I always trim the spent stalk with a sharp knife right down to the rhizome..yes it's full of water, not bacteria (I lost many my 1st year by leaving it on!) I rinse with a small paint brush dripping with a water/bleach solution, let air dry, then cover.
**Note. I've taken a newly planted rhizome out of the soil, when it showed rot (not sure why) sliced away nearly half & treated it,let dry out a day. It actually bloomed the following Spring, so don't underestimate those small rhizomes!
