Chapter 0_III

§13. Projective limits in homological algebra

13.1. The Mittag–Leffler condition

13.1.1.

Let $\mathcal{C}$ be an abelian category in which infinite products exist (axiom AB 3* of (T, 1.5)); then the infimum of a family of subobjects of an object of $\mathcal{C}$ exists, and every projective system of objects of $\mathcal{C}$ admits a projective limit, which is a left-exact functor of the projective system considered (T, 1.8). Let

$(A_{\alpha },f_{\alpha \beta })$ be a projective system of objects of $\mathcal{C}$ whose index set $I$ is right-filtered; set $A=\underleftarrow{\lim }{A}_{\alpha }$ and, for every $\alpha \in I$, let $f_{\alpha }:A\to A_{\alpha }$ be the canonical morphism. For every $\alpha \in I$, the $f_{\alpha \beta }(A_{\beta })$ for $\beta \ge \alpha$ form a filtered decreasing family of subobjects of $A_{\alpha }$; the subobject

  A'_α = inf_{β ≥ α} f_{αβ}(A_β)

is called the subobject of "universal images" in $A_{\alpha }$; it is clear that $f_{\alpha }(A)\subset A_{\alpha }^{\prime }$ and $f_{\alpha \beta }(A_{\beta }^{\prime })\subset A_{\alpha }^{\prime }$ for $\alpha \le \beta$, so $(A_{\alpha }^{\prime },f_{\alpha \beta }|A_{\beta }^{\prime })$ is a projective system, and $A=\underleftarrow{\lim }{A}_{\alpha }^{\prime }$.

13.1.2.

Given a projective system $(A_{\alpha },f_{\alpha \beta })$ in $\mathcal{C}$, the Mittag–Leffler condition is the following condition:

(ML) For every index $\alpha$, there exists $\beta \ge \alpha$ such that, for every $\gamma \ge \beta$, one has $f_{\alpha \gamma }(A_{\gamma })=f_{\alpha \beta }(A_{\beta })$.

It is clear that if the $f_{\alpha \beta }$ are epimorphisms, condition (ML) is satisfied. Conversely, if (ML) is satisfied, and if for every $\alpha \in I$, $A_{\alpha }^{\prime }$ is the subobject of "universal images" in $A_{\alpha }$, the restriction of $f_{\alpha \beta }$ to $A_{\beta }^{\prime }$ is an epimorphism $A_{\beta }^{\prime }\to A_{\alpha }^{\prime }$ for $\alpha \le \beta$: indeed, if $\gamma \ge \beta$ is such that $f_{\beta \delta }(A_{\delta })=f_{\beta \gamma }(A_{\gamma })$ for $\delta \ge \gamma$, one has $A_{\beta }^{\prime }=f_{\beta \gamma }(A_{\gamma })$, and this entails on the other hand $f_{\alpha \delta }(A_{\delta })=f_{\alpha \gamma }(A_{\gamma })$ for $\delta \ge \gamma$, so $A_{\alpha }^{\prime }=f_{\alpha \gamma }(A_{\gamma })=f_{\alpha \beta }(A_{\beta }^{\prime })$.

Note also that condition (ML) is satisfied whenever the objects $A_{\alpha }$ are artinian in $\mathcal{C}$, that is, every family of subobjects of $A_{\alpha }$ admits a minimal element: a minimal element of the filtered decreasing family $(f_{\alpha \beta }(A_{\beta }))$ of subobjects of $A_{\alpha }$ is then necessarily the smallest of these subobjects.

Translator's note. Throughout §13, EGA's (ML) denotes the Mittag–Leffler condition; we keep the EGA abbreviation. Modern accounts often write "ML condition" or "satisfies ML" interchangeably.

Remark (13.1.3).

Condition (ML) can also be formulated when $\mathcal{C}$ is, for example, the category of sets; one can then again define the subset of "universal images" in $A_{\alpha }$, and the remarks made on this subject in (13.1.1) and (13.1.2) remain valid.

13.2. The Mittag–Leffler condition for abelian groups

Proposition (13.2.1).

Let

  0 → A_α --u_α--> B_α --v_α--> C_α → 0

be an exact sequence of projective systems of abelian groups (relative to the same right-filtered index set $I$).

• (i) If $(B_{\alpha })$ satisfies (ML), so does $(C_{\alpha })$.
• (ii) If $(A_{\alpha })$ and $(C_{\alpha })$ satisfy (ML), so does $(B_{\alpha })$.

Let $(f_{\alpha \beta })$, $(g_{\alpha \beta })$, $(h_{\alpha \beta })$ be the systems of homomorphisms defining the projective systems $(A_{\alpha })$, $(B_{\alpha })$, $(C_{\alpha })$ respectively.

(i) Suppose $g_{\alpha \beta }(B_{\beta })=g_{\alpha \lambda }(B_{\lambda })$ for $\lambda \ge \beta$; since $v_{\beta }$ and $v_{\lambda }$ are surjective, one has $h_{\alpha \beta }(C_{\beta })=v_{\alpha }(g_{\alpha \beta }(B_{\beta }))=v_{\alpha }(g_{\alpha \lambda }(B_{\lambda }))=h_{\alpha \lambda }(C_{\lambda })$ for $\lambda \ge \beta$.

(ii) Let $\alpha \in I$, and let $\beta \ge \alpha$ be an index such that for $\lambda \ge \beta$, one has $f_{\alpha \beta }(A_{\beta })=f_{\alpha \lambda }(A_{\lambda })$; let also $\gamma \ge \beta$ be an index such that, for $\lambda \ge \gamma$, one has $h_{\beta \lambda }(C_{\lambda })=h_{\beta \gamma }(C_{\gamma })$. Let then $x_{\alpha }$ be an element of $g_{\alpha \gamma }(B_{\gamma })$; one has $x_{\alpha }=g_{\alpha \gamma }(y_{\gamma })$ with $y_{\gamma }\in B_{\gamma }$; set $y_{\beta }=g_{\beta \gamma }(y_{\gamma })$, so that $x_{\alpha }=g_{\alpha \beta }(g_{\beta \gamma }(y_{\gamma }))$. For every $\lambda \ge \gamma$, there exists by hypothesis $y_{\lambda }\in B_{\lambda }$ such that $h_{\beta \gamma }(v_{\gamma }(y_{\gamma }))=h_{\beta \lambda }(v_{\lambda }(y_{\lambda }))=v_{\beta }(g_{\beta \lambda }(y_{\lambda }))$, whence $v_{\beta }(y_{\beta }-g_{\beta \lambda }(y_{\lambda }))=0$, and consequently there exists $x_{\beta }\in A_{\beta }$

such that $y_{\beta }=g_{\beta \lambda }(y_{\lambda })+u_{\beta }(x_{\beta })$. One deduces $x_{\alpha }=g_{\alpha \lambda }(y_{\lambda })+u_{\alpha }(f_{\alpha \beta }(x_{\beta }))$; but since $\lambda \ge \beta$, there exists $x_{\lambda }\in A_{\lambda }$ such that $f_{\alpha \beta }(x_{\beta })=f_{\alpha \lambda }(x_{\lambda })$, and finally $x_{\alpha }=g_{\alpha \lambda }(y_{\lambda }+u_{\lambda }(x_{\lambda }))\in g_{\alpha \lambda }(B_{\lambda })$, which completes the demonstration.

