See this answer on Mathoverflow and this wikipedia section.
These links claim that one can construct $Spec \mathcal{A}$ for any sheaf of algebras over a scheme(and even for any locally ringed space). However, there are no references in either place. I tried to extend the construction in the quasi-coherent case, but it seems hopeless.Then I tried to imitate the construction of the Spectrum of a ring, and got something seemingly nonsensical.(My secondary question is about this.)
My Primary Question: Can anyone provide a reference for these facts?
This is something that occurred to me while trying to imitate the construction of the Ring spectrum. I was able to define sheaf-theoretic analogues for the usual things needed to construct an affine scheme from a ring(prime ideals, localization, etc.). But what I ended up with was a 'sheaf of sheaves'.
My Secondary(vague) question: To what extent can one extend the usual commutative algebra to (locally) ringed spaces? Does the a 'sheaf of sheaves' thing that I ended up with mean anything? If so, can anyone provide references?