Homological properties of torsion classes under change of rings (Q1326999)

For a ring \(R\) with identity let Mod-\(R\) denote the category of left unital \(R\)-modules and \(R\)-tors denote the family of hereditary torsion theories on Mod-\(R\). Let \(x\) be a central element of \(R\) which is neither a unit nor a zero divisor, let \(S\) denote the factor ring \(R/xR\) and let \(\varphi : R \to S\) be the natural epimorphism. Then each \(M \in \text{Mod-}S\) becomes a left \(R\)-module with \(ra = \varphi(r)a\) for all \(r \in R\), \(a \in M\). Now define the map \(\varphi_ * : R \text{-tors} \to S \text{-tors}\) as follows: for each \(\tau \in R \text{-tors}\), \(\varphi_ *(\tau) = \sigma\) where \(M \in \text{Mod-}S\) is \(\sigma\)- torsion if and only if \(M\) is \(\tau\)-torsion as a left \(R\)-module. This paper investigates relationships between \(R\)-tors and \(S\)-tors induced by \(\varphi_ *\). The first section considers the functors \(\text{Ext}^ n\). For example, using spectral sequences the author shows that if \(\sigma = \varphi_ *(\tau)\) is a perfect torsion theory and \(N\) is a \(\sigma\)-torsion \(S\)-module, then \(\text{Ext}^ 2_ R (N,M) \cong \text{Ext}^ 1_ S (N,M/ xM)\) for any \(\tau\)-closed \(R\)-module \(M\). In the second section he investigates induced relationships between localization functors, in particular when these commute with Hom and the tensor product.