Copied from the FAQ on the DMT site
How do I take care of my sharpener?
Clean your DMT® sharpener after each use with water and wipe dry.
After several sharpenings, use a mild kitchen abrasive cleanser and a nylon scrub brush (an old toothbrush works well too). Comet, Ajax, and Zud work well.
Make sure the sharpener is dry before putting it away"
I know it says to use the sharpener dry or with water, but can I use oil anyway? Can I store my DMT stone in kerosene?
While we recommend you use DMT® products dry or with water only, it is acceptable to use a variety of lubricants when sharpening with DMT® products.
Mineral oil, mineral spirits, kerosene, WD40 can all be used without harming your DMT® sharpener.
We do not recommend storing your DMT sharpener in kerosene. DMT® recommends that you always wipe your Diamond and Unbreakable Ceramic Sharpeners clean and store dry after use.