Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules

Citation:

Nan GAO,Xiaojing XU.Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules[J].Chinese Annals of Mathematics B,2018,39(1):47~58
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Authors:

Nan GAO; Xiaojing XU
Abstract: The aim of this paper is two-fold. Given a recollement $(\mathcal{T}', \mathcal{T}, \mathcal{T}'', i^*, i_*, i^!,$ $ j_!, j^*, j_*)$, where $\mathcal{T}', \ \mathcal{T}, \mathcal{T}''$ are triangulated categories with small coproducts and $\mathcal{T}$ is compactly generated. First, the authors show that the BBD-induction of compactly generated $t$-structures is compactly generated when $i_{*}$ preserves compact objects. As a consequence, given a ladder $(\mathcal{T}', \mathcal{T}, \mathcal{T}'', \mathcal{T}, \mathcal{T}')$ of height 2, then the certain BBD-induction of compactly generated $t$-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement $(D(B\mbox{-}{\rm Mod}), D(A\mbox{-}{\rm Mod}), D(C\mbox{-}{\rm Mod}), i^*, i_*, i^!,$ $ j_!, j^*, j_*)$ induced by a homological ring epimorphism, the last aim of this work is to show that if $A$ is Gorenstein, $_{A}B$ has finite projective dimension and $j_{!}$ restricts to $D^{b}(C\mbox{-}{\rm mod})$, then this recollement induces an unbounded ladder $(B\mbox{-}\underline{{\rm \mathcal{G}proj}}, A\mbox{-}\underline{{\rm \mathcal{G}proj}}, C\mbox{-}\underline{{\rm \mathcal{G}proj}})$ of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.

Keywords:

Compactly generated $t$-structure, Recollement,BBD-induction,& BPP-induction, Homological ring epimorphism, Gorenstein-& projective module

Classification:

18E30, 16E35
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