Proposition (13.2.2).

Let $I$ be a right-filtered ordered set having a countable cofinal part. Let

  0 → A_α --u_α--> B_α --v_α--> C_α → 0

be an exact sequence of projective systems of abelian groups having $I$ as index set. If $(A_{\alpha })$ satisfies condition (ML), the sequence

  0 → lim_← A_α → lim_← B_α → lim_← C_α → 0

is exact.

It comes down to proving that the homomorphism v = lim_← v_α : lim_← B_α → lim_← C_α is surjective. Let $z=(z_{\alpha })$ be an element of $\underleftarrow{\lim }{C}_{\alpha }$, and set $E_{\alpha }=v_{\alpha }^{-1}(z_{\alpha })$; it is clear that the $E_{\alpha }$ form a projective system of non-empty sets for the restrictions of the homomorphisms $g_{\alpha \beta }:B_{\beta }\to B_{\alpha }$. Let us show that this projective system satisfies condition (ML); identifying $A_{\alpha }$ with a part of $B_{\alpha }$ via $u_{\alpha }$ for every $\alpha \in I$, there exists $\beta \ge \alpha$ such that $g_{\alpha \beta }(A_{\beta })=g_{\alpha \lambda }(A_{\lambda })$ for $\lambda \ge \beta$; let us show that one also has $g_{\alpha \beta }(E_{\beta })=g_{\alpha \lambda }(E_{\lambda })$ for $\lambda \ge \beta$. Indeed, take $y_{\lambda }\in E_{\lambda }$ and set $y_{\beta }=g_{\beta \lambda }(y_{\lambda })$, $y_{\alpha }=g_{\alpha \lambda }(y_{\lambda })$; let $y_{\alpha }^{\prime }\in g_{\alpha \beta }(E_{\beta })$, so that $y_{\alpha }^{\prime }=g_{\alpha \beta }(y_{\beta }^{\prime })$ for some $y_{\beta }^{\prime }\in E_{\beta }$; one has $y_{\beta }^{\prime }-y_{\beta }=x_{\beta }\in A_{\beta }$, and by hypothesis there exists $x_{\lambda }\in A_{\lambda }$ such that $g_{\alpha \beta }(x_{\beta })=g_{\alpha \lambda }(x_{\lambda })$; therefore

  y'_α = g_{αβ}(y_β) + g_{αβ}(x_β) = g_{αλ}(y_λ) + g_{αλ}(x_λ)
       = g_{αλ}(y_λ + x_λ) ∈ g_{αλ}(E_λ),

which proves our assertion. That being so, one knows (Bourbaki, Top. gén., ch. II, 3rd ed., §3, th. 1) that under the hypotheses made on $I$, a projective system of non-empty sets satisfying (ML) has a non-empty projective limit; in consequence, there exists a point $y=(y_{\alpha })\in \underleftarrow{\lim }{E}_{\alpha }$, and since $v_{\alpha }(y_{\alpha })=z_{\alpha }$ by definition for every $\alpha$, one has $z=v(y)$. Q.E.D.

Proposition (13.2.3).

The hypotheses on $I$ being those of (13.2.2), let $(K_{\alpha }^{\bullet }{)}_{\alpha \in I}$ be a projective system of complexes of abelian groups $K_{\alpha }^{\bullet }=(K_{\alpha }^n{)}_{n\in \mathbb{Z}}$ whose differential operator is of degree +1. For each $n$, there exists a canonical functorial homomorphism

  h_n : H^n(lim_← K^•_α) → lim_← H^n(K^•_α).                                  (13.2.3.1)

If, for every degree $n$, the projective system of abelian groups $(K_{\alpha }^n{)}_{\alpha \in I}$ satisfies (ML), then all the homomorphisms $h_n$ are surjective. If in addition, for some degree $n$, the projective system $(H^{n-1}(K_{\alpha }^{\bullet }){)}_{\alpha \in I}$ satisfies (ML), the homomorphism $h_n$ is bijective.

Set, for every $n$, $K^n=\underleftarrow{\lim }{K}_{\alpha }^n$; the definition of the homomorphisms $h_n$ comes from the commutativity of the diagrams

  … → K^{n−1} ----> K^n ----> K^{n+1} → …
        ↓            ↓            ↓
  … → K^{n−1}_α → K^n_α → K^{n+1}_α → …

the differential operators in $K^{\bullet }$ being projective limits of the corresponding operators in the $K_{\alpha }^{\bullet }$.

Consider the exact sequences

  (*_n)     0 → B^n(K^•_α) → Z^n(K^•_α) → H^n(K^•_α) → 0
  (**_n)    0 → Z^{n−1}(K^•_α) → K^{n−1}_α → B^n(K^•_α) → 0

The hypothesis and Proposition (13.2.1, (i)) show that the projective system $(B^n(K_{\alpha }^{\bullet }){)}_{\alpha \in I}$ satisfies (ML) for every $n$; it therefore results from (13.2.2) that the sequence

  (***_n)   0 → lim_α B^n(K^•_α) → lim_α Z^n(K^•_α) → lim_α H^n(K^•_α) → 0

is exact. Now it is clear that $\underleftarrow{\lim }{B}^n(K_{\alpha }^{\bullet })$ identifies with a subgroup of $K^{n+1}$ containing $B^n(K^{\bullet })$, and that $\underleftarrow{\lim }{Z}^n(K_{\alpha }^{\bullet })$ identifies with a subgroup of $Z^n(K^{\bullet })$; in consequence, $h_n$ is surjective. If now one supposes furthermore that the projective system $(H^{n-1}(K_{\alpha }^{\bullet }){)}_{\alpha \in I}$ satisfies (ML), the exact sequences $({\ast }_{n-1})$ and Proposition (13.2.1, (ii)) show that the projective system $(Z^{n-1}(K_{\alpha }^{\bullet }){)}_{\alpha \in I}$ satisfies (ML); but then, (13.2.2) applied to the exact sequences $(\ast {\ast }_n)$ shows that the sequence

  0 → lim_α Z^{n−1}(K^•_α) → K^{n−1} --u--> lim_α B^n(K^•_α) → 0

is exact; since $\underleftarrow{\lim }{B}^n(K_{\alpha }^{\bullet })\supset B^n(K^{\bullet })$, and the composite of the injection $\underleftarrow{\lim }{B}^n(K_{\alpha }^{\bullet })\to K^n$ with $u$ is the differential operator $K^{n-1}\to K^n$, the fact that $u$ is surjective entails $\underleftarrow{\lim }{B}^n(K_{\alpha }^{\bullet })=B^n(K^{\bullet })$, so $h_n$ is injective. Q.E.D.

