I agree with Brian on one condition. That is if the tail breaks at a facet. The open facet is designed by God to heal without outside aid. That is why the "tail breaking" option is a good choice. But cutting also works, it just leaves a more bloody wound and you have to crunch through a vertebrae, which to me is a gagging experience. Either way, the amp is life-saving.
In the wild, when they are chased by a predator, they automatically drop part of their tails at a facet. These heal without antibiotics, even in the tropics. It's called autotomy (sp).
Either way, your ig's amputation is very likely to heal on it's own without special antibiotics and injections. Use antibiotic ointment, not antibiotic cream on the end of the tail + about 1/2 inch up the tail. Use gauze, iodine (not providone-iodine), and antibiotic ointment, and tape for the first week. This keeps the wound from gathering bacteria laden dust and other small stuff. I know I differ with Brian on this point. But it's non-invasive and can't harm the wound. The only thing that can happen is when you don't keep changing the dressing. Blood and body fluids collect in the gauze, giving bacteria a good environment. The ointment will prevent the gauze from sticking to the wound. I would change the dressing every day. Some tapes might not stick well to his skin. I used to use strips of packing tape to tape the gauze to the skin. The dressing looks like a clumsy lump of gauze, but it does shield the wound, and it does stay on long enough.
As the tail heals, in my experience, sometimes the vertebrae at the cut site will begin to protrude, possibly from the stump drying out, or from a reduction in swelling caused by the infection. When the stump has an obvious non-scab layer over all of it, I used fingernail clippers to snip it off. But to keep being truthful, that might not be necessary. The tail amp he had when you got him had a bone protrusion, yet it healed especially well.
All my best,