According to how long you want to keep your truffles, you have several possibilities.
If you want to keep them for a few days, put your truffle in a jam jar with a lead. A piece of tissue paper in it will absorb the excess of humidity It will then keep for about a week at the bottom of the fridge.
You can also slice your truffle, cover the slices with a good olive oil. It will keep for a few days and the oil will be flavoured.
For a longer keep : if you wish to keep the truffle whole, freeze it in a jam jar perfectly hermetic, the truffle will be wrapped in a piece of foil.
You can slice it, freeze the slices on a tray. When the slices are frozen, keep them in a zip bag. It will be easier to get just the quantity that you need.
In the same way you can mince your truffle, you will get a sort of « granite » that you will freeze on a tray and keep in a jam jar in the freezer. Just take a spoonfull when needed.
Another possibility is to preserve it in alcool : madeira, cognac or alcool for fruits. This process modifies the aromas. It is important for alcool to be over 50 ° otherwise some fermenttion will start.
These truffles can be used for sauces and stews.
Sterilization must be avoided , it kills the aromas