Pardon the delay. The glue I use is called PVA. It's archival quality, acid free, water soluable, and works great on paper, cardboard, cloth, leather, etc. I just bought a new bottle. The last one I had lasted 10 years. I use it on all my book repairs, except for some paper on paper repairs, which do better with paste. I love this stuff. If you use it on leather, use it sparingly. Too much and it'll soak through and darken the leather.
A torn/peeling spine is an easy repair. There are two main kinds of hardback spines, and you can tell them apart by opening the book wide and looking down at the spine. If it's tight against the back of the text, you can usually just glue down the torn/loose piece of the spine and it'll be fine. If, however, you look down and there's (by design) an open space between the spine and the text, then you need to do a bit more. I'm sure there are technical terms for all this. You can try Googling for photos and details of how to do this, but you need to glue down a section of paper or cloth to act as a backing and hinge, then glue the loose spine to that, to preserve that open space between the spine and the text. It's easy, once you get the idea of how it should work. Good luck.
