We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.

The purpose of this erratum is to point out a mistake in a previous article by the authors and to explain how this affects the results of that article.

In this paper, we construct some unirational Calabi–Yau threefolds in characteristic 3. We adopt the method by Schoen, but we use quasi-elliptic surfaces instead of elliptic surfaces. We find new examples which do not admit a lifting to characteristic zero.

After recalling briefly the main properties of the amalgamated duplication of a ring$R$along an ideal$I$, denoted by $R\bowtie I$ , see M.$D’Anna$and M.$Fontana$, to appear in$J. Algebra Appl.$, we restrict our attention to the study of the properties of $R\bowtie I$ , when$I$is a multiplicative canonical ideal of$R$, see W. J.$Heinzer$, J. A.$Huckaba$and I. J.$Papick$,$Comm. Algebra$. In particular, we study when every regular fractional ideal of $R\bowtie I$ is divisorial.

In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let$J$_{σ}be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [$n$]={1,2,...,$n$}. For any$d$-subset$S$⊂[$n$], let $m_{\preceq_{\textrm{rev}}S}(\sigma)$ denote the number of$d$-subsets$R$∈σ which are equal to or smaller than$S$with respect to the reverse lexicographic order. We will prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all$S$⊂[$n$]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all$S$⊂[$n$] and for all nonsingular matrices ϕ, where Δ_{ϕ}(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$ .