Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Algebraic Geometry (Definition)

The study of geometric objects defined by systems of polynomial equations (algebraic varieties).
📜

The statement of the theorem

Let RR be a commutative ring with unity. The spectrum of RR, denoted Spec(R)\text{Spec}(R), is the set of all prime ideals p\mathfrak{p} of RR, endowed with the Zariski topology, where the closed sets are of the form V(I)={p∈Spec(R)∣I⊆p}V(I) = \{\mathfrak{p} \in \text{Spec}(R) \mid I \subseteq \mathfrak{p}\} for any ideal I⊆RI \subseteq R. \n\nWe define the structure sheaf OSpec(R)\mathcal{O}_{\text{Spec}(R)} by setting the sections over an open set U=D(f)=Spec(R)∖V((f))U = D(f) = \text{Spec}(R) \setminus V((f)) to be the localization of RR at ff, i.e., OSpec(R)(D(f))=Rf\mathcal{O}_{\text{Spec}(R)}(D(f)) = R_f. \n\nAn **Algebraic Scheme** is a locally ringed space (X,OX)(X, \mathcal{O}_X) such that every point x∈Xx \in X has an open neighborhood UU such that the restricted space (U,OX∣U)(U, \mathcal{O}_X|_U) is isomorphic to Spec(R)\text{Spec}(R) for some commutative ring RR. \n\nSpecifically, if XX is an algebraic scheme, it is a scheme defined by the requirement that its local rings OX,x\mathcal{O}_{X, x} are of the form RpR_{\mathfrak{p}}, where RR is the ring of global sections and p\mathfrak{p} is the maximal ideal corresponding to xx.