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| Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules |
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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 |
| Page view： 2118
Net amount： 1448 |
Authors： |
Nan GAO; Xiaojing XU |
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| 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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