Exploring a local variant of the Buchsbaum--Eisenbud--Horrocks conjecture

The Buchsbaum--Eisenbud--Horrocks conjecture has attracted the attention of many researchers working in Commutative Algebra and Algebraic Geometry in the last fifty years. Quite recently, a variant of this conjecture has been formulated by Lima--Pereira, Nu\~no--Ballesteros, Orefice--Okamoto and Tomazella in their study of complete intersection singularities. The purpose of this paper is, on the one hand, to exhibit some cases where this conjecture holds. On the other hand, we show that the conjecture fails in general by exhibiting some counterexamples. Finally, we formulate a conjecture that can be regarded as a local variant of the Buchsbaum--Eisenbud--Horrocks' one.