Remarks (13.2.4).

(i) The reasoning of (13.2.2) (cf. Bourbaki, loc. cit.) shows that the conclusion of this proposition remains valid when one supposes only that the $A_{\alpha }$ can be equipped with structures of complete metrizable spaces, in which the translations are homeomorphisms, that the maps $f_{\alpha \beta }:A_{\beta }\to A_{\alpha }$ defining the projective system $(A_{\alpha })$ are uniformly continuous for the distances considered, and finally that the system $(A_{\alpha })$ satisfies the condition

(ML') For every index $\alpha$, there exists $\beta \ge \alpha$ such that, for every $\gamma \ge \beta$, $f_{\alpha \gamma }(A_{\gamma })$ is dense in $f_{\alpha \beta }(A_{\beta })$.

This allows one to add an analogous complement to (13.2.3): suppose that $K_{\alpha }^n=0$ for $n<0$ and for every $\alpha$; suppose moreover that $(K_{\alpha }^n{)}_{\alpha \in I}$ satisfies (ML) for $n\ge 0$ and that the $A_{\alpha }=H^0(K_{\alpha }^{\bullet })$ can be equipped with structures of metric spaces satisfying the above properties. Then the conclusions of (13.2.3) are unchanged for $n\ge 2$, and in addition $h_1$ is bijective, since the reasoning of (13.2.2) shows again that $(B^1(K_{\alpha }^{\bullet }){)}_{\alpha \in I}$ satisfies (ML), that the sequence $(\ast \ast {\ast }_1)$ is exact, and finally, by virtue of the foregoing, that $\underleftarrow{\lim }{B}^1(K_{\alpha }^{\bullet })=B^1(K^{\bullet })$. We have thus established, among others, the assertions of (T, 3.10.2).

(ii) It is possible to introduce the right-derived functors ${\lim }_{\leftarrow }^{(i)}$ of the functor $\underleftarrow{\lim }$, and to obtain more complete statements than the preceding ones [28].

13.3. Application: cohomology of a projective limit of sheaves

Proposition (13.3.1).

Let $X$ be a topological space, $({\mathcal{F}}_k{)}_{k\in \mathbb{N}}$ a projective system of sheaves of abelian groups on $X$, and let $\mathcal{F}=\underleftarrow{\lim }{\mathcal{F}}_k$. Suppose the following conditions are satisfied:

• (i) There exists a base $\mathfrak{B}$ of the topology of $X$ such that, for every $U\in \mathfrak{B}$ and every $i\ge 0$, the projective system $(H^i(U,{\mathcal{F}}_k){)}_{k\in \mathbb{N}}$ satisfies (ML).
• (ii) For every $x\in X$ and every $i>0$, one has ${\lim }_U(\underleftarrow{\lim }{H}^i(U,{\mathcal{F}}_k))=0$ as $U$ runs over the set of neighborhoods of $x$ belonging to $\mathfrak{B}$.
• (iii) The homomorphisms $u_{hk}:{\mathcal{F}}_k\to {\mathcal{F}}_h$ ($h\le k$) defining the projective system $({\mathcal{F}}_k)$ are surjective.

Under these conditions, for every $i>0$, the canonical homomorphism

  h_i : H^i(X, ℱ) → lim_← H^i(X, ℱ_k)

is surjective; if in addition, for some value of $i$, the projective system $(H^{i-1}(X,{\mathcal{F}}_k){)}_{k\in \mathbb{N}}$ satisfies (ML), $h_i$ is bijective.

a) We shall first suppose that the ${\mathcal{F}}_k$ are flasque as well as the kernels ${\mathcal{N}}_{hk}$ of the $u_{hk}$; we shall then show that condition (iii) of the statement entails $H^i(X,\mathcal{F})=0$ for $i>0$. It will suffice to prove that for every open $U$ of $X$ and every cover $\mathfrak{U}$ of $U$ by open subsets of $U$, one has $H^i(\mathfrak{U},\mathcal{F})=0$ for $i>0$. It will result, first, that for Čech cohomology, one has ${\check{H}}^i(U,\mathcal{F})=0$ for every $i>0$, then (by virtue of (G, II, 5.9.2) applied to the set of all open subsets of $X$) that $H^i(X,\mathcal{F})=0$ for every $i>0$. Since the ${\mathcal{F}}_k$ are flasque, one has $H^i(\mathfrak{U},{\mathcal{F}}_k)=0$ for $i>0$ (G, II, 5.2.3); consider for each $k$ the complex $C^{\bullet }(\mathfrak{U},{\mathcal{F}}_k)$ of alternating cochains of the nerve of the cover $\mathfrak{U}$ (G, II, 5.1), which evidently forms a projective system of complexes of abelian groups. Let us show that all the maps $C^{\bullet }(\mathfrak{U},{\mathcal{F}}_k)\to C^{\bullet }(\mathfrak{U},{\mathcal{F}}_h)$ ($h\le k$) are surjective. It clearly suffices, by definition, to show that for every open $V$ of $X$, the map $\Gamma (V,{\mathcal{F}}_k)\to \Gamma (V,{\mathcal{F}}_h)$ is surjective; but the sequence $0\to {\mathcal{N}}_{hk}\to {\mathcal{F}}_k\to {\mathcal{F}}_h\to 0$ being exact by hypothesis gives the exact cohomology sequence

  Γ(V, ℱ_k) → Γ(V, ℱ_h) → H^1(V, 𝒩_{hk}) = 0

since ${\mathcal{N}}_{hk}$ is flasque. The projective system $(C^{\bullet }(\mathfrak{U},{\mathcal{F}}_k){)}_{k\in \mathbb{N}}$ therefore satisfies (ML); the same holds for $(H^i(\mathfrak{U},{\mathcal{F}}_k){)}_{k\in \mathbb{N}}$ for every $i\ge 0$, since this is trivial for $i>0$, and since $H^0(\mathfrak{U},{\mathcal{F}}_k)=\Gamma (U,{\mathcal{F}}_k)$ (G, II, 5.2.2), condition (ML) is also met for $i=0$ by what precedes. One can therefore apply (13.2.3), which shows that $H^i(\mathfrak{U},\mathcal{F})=\underleftarrow{\lim }{H}^i(\mathfrak{U},{\mathcal{F}}_k)=0$ for every $i>0$.

