Situation 38.20.3. Let $(A, \mathfrak m_ A)$ be a local ring. Denote $\mathcal{C}$ the category whose objects are $A$-algebras $A'$ which are local rings such that the algebra structure $A \to A'$ is a local homomorphism of local rings. A morphism between objects $A', A''$ of $\mathcal{C}$ is a local homomorphism $A' \to A''$ of $A$-algebras. Let $A \to B$ be a local ring map of local rings and let $M$ be a $B$-module. If $A'$ is an object of $\mathcal{C}$ we set $B' = B \otimes _ A A'$ and we set $M' = M \otimes _ A A'$ as a $B'$-module. Given $A' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, consider the condition

38.20.3.1
$$\label{flat-equation-flat-at-primes} \forall \mathfrak q \in V(\mathfrak m_{A'}B' + \mathfrak m_ B B') \subset \mathop{\mathrm{Spec}}(B') : M'_{\mathfrak q}\text{ is flat over }A'.$$

Note the similarity with More on Algebra, Equation (15.19.1.1). In particular, if $A' \to A''$ is a morphism of $\mathcal{C}$ and (38.20.3.1) holds for $A'$, then it holds for $A''$, see More on Algebra, Lemma 15.19.2. Hence we obtain a functor

38.20.3.2
$$\label{flat-equation-flat-at-point} F_{lf} : \mathcal{C} \longrightarrow \textit{Sets}, \quad A' \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if }(05ML)\text{ holds}, \\ \emptyset & \text{else.} & \end{matrix} \right.$$

There are also:

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