Derived categories, resolutions, and Brown representability

Exercises on derived categories, resolutions, and Brown representability

These exercises complement my notes "Derived categories, resolutions, and Brown representability".

$Σ$-pure injectivity and Brown representability

We prove that a right $R$-module $M$ is $Σ$-pure injective if and only if $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$. Consequently, if $R$ is a unital ring, the homotopy category $\mathbf{K}({\mathrm{Mod}\text{-} R})$ satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply the main result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman.

Failure of Brown representability in derived categories

Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that…

Tate动机导出范畴的分层

Stratification of Derived Categories of Tate Motives

本研究对特定代数闭域上混合Tate动机导出范畴的局部化张量理想分类，证明其满足Barthel-Heard-Sanders意义下的分层，核心是提出借助Brown–Adams可表示性传递分层的新技术，具备独立研究价值。

On the abelianization of derived categories and a negative solution to Rosicky's problem

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky funct…

Cubical Resolutions and Derived Functors

We introduce pseudocubical objects with pseudoconnections in an arbitrary category, obtained from the Brown-Higgins structure of a cubical object with connections by suitably relaxing their identities, and construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors. The crucial point in the construction is that projective precubical resolutions which are naturally used to define our cubical derived functors possess pseudodegeneracies and pseudoconnections. The same fact is essentially used for proving that in the case of an additive functor between abelian categories,…