On the heart of a faithful torsion theory (Q865491)

An object \(V\) in an abelian category \(\mathcal{H}\) is said to be tilting if (i) \(\mathcal{H}\) has arbitrary coproducts \(V^{(\alpha)}\) of copies of \(V\), (ii) \(\text{Hom}(V, V^{(\alpha)}) \cong \text{Hom}(V, V)^{(\alpha)}\) for each \(\alpha\), (iii) the class \(\text{Gen} \, V\) of objects generated by \(V\) is the same as that of those \(E\) for which \(\text{Ext}^1(V,E)=0\), (iv) any object of \(\mathcal{H}\) embeds in an object of \(\text{Gen} \, V\). From the definition it follows that for any such \(V\), \(\text{Gen} \, V\) is a torsion class yielding a faithful torsion theory \((\mathcal{X},\mathcal{Y})\) in \({\mathcal M}od-\text{End}\, V\). Given any faithful torsion theory, \((\mathcal{X},\mathcal{Y})\), in a module category \({\mathcal M}od-R\), there is a suitable \(\mathcal{H}(\mathcal{X},\mathcal{Y})\), called the heart of the torsion theory such that \(\mathcal{H}(\mathcal{X},\mathcal{Y})\) has a tilting object with \((\mathcal{X},\mathcal{Y})\) as its associated torsion theory. This allows an analysis to be made of the abelian categories which have tilting objects. Several applications and examples are given.