Intervals of s-torsion pairs in extriangulated categories with negative first extensions

Comment

In triangulated categories, for a fixed $t$-structure, the HRS-tilt gives a poset isomorphism between the posets of (i) “intermediate” $t$-structures w.r.t. that $t$-structure and (ii) torsion pairs in the heart of that $t$-structure.

In abelian categories, Asai-Pfeifer and Tattar establish a poset isomorphism between the posets of (i) interval of a fixed torsion pair $[\mathcal{U},\mathcal{T}]$ (that is, the poset of torsion classes $\mathcal{T}’$ satisfying $\mathcal{U} \subseteq \mathcal{T}’ \subseteq \mathcal{T}$) and (ii) “torsion pairs” in the subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ (this category can be regarded as an exact category, and torsion pair is defined using this exact structure in a similar way to abelian categories).

In this paper, we show that these two bijections in triangulated categories and abelian categories are just special cases of a general bijection in extriangulated category. To treat these two examples, we need to introduce the negative first extension structure $\mathbf{E}^{-1}(X,Y)$ on extriangulated categories, which is an analogue of $\mathrm{Hom}(X,\Sigma^{-1} Y)$ in triangulated categories. The typical example of extriangulated categories with negative first extensions is an extension-closed subcategory of triangulated categories.

Our bijection is useful when we study extension-closed subcategories of triangulated categories. It often happens that an abelian category is equivalent to an extension-closed subcategory with a non-trivial negative first extension. (For example, the abelian category is embedded in a $m$-periodic derived category under some conditions, and its induced negative first extension is $(m-1)$-st Ext group in the original abelian category.)

In addition, I think it is enough interesting even when we are studying exact categories, since maybe this paper is the first one which studies torsion pairs in exact categories.

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