The Stacks project

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

  1. $M$ is flat.

  2. 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 $x \in \mathop{\mathrm{Ker}}(h)$.

  3. 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 $h'$.

  4. 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 flatness1. 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 $g$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 058D. Beware of the difference between the letter 'O' and the digit '0'.