3 More references

A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.

So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in arbitrary topoi. As any proposition (in the internal language) on a local ring (a commutative ring with $0 \neq 1$ and $s + t = 1 \implies s \in R^\times \lor t \in R^\times$) is true for any topos if and only if it can be derived intuitionistically (which is more or less the same as constructively), the original questions seems to boil down to:

"What are the constructively valid properties and constructions for a local ring?"

For example, the construction of Kähler differentials makes also sense constructively, which immediately implies (by the above reasoning) that every morphism $X \to Y$ of locally ringed spaces has an associated module $\Omega_{X/Y}$ of Kähler differentials.

And there is a lot of literature on constructive algebra. The book of Mines, Richman and Ruitenburg as well as many of the preprints on Fred Richman's homepage are a start. Some material can also be found in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk.

A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.

So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in arbitrary topoi. As any proposition (in the internal language) on a local ring (a commutative ring with $0 \neq 1$ and $s + t = 1 \implies s \in R^\times \lor t \in R^\times$) is true for any topos if and only if it can be derived intuitionistically (which is more or less the same as constructively), the original questions seems to boil down to:

"What are the constructively valid properties and constructions for a local ring?"

For example, the construction of Kähler differentials makes also sense constructively, which immediately implies (by the above reasoning) that every morphism $X \to Y$ of locally ringed spaces has an associated module $\Omega_{X/Y}$ of Kähler differentials.

And there is a lot of literature on constructive algebra. The book of Mines, Richman and Ruitenburg as well as many of the preprints on Fred Richman's homepage are a start.

1 [made Community Wiki]

A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.

So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in arbitrary topoi. As any proposition (in the internal language) on a local ring (a commutative ring with $0 \neq 1$ and $s + t = 1 \implies s \in R^\times \lor t \in R^\times$) is true for any topos if and only if it can be derived intuitionistically (which is more or less the same as constructively), the original questions seems to boil down to:

"What are the constructively valid properties and constructions for a local ring?"

For example, the construction of Kähler differentials makes also sense constructively, which immediately implies (by the above reasoning) that every morphism $X \to Y$ of locally ringed spaces has an associated module $\Omega_{X/Y}$ of Kähler differentials.

And there is a lot of literature on constructive algebra.