SQUARE45

Let

R

be a commutative ring with unity. The spectrum of

R

, denoted

\text{Spec}(R)

, is the set of all prime ideals

\mathfrak{p}

R

, endowed with the Zariski topology, where the closed sets are of the form

V(I) = \{\mathfrak{p} \in \text{Spec}(R) \mid I \subseteq \mathfrak{p}\}

for any ideal

I \subseteq R

. \n\nWe define the structure sheaf

\mathcal{O}_{\text{Spec}(R)}

by setting the sections over an open set

U = D(f) = \text{Spec}(R) \setminus V((f))

to be the localization of

R

f

, i.e.,

\mathcal{O}_{\text{Spec}(R)}(D(f)) = R_f

. \n\nAn **Algebraic Scheme** is a locally ringed space

(X, \mathcal{O}_X)

such that every point

x \in X

has an open neighborhood

U

such that the restricted space

(U, \mathcal{O}_X|_U)

is isomorphic to

\text{Spec}(R)

for some commutative ring

R

. \n\nSpecifically, if

X

is an algebraic scheme, it is a scheme defined by the requirement that its local rings

\mathcal{O}_{X, x}

are of the form

R_{\mathfrak{p}}

, where

R

is the ring of global sections and

\mathfrak{p}

is the maximal ideal corresponding to

x

Algebraic Geometry