Structure sheaf of a scheme (${\mathcal{O}}_X$)

The structure sheaf ${\mathcal{O}}_X$ records the regular functions on every open subset of a scheme $X$. Its stalk ${\mathcal{O}}_{X,x}$ is the local ring of functions defined near $x$.

A global idempotent on a scheme is zero or one in every local ring. Its one-locus is both open and closed. Conversely a clopen subset determines a global idempotent by gluing one on it and zero on its complement. Under this correspondence, decompositions into nonzero orthogonal idempotents correspond to disjoint clopen decompositions. When all connected components are open, the primitive idempotents correspond exactly to those components. Without orthogonality the assertion fails in characteristic two: $(1,0)=(1,1)+(0,1)$ in ${\mathbb{F}}_2\times {\mathbb{F}}_2$.

The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.

A regular function on an open subset $U$ of a scheme $X$ is a section of ${\mathcal{O}}_X(U)$. On an affine variety it is locally a quotient of polynomial functions with a denominator that does not vanish.

On a variety, a regular function is a regular map to the affine line; the arbitrary-target regular-map concept is broader.

A global regular function is a global section of the structure sheaf of a scheme, defined on the entire scheme. These functions form its ring $H^0(X,{\mathcal{O}}_X)$. On an affine scheme this recovers the defining ring; on a positive-dimensional projective complete intersection it consists only of constants over the base field.

Over an algebraically closed field, every regular function on a nonempty irreducible projective variety is constant. Regard the function as a morphism to ${\mathbb{P}}^1$ avoiding infinity. Its graph is closed; the closedness of projection from projective space makes its image closed in ${\mathbb{P}}^1$. A proper closed subset there is finite, and an irreducible finite image is a singleton. Irreducibility is the usual variety convention; on a disconnected projective algebraic set different components may have different constants.

For $W={\mathbb{A}}_k^2\setminus \{0\}$, restriction to $D(x)$ and $D(y)$ givesThe intersection is taken in $k(x,y)$ and the equality follows from unique factorization: a reduced denominator dividing powers of both $x$ and $y$ is a unit. The inclusion $W\hookrightarrow {\mathbb{A}}^2$ induces an isomorphism on global functions but is not an isomorphism, so $W$ is not an affine variety.