Lemma 15.16.1. Let R be a ring. Let M be an R-module. Let I_1, I_2 be ideals of R. If M/I_1M is flat over R/I_1 and M/I_2M is flat over R/I_2, then M/(I_1 \cap I_2)M is flat over R/(I_1 \cap I_2).
15.16 Flattening stratification
Let R \to S be a ring map and let M be an S-module. For any R-algebra R' we can consider the base changes S' = S \otimes _ R R' and M' = M \otimes _ R R'. We say R \to R' flattens M if the module M' is flat over R'. We would like to understand the structure of the collection of ring maps R \to R' which flatten M. In particular we would like to know if there exists a universal flattening R \to R_{univ} of M, i.e., a ring map R \to R_{univ} which flattens M and has the property that any ring map R \to R' which flattens M factors through R \to R_{univ}. It turns out that such a universal solution usually does not exist.
We will discuss universal flattenings and flattening stratifications in a scheme theoretic setting \mathcal{F}/X/S in More on Flatness, Section 38.21. If the universal flattening R \to R_{univ} exists then the morphism of schemes \mathop{\mathrm{Spec}}(R_{univ}) \to \mathop{\mathrm{Spec}}(R) is the universal flattening of the quasi-coherent module \widetilde{M} on \mathop{\mathrm{Spec}}(S).
In this and the next few sections we prove some basic algebra facts related to this. The most basic result is perhaps the following.
Proof. By replacing R with R/(I_1 \cap I_2) and M by M/(I_1 \cap I_2)M we may assume that I_1 \cap I_2 = 0. Let J \subset R be an ideal. To prove that M is flat over R we have to show that J \otimes _ R M \to M is injective, see Algebra, Lemma 10.39.5. By flatness of M/I_1M over R/I_1 the map
J/(J \cap I_1) \otimes _ R M = (J + I_1)/I_1 \otimes _{R/I_1} M/I_1M \longrightarrow M/I_1M
is injective. As 0 \to (J \cap I_1) \to J \to J/(J \cap I_1) \to 0 is exact we obtain a diagram
\xymatrix{ (J \cap I_1) \otimes _ R M \ar[r] \ar[d] & J \otimes _ R M \ar[r] \ar[d] & J/(J \cap I_1) \otimes _ R M \ar[r] \ar[d] & 0 \\ M \ar@{=}[r] & M \ar[r] & M/I_1M }
hence it suffices to show that (J \cap I_1) \otimes _ R M \to M is injective. Since I_1 \cap I_2 = 0 the ideal J \cap I_1 maps isomorphically to an ideal J' \subset R/I_2 and we see that (J \cap I_1) \otimes _ R M = J' \otimes _{R/I_2} M/I_2M. By flatness of M/I_2M over R/I_2 the map J' \otimes _{R/I_2} M/I_2M \to M/I_2M is injective, which clearly implies that (J \cap I_1) \otimes _ R M \to M is injective. \square
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