Oh please, spraying some aluminium on your skin is NOT going to cause Alzheimer's.
Injecting aluminium into your blood stream can cause a dementia that's similar to, but distinct from, Alzheimer's, and this happened in early dialysis patients when the dialysate contained high concentrations of aluminium.
Do you wrap food in tin-foil? Do you cook using non-stick aluminium pans? Do you drink tea? Do you take aspirin or antacids? These introduce far more aluminium into your body than the tiny amount absorbed by your sweat glands from antiperspirant. The cells of your skin are normally shed, taking with them any aluminium they've absorbed, as part of the natural life-cycle. The study you quoted concerned differences in drinking water, and even then had to conclude that "neither aluminum nor silica concentrations had any significant association with the values of the Mini-Mental State Examination scores at inception", despite the fact that most of the participants had been drinking the water there for most of their lives.