Proposition 4

Let B be an integral domain, A a subring of B and K and L the respective fields & fractions of A and B. Let V be a finite-dimensional vector space over K.

(i) For every lattice $M$ of $V$ with respect to $A$, the image $BM$ of $M_{(B)} = M \otimes_A B$ in $V_{(L)} = V \otimes_K L$ is a lattice of $V_{(L)}$ with respect to $B$.

(ii) Suppose further that $B$ is a flat $A$-module. Then the canonical mapping $M_{(B)} \to BM$ is bijective. If further $B$ is faithfully flat, the mapping which maps every lattice $M$ of $V$ with respect to $A$ to the lattice $BM$ of $V_{(L)}$ with respect to $B$ is injective.

(i) As $KM = V$, clearly $L . (BM) = V_{(L)}$; on the other hand $M$ is contained in a finitely generated sub-$A$-module $M_1$ of $V$ and hence $BM$ is contained in $BM_1$ which is a finitely generated $B$-module; whence assertion (i) (Proposition 1).

(ii) $V_{(L)} = V \otimes_K L = V \otimes_A L$ (Chapter II, § 2, no. 7, Proposition 18) and, as $L$ is a flat $B$-module, it is also a flat $A$-module (Chapter I, § 2, no. 7, Corollary 3 to Proposition 8). As $B$ is a flat $A$-module, the canonical mapping $M \otimes_A B \to V \otimes_A B$ is injective; on the other hand, since $V$ is a free $K$-module and $K$ a flat $A$-module, $V$ is a flat $A$-module (Chapter I, § 2, no. 7, Corollary 3 to Proposition 8) and hence the canonical mapping $V \otimes_A B \to V \otimes_A L$ is injective, which establishes the first assertion. To see also that the relation $BM_1 = BM_2$ implies $M_1 = M_2$ for two lattices $M_1, M_2$ of $V$ with respect to $A$ when $B$ is a faithfully flat $A$-module, note first that $BM_1 \cap BM_2 = B(M_1 \cap M_2)$ (Chapter I, § 2, no. 6, Proposition 6); we may therefore confine our attention to the case where $M_1 \subset M$, and our assertion then follows from Chapter I, § 3, no. 1, Proposition 3 applied to the canonical injection $M_1 \to M_2$.