b) Let us pass to the general case, and consider for each $k\in \mathbb{N}$ the canonical resolution ${\mathcal{C}}^{\bullet }(X,{\mathcal{F}}_k)=({\mathcal{C}}^i(X,{\mathcal{F}}_k){)}_{i\ge 0}$ of ${\mathcal{F}}_k$ by flasque sheaves (G, II, 4.3). For each $i\ge 0$, it is clear that $({\mathcal{C}}^i(X,{\mathcal{F}}_k){)}_{k\in \mathbb{N}}$ is a projective system of flasque sheaves; let us

show that it satisfies the conditions of a). Indeed, if ${\mathcal{N}}_{hk}$ is the kernel of $u_{hk}$ for $h\le k$, the sequence $0\to {\mathcal{N}}_{hk}\to {\mathcal{F}}_k\to {\mathcal{F}}_h\to 0$ is exact by (iii), and our assertion follows from the fact that the functor $\mathcal{A}⇝{\mathcal{C}}^i(X,\mathcal{A})$ is exact (G, II, 4.3). Let ${\mathcal{G}}^i=\underleftarrow{\lim }{\mathcal{C}}^i(X,{\mathcal{F}}_k)$; one therefore has $H^j(X,{\mathcal{G}}^i)=0$ for $j>0$ and $i\ge 0$ by virtue of a). We shall show that ${\mathcal{G}}^{\bullet }=({\mathcal{G}}^i{)}_{i\ge 0}$ is a resolution of the sheaf $\mathcal{F}$; since $H^j(X,{\mathcal{G}}^{\bullet })=0$ for $j>0$, the cohomology $H^{\bullet }(X,\mathcal{F})$ will equal $H^{\bullet }(\Gamma (X,{\mathcal{G}}^{\bullet }))$ (G, II, 4.7.1).

It is clear that, by passage to the projective limit, one deduces from the exact sequences

  0 → ℱ_k → 𝒞^0(X, ℱ_k) → 𝒞^1(X, ℱ_k) → …

a complex of sheaves of abelian groups

$$0\to \mathcal{F}\to {\mathcal{G}}^0\to {\mathcal{G}}^1\to \cdots .$$

To prove our assertion, one must establish that ${\mathcal{H}}^i({\mathcal{G}}^{\bullet })=0$ for $i>0$. This sheaf is generated by the presheaf $U\mapsto H^i(\Gamma (U,{\mathcal{G}}^{\bullet }))$ (G, II, 4.1); now, the complex $\Gamma (U,{\mathcal{G}}^{\bullet })$ is the projective limit of the projective system of complexes of abelian groups $(\Gamma (U,{\mathcal{C}}^{\bullet }(X,{\mathcal{F}}_k)){)}_{k\in \mathbb{N}}$ $(0_I,\mathrm{3.2.6})$. We have seen in a) that for each $i\ge 0$, the maps $\Gamma (U,{\mathcal{C}}^i(X,{\mathcal{F}}_k))\to \Gamma (U,{\mathcal{C}}^i(X,{\mathcal{F}}_h))$ ($h\le k$) are surjective; on the other hand, one has $H^i(U,{\mathcal{F}}_k)=H^i(\Gamma (U,{\mathcal{C}}^{\bullet }(X,{\mathcal{F}}_k)))$, the canonical resolution ${\mathcal{C}}^{\bullet }(U,{\mathcal{F}}_k|U)$ being induced on $U$ by ${\mathcal{C}}^{\bullet }(X,{\mathcal{F}}_k)$; by virtue of hypothesis (i), for every $U\in \mathfrak{B}$, one can apply (13.2.3) to the projective system of complexes $(\Gamma (U,{\mathcal{C}}^{\bullet }(X,{\mathcal{F}}_k)){)}_{k\in \mathbb{N}}$, and one therefore has $H^i(\Gamma (U,{\mathcal{G}}^{\bullet }))=\underleftarrow{\lim }{H}^i(U,{\mathcal{F}}_k)$ for every $i\ge 0$. Hypothesis (ii) then proves indeed, by definition, that the sheaves ${\mathcal{H}}^i({\mathcal{G}}^{\bullet })$ are zero for $i>0$.

One has then for every $i\ge 0$, $H^i(X,\mathcal{F})=H^i(\Gamma (X,{\mathcal{G}}^{\bullet }))$ and

  Γ(X, 𝒢^•) = lim_← Γ(X, 𝒞^•(X, ℱ_k)).

We have just remarked that the maps $\Gamma (X,{\mathcal{C}}^i(X,{\mathcal{F}}_k))\to \Gamma (X,{\mathcal{C}}^i(X,{\mathcal{F}}_h))$ ($h\le k$) are all surjective; the conclusion therefore again results from (13.2.3).

Remarks (13.3.2).

(i) The statement (13.3.1) is of interest only for $i>0$, since for $i=0$, $h_0$ is always an isomorphism without hypothesis $(0_I,\mathrm{3.2.6})$.

(ii) Conditions (i) and (ii) of (13.3.1) will in particular be satisfied if $H^i(U,{\mathcal{F}}_k)=0$ for every $k$, every $i>0$ and every $U\in \mathfrak{B}$, and if for $U\in \mathfrak{B}$, the maps $\Gamma (U,{\mathcal{F}}_k)\to \Gamma (U,{\mathcal{F}}_h)$ are surjective. This will be the most frequent case of application of (13.3.1).

13.4. The Mittag–Leffler condition and graded objects associated to projective systems

13.4.1.

Let $\mathbf{A}=(A_k,u_{hk}{)}_{k\in \mathbb{Z}}$ be a projective system in an abelian category $\mathcal{C}$; we shall say that it is bounded below if there exists $k_0$ such that $A_k=0$ for $k<k_0$.

We shall define on each $A_k$ a filtration $(F^p(A_k){)}_{p\in \mathbb{Z}}$ by the formulas

  F^p(A_k) = Ker(A_k → A_{p−1})    for p ≤ k+1                              (13.4.1.1)
  F^p(A_k) = 0                      for p ≥ k+1

One has therefore by hypothesis $F^k(A_k)=A_k$ and $F^{k+1}(A_k)=0$, in other words the filtration considered is finite (11.1.3). The graded objects associated to this filtration are therefore

  gr^p(A_k) = Ker(A_k → A_{p−1}) / Ker(A_k → A_p)

and consequently $g{r}^p(A_k)$ is isomorphic to the image under $A_k\to A_p$ of $Ker(A_k\to A_{p-1})$; by virtue of the transitivity of the morphisms defining a projective system, one therefore has

  gr^p(A_k) = Ker(A_p → A_{p−1}) ∩ Im(A_k → A_p)                            (13.4.1.2)

but since, by virtue of (13.4.1.1), one has $Ker(A_p\to A_{p-1})=g{r}^p(A_p)$, one also has

  gr^p(A_k) = gr^p(A_p) ∩ Im(A_k → A_p).                                    (13.4.1.3)

The preceding definitions show, moreover, that one has for $k\le h$

$$u_{hk}(F^p(A_k))\subset F^p(A_h)$$

and consequently that the $g{r}^p(u_{hk})$ define a projective system $(g{r}^p(A_k){)}_{k\in \mathbb{Z}}$ for every $p\in \mathbb{Z}$.

