You certainly can compost the the scrapes from the kitchen, but the desert conditions will probably make composting more difficult.
Try making a mound of kitchen scrapes and lightly mixed it with the surrounding soil (don't bury it too deep, I'd say about 4-5 inches down). If the soil is really dry give it a good drink of water and shade it from the desert sun. Check it once a week, or whenever it is dry, and re-flip/remix the soil so you can aerate it. You want the pile to be moist and aerated so the scrapes break down faster. The rate of decomposition will depend on the number of microbes and insect decomposers present there. If the pile ever starts smelling putrid, then that means there was a lack of oxygen and you need to flip the pile more.
Gook luck and keep us posted!