I'm currently reading scheme theory and i'm trying to figure out the reason of introducing additionally the sheaf of a ring A together with its spectrum.I can understand all the basic properties of the spectrum,understand all the proofs but i can't see the importance of the sheaf.Is there any historical example that led to the definition of the spectrum?Is there any example of two spectrums of rings A,R such that SpecA is homeomorphic to SpecR but with the corresponding sheaves to be "different" in the sence that they encode a different information from the rings and thus showing the importance of introducing the sheaf of the ring together with its spectrum?
Importance of the structure sheaf of the spectrum
algebraic-geometryschemessoft-question
Best Answer
For sure: If $k$ is a field, both $\operatorname{Spec}(k[x]/x^2)$ and $\operatorname{Spec}(k[x]/x)$ are homeomorphic (both are only a point) but not isomorphic as schemes. The crucial difference lies in the fact that $k[x]/x^2$ contains nilpotent elements, while $k[x]/x$ does not. This translates "geometrically" to infinitesimal information.
Every textbook on basic scheme theory should contain information (or at least exercises) regarding the importance of the structure sheaf in the "scheme-theoretic" picture. From the top of my head both Eisenbud-Harris' book and Vakil's notes should have some depth regarding this fundamental issue.