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I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following. When does the dualizing sheaf of a projective scheme is an locally ...

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...