Proposition 2

Let G be a filtered commutative group and H a subgroup of G; let H be given the induced filtration and G/H the quotient filtration. If j : H → G is the canonical injection and p : G → G/H the canonical surjection, j and p are compatible with the filtrations and the sequence

(13)

is exact.

The first assertion is obvious; if (G_n) is the filtration on G, then

and hence gr(j) is injective; moreover the canonical mapping is surjective, hence so is gr(p) and gr(p) ∘ gr(j) = 0 by (12). Finally, let $\xi \in G_n/G_{n+1}$ belong to the kernel of gr(p); then there exists x ∈ ξ such that x ∈ H + G_{n+1}; but as G_{n+1} ⊂ G, and hence x = y + z where y ∈ H ∩ G, and z ∈ G, +_1; this proves that ξ is the class mod. G, +_1 of j(y), in other words it belongs to the image of gr(H) under gr(j).

Note that, given an exact sequence 0 → G' \xrightarrow{u} G \xrightarrow{v} G'' → 0 of filtered commutative groups, where u and v are compatible with the filtrations, the sequence 0 \longrightarrow \operatorname{gr}(G') \xrightarrow{\operatorname{gr}(u)} \operatorname{gr}(G) \xrightarrow{\operatorname{gr}(v)} \operatorname{gr}(G'') \longrightarrow 0 is not necessarily exact (Exercise 4).

If now A and B are two filtered rings and h: A → B a ring homomorphism which is compatible with the filtrations, it is immediately verified that the graded group homomorphism gr(h): gr(A) → gr(B) is also a ring homomorphism. In particular, if A' is a subring of A with the induced filtration, gr(A') is canonically identified with a graded subring of gr(A) (Proposition 2); if b is a two-sided ideal of A and A/b is given the quotient filtration, gr(A/b) is canonically identified with the quotient graded ring gr(A)/gr(b) (Proposition 2).

Finally, let A be a filtered ring, E, F two filtered A-modules and u : E → F an homomorphism compatible with the filtrations. Then it is immediate that gr(u) : gr(E) → gr(F) is a gr(A)-linear mapping and hence a homogeneous homomorphism of degree 0 of graded gr(A)-modules. Moreover, if u' : E → F is another A-homomorphism compatible with the filtrations, so is u + u' and

Remarks

(1) Clearly filtered ring homomorphisms (resp. homomorphisms of filtered modules over a given filtered ring A) compatible with the filtrations can be taken as morphisms for the filtered ring structure (resp. filtered A-module structure) (Set Theory, Chapter IV, § 2, no. 1).

(2) Let E and F be two modules over a filtered ring A and let them have the filtrations derived from the filtration (A_n) on A (no. 1, Example 2). Then every A-linear mapping u : E → F is compatible with the filtrations, since

(3) Note that a filtered A-module homomorphism u : E → F which is compatible with the filtrations may satisfy gr(u) = 0 without being zero; this is so for example of the endomorphism x ↦ nx of the additive group $\mathbf{Z}$ with the (n)-adic filtration (for any integer n > 1). The relation gr(u) = gr(v) for two homomorphisms u, v of E to F, compatible with the filtrations, does not therefore imply necessarily u = u.

(4) The definitions at the beginning of this no. extend immediately to two groups G, G', which are not necessarily commutative and are filtered by subgroups $G_n, G'_n$ such that $G_{n+1}$ (resp. $G'_{n+1}$) is normal in G, (resp. $G'_n$). Proposition 2 is also valid with the same hypotheses on the G, and assuming that H is invariant in G, the proof remaining unchanged except for notation.