= Sheaf of total quotient rings
{title2=$\mathcal K_X$}
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 presheaf
$$
U\longmapsto S(U)^{-1}\mathcal O_X(U)
$$
gives $\mathcal K_X$. 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.<The Stacks project>[Tag01X1, Definition31.24.1]{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 $\operatorname{Spec}A$ its sections are the total quotient ring of $A$. These identifications require the stated hypothesis.<The Stacks project>[Tag0EMG, Lemma31.25.2]{ref}
For an integral scheme this recovers its field of rational functions locally. For $A=k[\varepsilon]/(\varepsilon^2)$, 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.
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