13.4.2.

We shall say that the projective system $\mathbf{A}$ is essentially constant if the morphisms $A_{k+1}\to A_k$ are isomorphisms for $k$ large enough. We shall say that the projective system $\mathbf{A}$ is strict if the morphisms $A_i\to A_j$ ($j\le i$) are epimorphisms. When $\mathbf{A}$ is strict, it follows from (13.4.1.3) that for $p\le k\le h$, the canonical morphism $g{r}^p(A_h)\to g{r}^p(A_k)$ is an isomorphism, in other words, the projective system $(g{r}^p(A_k){)}_{k\in \mathbb{Z}}$ is essentially constant. The sequence of objects $g{r}^p(A_p)$ (identified with $\underleftarrow{\lim }g{r}^p(A_k)$ for every $p$) is then denoted $g{r}^{\bullet }(\mathbf{A})$ and called the graded object associated to the strict projective system $\mathbf{A}=(A_k)$.

If one now supposes that the projective system $\mathbf{A}$ (bounded below) satisfies (ML), one knows (13.1.2) that the projective system ${\mathbf{A}}^{\prime }=(A_k^{\prime })$ of objects of "universal images" is strict, and is moreover bounded below; the graded object $g{r}^{\bullet }({\mathbf{A}}^{\prime })$ associated to ${\mathbf{A}}^{\prime }$ is then again called the graded object associated to $\mathbf{A}$ and denoted $g{r}^{\bullet }(\mathbf{A})$.

Proposition (13.4.3).

Let $\mathbf{A}=(A_k{)}_{k\in \mathbb{Z}}$ be a projective system bounded below in an abelian category $\mathcal{C}$. The following two conditions are equivalent:

• a) $\mathbf{A}$ satisfies condition (ML).
• b) For every $p\in \mathbb{Z}$, the projective system $(g{r}^p(A_k){)}_{k\in \mathbb{Z}}$ is essentially constant.

In addition, when these conditions are satisfied, one has for every $p\in \mathbb{Z}$ a canonical isomorphism

  gr^p(𝐀) ⥲ lim_← gr^p(A_k).                                                (13.4.3.1)
            k

It follows immediately from (13.4.1.2) that a) implies b); the same formula applied to the projective system ${\mathbf{A}}^{\prime }$ (notations of (13.4.2)) gives the isomorphism (13.4.3.1) by definition. For $k\le h$, set $A_{kh}=Im(A_h\to A_k)$; if $k\le h\le j$, one has $A_{kj}\subset A_{kh}\subset A_k$. Equip $A_{kh}$ with the filtration induced by $(F^p(A_k))$; one verifies immediately, by virtue of the transitivity of the morphisms defining $\mathbf{A}$, that this filtration is also the quotient filtration of $(F^p(A_h))$; consequently, one has

$$g{r}^p(A_{kh})=Im(g{r}^p(A_h)\to g{r}^p(A_k)).(\mathrm{13.4.3.2})$$

That being so, suppose b) verified; for every $p\in \mathbb{Z}$ and every $k\ge p$, there exists an integer $L(p,k)$ such that the right-hand side of (13.4.3.2) is constant for $h\ge L(p,k)$; since $g{r}^p(A_k)=0$ for $p<k_0$, there is only (for $k$ given) a finite number of $L(p,k)$ non-zero when $p$ runs over the set of integers $\le k$. Let $L(k)=m$ be the largest of these integers; for every $h\ge m$, one has $A_{kh}\subset A_{km}$, and by definition of $m$, the canonical injection $A_{kh}\to A_{km}$ defines an isomorphism $g{r}^{\bullet }(A_{kh})\stackrel{\sim }{\to }g{r}^{\bullet }(A_{km})$; since the filtrations are finite, one concludes that the preceding injection is itself bijective (Bourbaki, Alg. comm., ch. III, §2, n° 8, th. 1), which proves that $\mathbf{A}$ satisfies (ML).

13.4.4.

Suppose that in $\mathcal{C}$ the projective limit $A=\underleftarrow{\lim }{A}_k$ exists. In the definitions of (13.4.1), one can then replace $A_k$ by $A$, and the filtration thus defined on $A$ is again such that

  gr^p(A) = gr^p(A_p) ∩ Im(A → A_p).                                         (13.4.4.1)

Corollary (13.4.5).

Suppose that $\mathcal{C}$ is the category of abelian groups. If the projective system $\mathbf{A}$ satisfies (ML) and if $A=\underleftarrow{\lim }{A}_k$, one has for every $p\in \mathbb{Z}$ a canonical isomorphism

  gr^p(A) ⥲ lim_← gr^p(A_k).                                                 (13.4.5.1)
            k

Indeed, one has $Im(A_k\to A_p)=Im(A\to A_p)$ whenever $k$ is large enough (Bourbaki, Top. gén., ch. II, 3rd ed., §3, n° 5, th. 1), and the conclusion results from (13.4.1.3) and (13.4.4.1).

13.5. Projective limits of spectral sequences of filtered complexes

13.5.1.

Let $\mathcal{C}$ be an abelian category, $X^{\bullet }$ a complex of objects of $\mathcal{C}$ equipped with a filtration $(F^p(X^{\bullet }){)}_{p\in \mathbb{Z}}$ such that $F^{p_0}(X^{\bullet })=X^{\bullet }$ for some index $p_0$. Consider for each $k\in \mathbb{Z}$ the complex $X_k^{\bullet }=X^{\bullet }/F^{k+1}(X^{\bullet })$; it is canonically equipped with the filtration formed by the $F^p(X_k^{\bullet })=F^p(X^{\bullet })/F^{k+1}(X^{\bullet })$ for $p\le k$ and $F^p(X_k^{\bullet })=0$ for $p\ge k+1$. Moreover, one has canonical morphisms $X_{k+1}^{\bullet }\to X_k^{\bullet }$, which make ${\mathbf{X}}^{\bullet }=(X_k^{\bullet }{)}_{k\in \mathbb{Z}}$ a projective system of filtered complexes of objects of $\mathcal{C}$. Note that this projective system is strict and is such that $X_k^{\bullet }=0$ for $k<p_0$.

13.5.2.

