Schanuel’s Lemma In a Localization Framework: The S-Injective Case

Speaker: Dr. Hanni Garminia

Let R be a commutative ring and S a multiplicative subset of R. The classical Schanuel Lemma for injective modules is a fundamental result in homological algebra: whenever two short exact sequences of R-modules share a common left term and both middle terms are injective, the two right terms are isomorphic up to a direct sum with the respective opposite middle terms. This reflects a deep structural symmetry that injective resolutions of a fixed module share at their initial stage. Motivated by the notion of S-injective modules recently introduced by Bennis and Bouziri, which weakens classical injectivity by requiring the extension property only after passing to the localization R_S, we examine how far this structural symmetry persists when the middle terms are merely S-injective rather than injective. We present two analogues of Schanuel’s Lemma in this setting. The first works directly within the category of R-modules and recovers the isomorphism between the right terms under mild additional conditions relating the modules to the set S. The second approach localizes both sequences at S, where S-injectivity becomes classical injectivity over R_S, and the conclusion then follows from the original lemma applied in the localized category. As an immediate consequence, a localized form of the isomorphism holds between the right terms for any two such sequences, without further hypotheses. Taken together, these results show that the comparison principle at the heart of Schanuel’s Lemma remains intact within the broader framework of S-relative homological algebra.