 Zariski coverage on finitely presented rings

On the opposite of the category of finitely presented unital commutative rings, a basic covering family is $A\to A[f_i^{-1}]$, where finitely many $f_i$ generate the unit ideal. Include the empty cover of the zero ring. Localization commutes with base change, so these families define a coverage on a category. Their sheaf topos classifies the coherent theory of local rings. This coverage is subcanonical: representable functors obey the usual localization gluing equations.