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In CMPs, coherence protocols are used to maintain data coherence among the multiple local caches. In this paper, we focus on CMPs using write-through local caches, and a directory-based coherence protocol implemented as a duplicate of the local cache tags. A large fraction of directory lookups is due to stores performed on private data local to the processor performing the store. We propose to add a filter before the directory in order to either reduce the associativity of the lookups or even eliminate those that are unnecessary. When a block from the shared cache has only one copy in the local caches, the filter identifies the processor and allows for reducing the number of comparisons performed in the corresponding directory lookup. When that is not possible, the filter bits are used to code other situations that can also reduce the number of directory lookups or their associativity. We evaluate the fillter in a CMP with 8 in-order processors with 4 threads each and a memory hierarchy with local caches and a shared cache. We show that a filter representing 0.7% of the size of the shared cache can avoid, on average, 97% and 93% of all comparisons performed by directory lookups for SPLASH2 and Specweb2005, respectively. Only for SPLASH2, there is a small performance loss of 0.3%. As a result, on average, directory power is reduced 30.8% and 22.4% for SPLASH2 and Specweb2005, respectively.