Vanishing of Tor modules and homological dimensions of unions of aCM schemes

We study the vanishing of some Tori(M,R/J ) when R is a local Cohen–Macaulay ring, J any ideal of R with R/J Cohen–Macaulay and M a finitely generated R-module. We use this result to study the homological dimension of unions X ∪ Y of arithmetically Cohen–Macaulay closed subschemes of Pr. In particular, we show that “generically” such a homological dimension is the expected one. We give some generalization when one of the two schemes has codimension 2 and we apply this result to the monomial case.