The tarp lays or lies flat on the ground

It lies flat on the ground

So you lie down on the sofa (no direct object), but you lay the book down on the table (the book is the direct object). This is in the present tense, where you are talking about doing something now: you lie down on the sofa, and you lay down a book. There are a bunch of ways to remember this part. So the answer is Lays.