Granting exclusive cache access using locality cache coherency state

A cache coherency management facility to reduce latency in granting exclusive access to a cache in certain situations. A node requests exclusive access to a cache line of the cache. The node is in one region of nodes of a plurality of regions of nodes. The one region of nodes includes the node requesting exclusive access and another node of the computing environment, in which the node and the another node are local to one another as defined by a predetermined criteria. The node requesting exclusive access checks a locality cache coherency state of the another node, the locality cache coherency state being specific to the another node and indicating whether the another node has access to the cache line. Based on the checking indicating that the another node has access to the cache line, a determination is made that the node requesting exclusive access is to be granted exclusive access to the cache line. The determining being independent of transmission of information relating to the cache line from one or more other nodes of the one or more other regions of nodes.