Lemma 10.81.1 (058D)—The Stacks project

Lemma 10.81.1. Let $M$ be an $R$ -module. The following are equivalent:

$M$ is flat.
If $f: R^ n \to M$ is a module map and $x \in \mathop{\mathrm{Ker}}(f)$ , then there are module maps $h: R^ n \to R^ m$ and $g: R^ m \to M$ such that $f = g \circ h$ and $x \in \mathop{\mathrm{Ker}}(h)$ .
Suppose $f: R^ n \to M$ is a module map, $N \subset \mathop{\mathrm{Ker}}(f)$ any submodule, and $h: R^ n \to R^{m}$ a map such that $N \subset \mathop{\mathrm{Ker}}(h)$ and $f$ factors through $h$ . Then given any $x \in \mathop{\mathrm{Ker}}(f)$ we can find a map $h': R^ n \to R^{m'}$ such that $N + Rx \subset \mathop{\mathrm{Ker}}(h')$ and $f$ factors through $h'$ .
If $f: R^ n \to M$ is a module map and $N \subset \mathop{\mathrm{Ker}}(f)$ is a finitely generated submodule, then there are module maps $h: R^ n \to R^ m$ and $g: R^ m \to M$ such that $f = g \circ h$ and $N \subset \mathop{\mathrm{Ker}}(h)$ .

Proof. That (1) is equivalent to (2) is just a reformulation of the equational criterion for flatness ¹. To show (2) implies (3), let $g: R^ m \to M$ be the map such that $f$ factors as $f = g \circ h$ . By (2) find $h'': R^ m \to R^{m'}$ such that $h''$ kills $h(x)$ and $g: R^ m \to M$ factors through $h''$ . Then taking $h' = h'' \circ h$ works. (3) implies (4) by induction on the number of generators of $N \subset \mathop{\mathrm{Ker}}(f)$ in (4). Clearly (4) implies (2). $\square$

[1] In fact, a module map

$f : R^ n \to M$ corresponds to a choice of elements

$x_1, x_2, \ldots , x_ n$ of

$M$ (namely, the images of the standard basis elements

$e_1, e_2, \ldots , e_ n$ ); furthermore, an element

$x \in \mathop{\mathrm{Ker}}(f)$ corresponds to a relation between these

$x_1, x_2, \ldots , x_ n$ (namely, the relation

$\sum _ i f_ i x_ i = 0$ , where the

$f_ i$ are the coordinates of

$x$ ). The module map

$h$ (represented as an

$m \times n$ -matrix) corresponds to the matrix

$(a_{ij})$ from Lemma 10.39.11, and the

$y_ j$ of Lemma 10.39.11 are the images of the standard basis vectors of

$R^ m$ under

$g$ .

The Stacks project

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