Exercise 6, § 2

Let $A$ be a commutative ring, $E$ and $F$ two $A$-modules and $(E_n)$ (resp. $(F_j)$) an exhaustive filtration on $E$ (resp. $F$) consisting of sub-$A$-modules. On the tensor product $G = E \otimes_A F$, consider the exhaustive filtration consisting of the $G_n = \sum_{i+j=n} \mathrm{Im}(E_i \otimes_A F_j)$.

(a) Show that the composite canonical homomorphisms

are the restrictions of a graded homomorphism of degree 0 (called canonical) $\mathrm{gr}(E) \otimes_A \mathrm{gr}(F) \to \mathrm{gr}(E \otimes_A F)$, which is surjective.

(b) Show that, if $\mathrm{gr}(E)$ is a flat $A$-module, the $A$-modules $E_m/E_n$ for $m \leq n$ and $E/E_n$ for $n \in \mathbf{Z}$ are flat.

(c) Suppose in the following that $\mathrm{gr}(E)$ is a flat $A$-module and that $E$ is a flat $A$-module (the second hypothesis being a consequence of the first if $(E_n)$ is a discrete filtration). Then the $H_{ij} = E_i \otimes_A F_j$ are canonically identified with submodules of $G = E \otimes_A F$ (Chapter I, § 2, no. 5, Proposition 4). Show that, for any two finite subsets $R, S$ of $\mathbf{Z} \times \mathbf{Z}$,

where in the second sum $(i,j, h, k)$ run through $R \times S$. (If $R$ and $S$ are each reduced to a single element, use Chapter I, § 2, no. 6, Proposition 7. If $p$ is the greatest of the indices $i$ and $h$ appearing in the elements of $R$ or $S$, $q$ the least of the indices $j$ and $k$ appearing in the elements of $R$ or $S$, consider the images of the two sides of (*) under the canonical homomorphism

and argue by induction on Card(R) + Card(S).)

(d) Deduce from (c) that the canonical homomorphism defined in (a) is then bijective.