Deformations of modules through butterflies and gerbes

Abstract

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category A-mod of modules over a fixed ring A in a topos E. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site A-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by llusie.

Deformations of Algebras with 2-Extensions

“…• obtains the theorems of [Her20] about modules for algebras, • fixes a subtle flaw (Example A.3) in [Wis12]. This class vanishes if and only if there is a solution B ′ to Question 0.1.…”

“…Remark 3.4. One can obtain the same map Hom A (I, M ) → Exal 2 A (B, M ) as in [Her20,§4]. Choose a flat A ′ -algebra with a surjection P → B:…”

“…To explore ð M -gerbes and move further down the long exact sequence of sheaf cohomology, we need a 2-category enriched over extensions of algebras. As for extensions of modules [Her20], [Wis21], objects will be longer sequences 0 → M → N → R → B → 0. We need to generalize the condition M 2 = 0.…”

“…The author was trying to guess the correct notion of 2-extension when B. Briggs told him it could be found in [LS67] and even [Ger66]. This notion obtained the results of [Her20] in the same fashion and restated some of those of [Wis15].…”