This paper presents a novel static analysis technique to detect XML query-update independence, in the presence of a schema. Rather than types, our system infers chains of types. Each chain represents a path that can be traversed on a valid document during query/update evaluation. The resulting independence analysis is precise, although it raises a challenging issue: recursive schemas may lead to infer infinitely many chains. A sound and complete approximation technique ensuring a finite analysis in any case is presented, together with an efficient implementation performing the chain-based analysis in polynomial space and time.