Sheaf of total quotient rings (${\mathcal{K}}_X$)

The sheaf of total quotient rings, or sheaf of meromorphic functions, of a scheme $X$ is constructed by locally allowing denominators that are nonzerodivisors. For an open set $U$, let $S(U)$ consist of sections of ${\mathcal{O}}_X(U)$ whose germ is a nonzerodivisor at every point of $U$. Sheafifying the presheafgives ${\mathcal{K}}_X$. The stalkwise regular-denominator condition ensures that restriction to smaller open sets is defined. Merely taking arbitrary fraction rings of section rings is not a substitute for this construction.^[ref]

For a locally Noetherian scheme, ${\mathcal{K}}_{X,x}$ is the total quotient ring of ${\mathcal{O}}_{X,x}$, and on an affine open $\mathrm{Spec}A$ its sections are the total quotient ring of $A$. These identifications require the stated hypothesis.^[ref]

For an integral scheme this recovers its field of rational functions locally. For $A=k[\epsilon ]/({\epsilon }^2)$, every nonzerodivisor is already a unit, so its total quotient ring is $A$ itself and retains a nonzero nilpotent. Total quotient rings therefore need not be fields or remove infinitesimal structure.