Ok, they are finally done. I waited for all the foliage to die before I dug them up. Now I have a bunch of tiny pearl-onion-sized bulbs. Do I save these for next year?
What you could do is leave some in the ground, and take the rest out - to see which method works better in your particular climate.
Mine stay in all year unless they're being thinned. However, with your more continental summer (hot, dry) it might be too warm for them.
Don't fridge them but do keep them in a cool and shaded place over the summer while they're out of the ground. Protect from mousies and other pests.
If you saved seed - sow in July-August and you may well get flowers for spring from the seedlings.
Plant the bulbs again in September-October, as soon as the weather starts cooling. Leaves will be coming through October-November.
I keep mine in a large, about 14 inch pot, filled with Freesia bulbs. They were through blooming about 2 weeks ago, and are now moved to out of sight location to be ignored until they revive next year. The pot is in the shade, but other wise not protected. The same mix has been in the pot for 3 years now and next year I may replace the mix with new, if I think of it. They were outstanding this spring. I have a grid made out of 2x2 inch hardware cloth mounted about 10 inches above the pot, and feed the foliage up through as it grows to minimize the flopping of the stems as they bloom. Al