photo library

can you delete a picture on your phone but have it remain in iCloud photo library??

No, only, if you do not enable iCloud Photo Library on your iPhone. If your iPhone is syncing with iCloud Photo Library, any photo you delete from the iPhone will be removed from iCloud Photo Library as well. The purpose of iCloud Photo Library is to update any changes across all devices. To save storage on your iPhone enable "Optimize Storage". Then your iPhone will only store smaller versions of the photos.