Beilinson ’ s construction of nearby cycles and gluing

We fix a field k. All the schemes we will consider will be separated and of finite type over k. If X is a scheme, we write D c(X) for the category of bounded constructible `-adic complexes on X or, if k = C, for bounded constructible complexes on X(C). (The formalism will work in both cases.) In both cases, we’ll write Perv(X) for the heart of the selfdual perverse tstructure on D c(X). When we say “exact” later, it is always understood to mean “exact for the perverse t-structure”. The standard reference for this paragraph is the book [2] by BeilinsonBernstein-Deligne.