1. A computer system for relinquishing, by a selected processor, the selected processor's own exclusive ownership of a specified cache line, in, the system comprising: a multi-processor system comprising the selected processor, each processor of the multi-processor system comprising a private cache, the selected processor comprising a selected private cache, the selected processor configured to perform a method comprising: executing, by the selected processor, a demote instruction of a software program, the demote instruction specifying a logical address of the specified cache line, the executing the demote instruction by the selected processor causing a method to be performed comprising: based on the demote instruction specified logical address, determining, by the selected processor, whether the specified cache line is exclusively owned in the selected private cache of the selected processor; and based on the selected processor determining that the specified cache line is exclusively owned in the selected private cache of the selected processor, relinquishing exclusive ownership, in the selected private cache, of the specified cache line by the selected processor.