Existence of unliftable modules

Abstract

Let ( Q , n ) (Q,\operatorname {\mathfrak {n}}) be a commutative Noetherian local ring, and let R = Q / ( x ) R=Q/(x) where x x is a non-zerodivisor of Q Q contained in n \operatorname {\mathfrak {n}} . Then a finitely generated R R -module M M is said to lift to Q Q if there exists a finitely generated Q Q -module M ′ M’ such that x x is M ′ M’ -regular and M ≅ M ′ / x M ′ M \cong M’/xM’ . In this paper we give a general construction of finitely generated R R -modules of finite projective dimension over R R which fail to lift to Q Q provided x ∈ n 2 x \in \operatorname {\mathfrak {n}}^{2} and the depth of R R is at least 2.