Homological epimorphisms, compactly generated \(t\)-structures and Gorenstein-projective modules
Given a set \(S\) of compact objects in a cocomplete triangulated category \(\mathcal{T}\), let \(\mathcal{D}'\) be the smallest full subcategory including \(S\) and closed under the suspension, extensions, coproducts and direct summands, and let \(\mathcal{D}''\) be the full subcategory of \(\mathcal{T}\) formed by the objects \(X''\) such that \(\text{Hom}_{\mathcal{T}}(X',X''[n]) = 0\), for all \(X' \in S\) and \(n \leq 0\). Then, \((\mathcal{D}',\mathcal{D}'')\) forms a \(t\)-structure on \(\mathcal{T}\), called \textit{compactly generated} (see [\textit{L. Alonso Tarrío} et al., Trans. Am. Math. Soc. 355, No. 6, 2523--2543 (2003; Zbl 1019.18007)], Theorem A.1). In the article [\textit{N. Broomhead} et al., J. Algebra 394, 51--78 (2013; Zbl 1322.18003)], the authors proved that, given a finite collection \(\{ (\mathcal{D}'_{i},\mathcal{D}''_{i}) \}\) of compactly generated \(t\)-structures on \(\mathcal{T}\), then the pair \((\mathcal{D}',\mathcal{D}'')\), where \(\mathcal{D}'\) is the smallest full subcategory of \(\mathcal{T}\) including all \(\mathcal{D}'_{i}\) and closed under extensions and direct summands, and \(\mathcal{D}''\) is the intersection of \(\{ \mathcal{D}''_{i} \}\), is a \(t\)-structure (see loc. cit. Theorem A). The latter is called the \textit{BPP-induction} of the previous collection of compactly generated \(t\)-structures in the article under review. On the other hand, it is a classical result that, given a recollement of triangulated categories of \(\mathcal{T}\) with respect to \(\mathcal{T}'\) and \(\mathcal{T}''\), and \(t\)-structures \((\mathcal{X}',\mathcal{X}'')\) on \(\mathcal{T}'\) and \((\mathcal{Y}',\mathcal{Y}'')\) on \(\mathcal{T}''\), then there is an induced \(t\)-structure on \((\mathcal{D}',\mathcal{D}'')\) on \(\mathcal{T}\) (see [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)] , Theorem 1.4.10). The authors called this procedure the \textit{BBD-induction} of the \(t\)-structures \((\mathcal{X}',\mathcal{X}'')\) and \((\mathcal{Y}',\mathcal{Y}'')\). In the article under review, the authors prove that, given a recollement of cocomplete triangulated categories of \(\mathcal{T}\) with respect to \(\mathcal{T}'\) and \(\mathcal{T}''\), such that \(\mathcal{T}\) is compactly generated and the functor from \(\mathcal{T}'\) to \(\mathcal{T}\) preserves compact objects, and collections \(\{ (\mathcal{X}'_{i},\mathcal{X}''_{i})\}\) and \(\{ (\mathcal{Y}'_{i},\mathcal{Y}''_{i})\}\) of compactly generated \(t\)-structures on \(\mathcal{T}'\) and \(\mathcal{T}''\), resp., indexed by the same finite set \(I\), then the corresponding \(i\)-th BBD-induction \((\mathcal{D}'_{i},\mathcal{D}''_{i})\) is a compactly generated \(t\)-structure, for all \(i \in I\), and in fact the BPP-induction and the BBD-induction procedures commute (see Prop. 2.1). The main result of the article is just a direct consequence of (the first part of) this statement, namely that the BBD-induction of two compactly generated \(t\)-structures on a ladder of height \(2\) of cocomplete triangulated categories with a compactly generated category in the middle gives a compactly generated \(t\)-structure (see Theorem 2.1). The authors then prove that, given a homological epimorphism \(\lambda : A \rightarrow B\) of finite dimensional algebras such that \(A\) is Gorenstein and \(B\) has finite projective dimension over \(A\), the usual recollement \((D(B\text{-}\text{Mod}), D(A\text{-}\text{Mod}), D(C\text{-}\text{Mod}), i^{*},i_{*},i^{!},j_{!},j^{*},j_{*})\) of unbounded derived categories induces an unbounded ladder for the corresponding stable categories of finitely generated Gorenstein-projective modules, if \(j_{!}\) restricts to the bounded derived category of \(C\) (see Theorem 3.1). They then provide some examples.
scientific article; zbMATH DE number 6838912
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