Consider more generally a strict projective system ${\mathbf{X}}^{\bullet }=(X_k^{\bullet }{)}_{k\in \mathbb{Z}}$ of complexes of objects of $\mathcal{C}$, bounded below; consider on each $X_k^{\bullet }$ the filtration defined in (13.4.1) (placing oneself in the abelian category of complexes of $\mathcal{C}$ bounded below). The $X_k^{\bullet }\to X_p^{\bullet }$ ($p\le k$) become morphisms of filtered complexes, with finite filtrations. The functorial character of the spectral sequences of filtered complexes (11.2.3) shows that the morphisms defining the projective system ${\mathbf{X}}^{\bullet }$ furnish morphisms making $E({\mathbf{X}}^{\bullet })=(E(X_k^{\bullet }))$ a projective system of spectral sequences.

Lemma (13.5.3).

Suppose that the projective system ${\mathbf{X}}^{\bullet }=(X_k^{\bullet }{)}_{k\in \mathbb{Z}}$ of filtered complexes is obtained as in (13.5.2). Then:

• a) For $r\ge p-p_0$, one has $B_r^{pq}(X_k^{\bullet })=B_{\infty }^{pq}(X_k^{\bullet })$ for every $k\in \mathbb{Z}$.
• b) For $k+1\ge p+r$, one has $Z_r^{pq}(X_k^{\bullet })=Z_{\infty }^{pq}(X_k^{\bullet })$.
• c) For $k+1\ge p+r$, the morphisms $Z_r^{pq}(X_h^{\bullet })\to Z_r^{pq}(X_k^{\bullet })$ and $B_r^{pq}(X_h^{\bullet })\to B_r^{pq}(X_k^{\bullet })$ are isomorphisms for every $h\ge k$.

These three properties result immediately from the definitions of (11.2.2), taking into account that $F^{p-r+1}(X_k^{\bullet })=X_k^{\bullet }$ for $p-r<p_0$.

13.5.4.

Suppose the hypotheses of (13.5.3) are satisfied. Then, for $p$, $q$, $r$ fixed ($r$ finite), the projective systems $(Z_r^{pq}(X_k^{\bullet }){)}_{k\in \mathbb{Z}}$, $(B_r^{pq}(X_k^{\bullet }){)}_{k\in \mathbb{Z}}$, $(E_r^{pq}(X_k^{\bullet }){)}_{k\in \mathbb{Z}}$ are essentially constant; one will denote by $Z_r^{pq}({\mathbf{X}}^{\bullet })$, $B_r^{pq}({\mathbf{X}}^{\bullet })$ and $E_r^{pq}({\mathbf{X}}^{\bullet })=Z_r^{pq}({\mathbf{X}}^{\bullet })/B_r^{pq}({\mathbf{X}}^{\bullet })$ their respective projective limits. The $Z_r^{pq}({\mathbf{X}}^{\bullet })$ and $B_r^{pq}({\mathbf{X}}^{\bullet })$ identify canonically with subobjects of $E_1^{pq}({\mathbf{X}}^{\bullet })$. The definition of the $d_r^{pq}$ (M, XV, 1) shows that these morphisms (relative to the $X_k^{\bullet }$) are also essentially constant, and consequently define morphisms

  d^{pq}_r : E^{pq}_r(𝐗^•) → E^{p+r, q−r+1}_r(𝐗^•)                          (13.5.4.1)

such that $d_r^{p+r,q-r+1}\circ d_r^{pq}=0$; moreover, one has canonical isomorphisms of $Ker(d_r^{pq})$ onto $Z_{r+1}^{pq}({\mathbf{X}}^{\bullet })/B_r^{pq}({\mathbf{X}}^{\bullet })$ and of $Im(d_r^{pq})$ onto $B_{r+1}^{p+r,q-r+1}({\mathbf{X}}^{\bullet })/B_r^{p+r,q-r+1}({\mathbf{X}}^{\bullet })$.

Lemma (13.5.5).

Under the hypotheses of (13.5.3), one has, for $s\ge r>p-p_0$, a canonical monomorphism

$$i:E_s^{pq}({\mathbf{X}}^{\bullet })\to E_r^{pq}({\mathbf{X}}^{\bullet })(\mathrm{13.5.5.1})$$

and a canonical isomorphism

$$j_r:E_r^{pq}({\mathbf{X}}^{\bullet })\stackrel{\sim }{\to }{E}_{\infty }^{pq}(X_{p+r-1}^{\bullet })(\mathrm{13.5.5.2})$$

such that the diagram

                       j_s
  E^{pq}_s(𝐗^•) ----------> E^{pq}_∞(X^•_{p+s−1})
       ↓ i                          ↓
  E^{pq}_r(𝐗^•) ----------> E^{pq}_∞(X^•_{p+r−1})                           (13.5.5.3)
                       j_r

is commutative (the right-hand vertical arrow coming from the morphism $X_{p+s-1}^{\bullet }\to X_{p+r-1}^{\bullet }$).

The existence of $i$ comes from the fact that $B_r^{pq}(X_k^{\bullet })=B_{\infty }^{pq}(X_k^{\bullet })$ for $r>p-p_0$ (13.5.3, a)); one has $Z_r^{pq}(X_k^{\bullet })=Z_{\infty }^{pq}(X_k^{\bullet })$ for $k+1\ge p+r$ (13.5.3, b)), whence in particular $Z_r^{pq}(X_{p+r-1}^{\bullet })=Z_{\infty }^{pq}(X_{p+r-1}^{\bullet })$, and on the other hand $Z_r^{pq}(X_{p+r-1}^{\bullet })$ and $B_r^{pq}(X_{p+r-1}^{\bullet })$ identify canonically with $Z_r^{pq}({\mathbf{X}}^{\bullet })$ and $B_r^{pq}({\mathbf{X}}^{\bullet })$ by virtue of (13.5.3, c)), whence the existence of $j_r$ and the commutativity of (13.5.5.3).

Corollary (13.5.6).

Under the hypotheses of (13.5.3), if one of the projective limits $\underleftarrow{\lim }{E}_r^{pq}({\mathbf{X}}^{\bullet })$, $\underleftarrow{\lim }{E}_{\infty }^{pq}(X_k^{\bullet })$ exists, so does the other, and one has a canonical isomorphism

  j_∞ : lim_r E^{pq}_r(𝐗^•) ⥲ lim_k E^{pq}_∞(X^•_k).                        (13.5.6.1)

In addition, for the projective system $(E_r^{pq}({\mathbf{X}}^{\bullet }){)}_{r\in \mathbb{Z}}$ to be essentially constant (13.4.2), it is necessary and sufficient that the projective system $(E_{\infty }^{pq}(X_k^{\bullet }){)}_{k\in \mathbb{Z}}$ be so.

13.5.7.

