Is there currently a good abstract theory (derived from algebraic geometry and cohomological theories) to study binary operations on arithmetical functions like the Dirichlet convolution $$f\star g = \sum_{d|n} f\left(\frac{n}{d}\right) g(d)\,\, ?$$ I don't know anything in arithmetic geometry, but I know the basic stuff on algebraic geometry (mainly the book of Hartshorne) and I was looking for more advanced cohomological stuff do deal with such operations on arithmetic functions. A collegue told me about crystalline cohomology but was not sure. If it is the case, is there any good book to study theses theories (except SGA) ?

Any helpful comment or book suggestion will be highly appreciated.