{D}$ is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $\mathcal{D}$ to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $\mathcal{D}$. This allows us to show that any standard well generated triangulated category $\mathcal{D}$ possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.
