We investigate periodic review inventory management problems for a manufacturer in a decentralized supply chain under information sharing. The manufacturer orders from its upstream supplier and fulfills demand ordered by its downstream retailer. The upstream supplier has infinite capacity, provides a constant order lead time, and charges zero fixed ordering cost to the manufacturer. The downstream retailer faces independent and identically distributed customer demand, orders from the manufacturer according to a given (s, S) policy, and shares both the realized customer demand and its own inventory position with the manufacturer. The manufacturer incurs linear inventory holding and backorder penalty costs applied to overage and underage scenarios, respectively, and aims to minimize the long-run average holding and penalty costs. We propose an exact optimization algorithm to derive the optimal state-dependent base-stock policy. Furthermore, through comparative analysis with the case without information sharing, we provide insights into the value of information.