Lemma 10.159.4. Let A be a ring. Let \kappa = \max (|A|, \aleph _0). Then every flat A-algebra B is the filtered colimit of its flat A-subalgebras B' \subset B of cardinality |B'| \leq \kappa . (Observe that B' is faithfully flat over A if B is faithfully flat over A.)
Proof. If B has cardinality \leq \kappa then this is true. Let E \subset B be an A-subalgebra with |E| \leq \kappa . We will show that E is contained in a flat A-subalgebra B' with |B'| \leq \kappa . The lemma follows because (a) every finite subset of B is contained in an A-subalgebra of cardinality at most \kappa and (b) every pair of A-subalgebras of B of cardinality at most \kappa is contained in an A-subalgebra of cardinality at most \kappa . Details omitted.
We will inductively construct a sequence of A-subalgebras
E = E_0 \subset E_1 \subset E_2 \subset \ldots
each having cardinality \leq \kappa and we will show that B' = \bigcup E_ k is flat over A to finish the proof.
The construction is as follows. Set E_0 = E. Given E_ k for k \geq 0 we consider the set S_ k of relations between elements of E_ k with coefficients in A. Thus an element s \in S_ k is given by an integer n \geq 1 and a_1, \ldots , a_ n \in A, and e_1, \ldots , e_ n \in E_ k such that \sum a_ i e_ i = 0 in E_ k. The flatness of A \to B implies by Lemma 10.39.11 that for every s = (n, a_1, \ldots , a_ n, e_1, \ldots , e_ n) \in S_ k we may choose
(m_ s, b_{s, 1}, \ldots , b_{s, m_ s}, a_{s, 11}, \ldots , a_{s, nm_ s})
where m_ s \geq 0 is an integer, b_{s, j} \in B, a_{s, ij} \in A, and
e_ i = \sum \nolimits _ j a_{s, ij} b_{s, j}, \forall i, \quad \text{and}\quad 0 = \sum \nolimits _ i a_ i a_{s, ij}, \forall j.
Given these choicse, we let E_{k + 1} \subset B be the A-subalgebra generated by
E_ k and
the elements b_{s, 1}, \ldots , b_{s, m_ s} for every s \in S_ k.
Some set theory (omitted) shows that E_{k + 1} has at most cardinality \kappa (this uses that we inductively know |E_ k| \leq \kappa and consequently the cardinality of S_ k is also at most \kappa ).
To show that B' = \bigcup E_ k is flat over A we consider a relation \sum _{i = 1, \ldots , n} a_ i b'_ i = 0 in B' with coefficients in A. Choose k large enough so that b'_ i \in E_ k for i = 1, \ldots , n. Then (n, a_1, \ldots , a_ n, b'_1, \ldots , b'_ n) \in S_ k and hence we see that the relation is trivial in E_{k + 1} and a fortiori in B'. Thus A \to B' is flat by Lemma 10.39.11. \square
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