One denotes by $B_{\infty }^{pq}({\mathbf{X}}^{\bullet })$ and $Z_{\infty }^{pq}({\mathbf{X}}^{\bullet })$ the subobjects of $E_1^{pq}({\mathbf{X}}^{\bullet })$ respectively equal to $B_r^{pq}({\mathbf{X}}^{\bullet })$ for $r>p-p_0$ and to ${\inf }_r{Z}_r^{pq}({\mathbf{X}}^{\bullet })$ (when the latter exists), so that $\underleftarrow{\lim }{E}_r^{pq}({\mathbf{X}}^{\bullet })$

identifies canonically with $E_{\infty }^{pq}({\mathbf{X}}^{\bullet })=Z_{\infty }^{pq}({\mathbf{X}}^{\bullet })/B_{\infty }^{pq}({\mathbf{X}}^{\bullet })$. One will note that the objects $Z_{\infty }^{pq}({\mathbf{X}}^{\bullet })$, $B_{\infty }^{pq}({\mathbf{X}}^{\bullet })$, $E_r^{pq}({\mathbf{X}}^{\bullet })$ ($1\le r\le +\infty$) and $d_r^{pq}$ depend functorially on the projective system ${\mathbf{X}}^{\bullet }$ submitted to the restrictions of (13.5.5), and that the morphisms defined in (13.5.5) and (13.5.6) are functorial.

13.6. Spectral sequence of a functor relative to an object equipped with a finite filtration

13.6.1.

Let $\mathcal{C}$, ${\mathcal{C}}^{\prime }$ be two abelian categories, $T:\mathcal{C}\to {\mathcal{C}}^{\prime }$ a covariant additive functor. Suppose that every object of $\mathcal{C}$ is isomorphic to a subobject of an injective object, so that the right-derived functors $R^{p}T$ ($p\ge 0$) exist.

Lemma (13.6.2).

Let $A$ be an object of $\mathcal{C}$, equipped with a finite filtration $(F^i(A){)}_{i\in \mathbb{Z}}$. There exists an injective resolution $X^{\bullet }=(X^i{)}_{i\ge 0}$ of $A$ equipped with a finite filtration $(F^i(X^{\bullet }){)}_{i\in \mathbb{Z}}$ such that the relation $F^i(A)=A$ (resp. $F^i(A)=0$) entails $F^i(X^{\bullet })=X^{\bullet }$ (resp. $F^i(X^{\bullet })=0$) and such that, for every $i\in \mathbb{Z}$, $F^i(X^{\bullet })$ is an injective resolution of $F^i(A)$.

Let $p$ (resp. $q>p$) be the largest index such that $F^p(A)=A$ (resp. the smallest index for which $F^q(A)=0$). One reasons by induction on $q-p$, the lemma being evident for $q-p=1$. Having formed an injective resolution $X^{\prime \prime \prime \bullet }$ of $A/F^{q-1}(A)$ having the desired properties, one considers the exact sequence $0\to F^{q-1}(A)\to A\to A/F^{q-1}(A)\to 0$, one takes an injective resolution $X^{\prime \prime \bullet }$ of $F^{q-1}(A)$, then one determines an injective resolution $X^{\bullet }$ of $A$ so as to have an exact sequence $0\to X^{\prime \prime \bullet }\to X^{\bullet }\to X^{\prime \prime \prime \bullet }\to 0$ compatible with the preceding (M, V, 2.2); it is clear that $X^{\bullet }$ answers the question.

Corollary (13.6.3).

Let $B$ be a second object of $\mathcal{C}$, equipped with a finite filtration $(F^i(B){)}_{i\in \mathbb{Z}}$, $s$ an integer, and let $u:A\to B$ be a morphism such that $u(F^i(A))\subset F^{i+s}(B)$ for every $i\in \mathbb{Z}$. If $Y^{\bullet }=(Y^i{)}_{i\ge 0}$ is an injective resolution of $B$ equipped with a filtration $(F^i(Y^{\bullet }){)}_{i\in \mathbb{Z}}$ having the properties stated in (13.6.2), there exists a morphism $v:X^{\bullet }\to Y^{\bullet }$ compatible with $u$ and such that $v(F^i(X^{\bullet }))\subset F^{i+s}(Y^{\bullet })$ for every $i\in \mathbb{Z}$. In addition, two such morphisms $v$, $v^{\prime }$ are homotopic.

This results immediately by induction on $q-p$ from the preceding construction and from (M, V, 2.3).

13.6.4.

Under the hypotheses of (13.6.2), consider now the complex $T(X^{\bullet })$ in ${\mathcal{C}}^{\prime }$, which is evidently filtered by the complexes $T(F^i(X^{\bullet }))$, since $F^i(X^{\bullet })$ is a direct factor of $X^{\bullet }$. It follows from (13.6.3) that the spectral sequence of this filtered complex depends only on the filtered object $A$, up to isomorphism. Its abutment is the cohomology $R^{\bullet }T(A)$, with the filtration

  F^p(R^n T(A)) = Im(R^n T(F^p(A)) → R^n T(A))
                = Ker(R^n T(A) → R^n T(A/F^p(A)))                           (13.6.4.1)

(11.2.2), and its term E_1 is given by

$$E_1^{pq}=R^{p+q}T(g{r}^p(A))(\mathrm{13.6.4.2})$$

$g{r}^p(A)$ denoting as usual $F^p(A)/F^{p+1}(A)$. It is clear, by (11.2.2), that the filtration of the abutment is finite, and that for $p$, $q$ given, the sequences of

$B_r^{pq}(A)=B_r^{pq}(T(X^{\bullet }))$ and $Z_r^{pq}(A)=Z_r^{pq}(T(X^{\bullet }))$ are stationary, so the preceding spectral sequence is biregular (11.1.3). We shall denote this sequence $E(A)=(E_r^{pq}(A))$ and we shall say that it is the spectral sequence of the functor $T$ relative to the filtered object $A$.

13.6.5.

Suppose now the hypotheses of (13.6.3) are satisfied, whose notations we keep. Since $F^i(X^{\bullet })$ (resp. $F^i(Y^{\bullet })$) is a direct factor of $X^{\bullet }$ (resp. $Y^{\bullet }$), one has $(Tv)(T(F^i(X^{\bullet })))\subset T(F^{i+s}(Y^{\bullet }))$ for every $i\in \mathbb{Z}$; the definitions of (11.2.2) show then that for $1\le r\le +\infty$, Tv defines a morphism $B_r^{pq}(T(X^{\bullet }))\to B_r^{p+s,q-s}(T(Y^{\bullet }))$ and a morphism $Z_r^{pq}(T(X^{\bullet }))\to Z_r^{p+s,q-s}(T(Y^{\bullet }))$, whence a morphism

  w_r : E^{pq}_r(A) → E^{p+s, q−s}_r(B);

similarly, one has for the abutment morphisms $u_n:R^{n}T(A)\to R^{n}T(B)$ such that $u_n(F^p(R^{n}T(A)))\subset F^{p+s}(R^{n}T(B))$.

