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In a multi-level cache such as those used for web caching, a hit at level l leads to the caching of the requested object in all intermediate caches on the reverse path (levels l - 1 ..... 1). This paper shows that a simple modification to this de facto behavior, in which only the l - 1 level cache gets to store a copy, can lead to significant performance gains. The modified caching behavior is called Leave Copy Down (LCD); it has the merit of being able to avoid the amplification of replacement errors and also the unnecessary repetitious caching of the same objects at multiple levels. Simulation results against other cache interconnections show that when LCD is applied under typical web workloads, it reduces the average hit distance. We construct an approximate analytic model for the case of LCD interconnection of LRU caches and use it to gain a better insight as to why the LCD interconnection yields an improved performance.