The definition of the $d_r^{pq}$ (M, XV, 1) shows moreover that the diagrams

                       d^{pq}_r
       E^{pq}_r(A) ----------------> E^{p+r, q−r+1}_r(A)
            ↓ w_r                            ↓ w_r
  E^{p+s, q−s}_r(B) -------------> E^{p+r+s, q−r−s+1}_r(B)
                  d^{p+s, q−s}_r

are commutative; one deduces an analogous commutative diagram for the isomorphisms ${\alpha }_r^{pq}$, which we shall leave to the reader the care of making explicit. Finally (loc. cit.), one also has commutative diagrams for the abutments

                    β^{pq}
       E^{pq}_∞(A) -------> gr^p(R^{p+q} T(A))
            ↓ w_∞                  ↓ u_{p+q}
  E^{p+s, q−s}_∞(B) ----> gr^{p+s}(R^{p+q} T(B))
                  β^{p+s, q−s}

13.6.6.

Suppose in particular that there exists a ring $\mathcal{S}$, equipped with a filtration $(F^i(\mathcal{S}){)}_{i\in \mathbb{Z}}$, and a ring homomorphism

  h : 𝒮 → Hom_𝒞(A, A)                                                       (13.6.6.1)

such that for every $t\in F^i(\mathcal{S})$, one has $h_t(F^j(A))\subset F^{i+j}(A)$ for every pair $i$, $j$. We shall say for brevity that $A$ is then equipped with a structure of $\mathcal{S}$-$\mathcal{C}$-module filtered over the filtered ring $\mathcal{S}$. By passage to the associated graded objects, every $h_t$ for $t\in F^j(\mathcal{S})$ defines a graded endomorphism ${\bar{h}}_t$ of $g{r}^{\bullet }(A)$, homogeneous of degree $j$; moreover, this morphism depends only on the class of $t$ in $g{r}^j(\mathcal{S})$, and one thus defines a homomorphism of graded rings

  h̄ : gr^•(𝒮) → Hom_𝒞(gr^•(A), gr^•(A))

where the right-hand side is the ring of graded endomorphisms of $g{r}^{\bullet }(A)$. We shall say that $g{r}^{\bullet }(A)$ is equipped with a structure of $g{r}^{\bullet }(\mathcal{S})$-$\mathcal{C}$-module graded. It follows then from (13.6.5) that for $1\le r\le +\infty$, every $\bar{t}\in g{r}^j(\mathcal{S})$ canonically defines in the bigraded objects $(B_r^{pq}(A){)}_{(p,q)\in \mathbb{Z}\times \mathbb{Z}}$, $(Z_r^{pq}(A){)}_{(p,q)\in \mathbb{Z}\times \mathbb{Z}}$ and $E_r(A)=(E_r^{pq}(A){)}_{(p,q)\in \mathbb{Z}\times \mathbb{Z}}$ bigraded endomorphisms of degrees $(j,-j)$; in $E_r(A)$ (for $r$ finite), this endomorphism commutes with the bigraded endomorphism defined by the $d_r^{pq}$. Since these endomorphisms satisfy the usual conditions of associativity and distributivity with respect to the addition in $g{r}^{\bullet }(\mathcal{S})$ and in the bigraded objects considered, we shall say for brevity that the latter are $g{r}^{\bullet }(\mathcal{S})$-${\mathcal{C}}^{\prime }$-modules bigraded; it is immediate that the ${\alpha }_r^{pq}$ define an isomorphism for this type of structures. For every integer $n$, one will denote by $B_r^{(n)}(A)$ (resp. $Z_r^{(n)}(A)$, $E_r^{(n)}(A)$) the graded subobject of $B_r^{\bullet \bullet }(A)$ (resp. $Z_r^{\bullet \bullet }(A)$, $E_r^{\bullet \bullet }(A)$) formed by the $B_r^{pq}(A)$ (resp. $Z_r^{pq}(A)$, $E_r^{pq}(A)$) such that $p+q=n$ (for $1\le r<+\infty$); it is immediate that these are $g{r}^{\bullet }(\mathcal{S})$-${\mathcal{C}}^{\prime }$-modules graded. Finally, every $\bar{t}\in g{r}^j(\mathcal{S})$ defines for every $n$ a graded endomorphism of degree $j$ in the graded object $g{r}^{\bullet }(R^{n}T(A))$, which is thus equipped with a structure of $g{r}^{\bullet }(\mathcal{S})$-${\mathcal{C}}^{\prime }$-module graded, the ${\beta }^{pq}$ (for $p+q=n$) define an isomorphism of $E_{\infty }^{(n)}(A)$ onto $g{r}^{\bullet }(R^{n}T(A))$ for this kind of structure.

Note that when ${\mathcal{C}}^{\prime }$ is the category of abelian groups, the structures of $\mathcal{S}$-${\mathcal{C}}^{\prime }$-module (resp. of $g{r}^{\bullet }(\mathcal{S})$-${\mathcal{C}}^{\prime }$-module graded or bigraded) are none other than the usual structures of $\mathcal{S}$-module (resp. $g{r}^{\bullet }(\mathcal{S})$-module graded, bigraded).

13.7. Derived functors of a projective limit of arguments

13.7.1.

Let $\mathcal{C}$, ${\mathcal{C}}^{\prime }$ be two abelian categories, $\mathcal{C}$ being supposed such that every object of $\mathcal{C}$ is a subobject of an injective object; let $T:\mathcal{C}\to {\mathcal{C}}^{\prime }$ be a covariant additive functor. Consider a strict projective system $\mathbf{A}=(A_k{)}_{k\in \mathbb{Z}}$ in $\mathcal{C}$, bounded below; to be precise, we shall suppose that $A_k=0$ for $k<k_0$. We associate canonically to this system a filtration $(F^p(A_k){)}_{p\in \mathbb{Z}}$ on each $A_k$ by the formulas (13.4.1.1), and since this is a strict projective system, the canonical morphisms

  F^i(A_h)/F^j(A_h) → F^i(A_k)/F^j(A_k)    (h ≥ k)                           (13.7.1.1)

for $i\le j\le k+1$ are isomorphisms. Recall in addition that one has $F^k(A_k)=A_k$ and $F^{k+1}(A_k)=0$ for every $k$.

13.7.2.

Let us construct now for each $k$ an injective resolution $X_k^{\bullet }=(X_k^i{)}_{i\ge 0}$ of $A_k$ having the properties of (13.6.2). The canonical morphisms $A_{k+1}\to A_k$ allow (13.6.3) to define for each $k$ a morphism of complexes $X_{k+1}^{\bullet }\to X_k^{\bullet }$ compatible