Lemma 13.42.1. Let \mathcal{D} be a triangulated category. Let (A_ i) be an inverse system in \mathcal{D}. Then (A_ i) is essentially constant (see Categories, Definition 4.22.1) if and only if there exists an i and for all j \geq i a direct sum decomposition A_ j = A \oplus Z_ j such that (a) the maps A_{j'} \to A_ j are compatible with the direct sum decompositions and identity on A, (b) for all j \geq i there exists some j' \geq j such that Z_{j'} \to Z_ j is zero.
13.42 Essentially constant systems
Some preliminary lemmas on essentially constant systems in triangulated categories.
Proof. Assume (A_ i) is essentially constant with value A. Then A = \mathop{\mathrm{lim}}\nolimits A_ i and there exists an i and a morphism A_ i \to A such that (1) the composition A \to A_ i \to A is the identity on A and (2) for all j \geq i there exists a j' \geq j such that A_{j'} \to A_ j factors as A_{j'} \to A_ i \to A \to A_ j. From (1) we conclude that for j \geq i the maps A \to A_ j and A_ j \to A_ i \to A compose to the identity on A. It follows that A_ j \to A has a kernel Z_ j and that the map A \oplus Z_ j \to A_ j is an isomorphism, see Lemmas 13.4.12 and 13.4.11. These direct sum decompositions clearly satisfy (a). From (2) we conclude that for all j there is a j' \geq j such that Z_{j'} \to Z_ j is zero, so (b) holds. Proof of the converse is omitted. \square
Lemma 13.42.2. Let \mathcal{D} be a triangulated category. Let
A_ n \to B_ n \to C_ n \to A_ n[1]
be an inverse system of distinguished triangles in \mathcal{D}. If (A_ n) and (C_ n) are essentially constant, then (B_ n) is essentially constant and their values fit into a distinguished triangle A \to B \to C \to A[1] such that for some n \geq 1 there is a map
\xymatrix{ A_ n \ar[d] \ar[r] & B_ n \ar[d] \ar[r] & C_ n \ar[d] \ar[r] & A_ n[1] \ar[d] \\ A \ar[r] & B \ar[r] & C \ar[r] & A[1] }
of distinguished triangles which induces an isomorphism \mathop{\mathrm{lim}}\nolimits _{n' \geq n} A_{n'} \to A and similarly for B and C.
Proof. After renumbering we may assume that A_ n = A \oplus A'_ n and C_ n = C \oplus C'_ n for inverse systems (A'_ n) and (C'_ n) which are essentially zero, see Lemma 13.42.1. In particular, the morphism
C \oplus C'_ n \to (A \oplus A'_ n)[1]
maps the summand C into the summand A[1] for all n by a map \delta : C \to A[1] which is independent of n. Choose a distinguished triangle
A \to B \to C \xrightarrow {\delta } A[1]
Next, choose a morphism of distinguished triangles
(A_1 \to B_1 \to C_1 \to A_1[1]) \to (A \to B \to C \to A[1])
which is possible by TR3. For any object D of \mathcal{D} this induces a commutative diagram
\xymatrix{ \ldots \ar[r] & \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(C, D) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(B, D) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(A, D) \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(C_ n, D) \ar[r] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(B_ n, D) \ar[r] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(A_ n, D) \ar[r] & \ldots }
The left and right vertical arrows are isomorphisms and so are the ones to the left and right of those. Thus by the 5-lemma we conclude that the middle arrow is an isomorphism. It follows that (B_ n) is isomorphic to the constant inverse system with value B by the discussion in Categories, Remark 4.22.7. Since this is equivalent to (B_ n) being essentially constant with value B by Categories, Remark 4.22.5 the proof is complete. \square
Lemma 13.42.3. Let \mathcal{A} be an abelian category. Let A_ n be an inverse system of objects of D(\mathcal{A}). Assume
there exist integers a \leq b such that H^ i(A_ n) = 0 for i \not\in [a, b], and
the inverse systems H^ i(A_ n) of \mathcal{A} are essentially constant for all i \in \mathbf{Z}.
Then A_ n is an essentially constant system of D(\mathcal{A}) whose value A satisfies that H^ i(A) is the value of the constant system H^ i(A_ n) for each i \in \mathbf{Z}.
Proof. By Remark 13.12.4 we obtain an inverse system of distinguished triangles
\tau _{\leq a}A_ n \to A_ n \to \tau _{\geq a + 1}A_ n \to (\tau _{\leq a}A_ n)[1]
Of course we have \tau _{\leq a}A_ n = H^ a(A_ n)[-a] in D(\mathcal{A}). Thus by assumption these form an essentially constant system. By induction on b - a we find that the inverse system \tau _{\geq a + 1}A_ n is essentially constant, say with value A'. By Lemma 13.42.2 we find that A_ n is an essentially constant system. We omit the proof of the statement on cohomologies (hint: use the final part of Lemma 13.42.2). \square
Lemma 13.42.4. Let \mathcal{D} be a triangulated category. Let
A_ n \to B_ n \to C_ n \to A_ n[1]
be an inverse system of distinguished triangles. If the system C_ n is pro-zero (essentially constant with value 0), then the maps A_ n \to B_ n determine a pro-isomorphism between the pro-object (A_ n) and the pro-object (B_ n).
Proof. For any object X of \mathcal{D} consider the exact sequence
\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (C_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (B_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (A_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (C_ n[-1], X) \to
Exactness follows from Lemma 13.4.2 combined with Algebra, Lemma 10.8.8. By assumption the first and last term are zero. Hence the map \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (B_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (A_ n, X) is an isomorphism for all X. The lemma follows from this and Categories, Remark 4.22.7. \square
Lemma 13.42.5. Let \mathcal{A} be an abelian category.
A_ n \to B_ n
be an inverse system of maps of D(\mathcal{A}). Assume
there exist integers a \leq b such that H^ i(A_ n) = 0 and H^ i(B_ n) = 0 for i \not\in [a, b], and
the inverse system of maps H^ i(A_ n) \to H^ i(B_ n) of \mathcal{A} define an isomorphism of pro-objects of \mathcal{A} for all i \in \mathbf{Z}.
Then the maps A_ n \to B_ n determine a pro-isomorphism between the pro-object (A_ n) and the pro-object (B_ n).
Proof. We can inductively extend the maps A_ n \to B_ n to an inverse system of distinguished triangles A_ n \to B_ n \to C_ n \to A_ n[1] by axiom TR3. By Lemma 13.42.4 it suffices to prove that C_ n is pro-zero. By Lemma 13.42.3 it suffices to show that H^ p(C_ n) is pro-zero for each p. This follows from assumption (2) and the long exact sequences
H^ p(A_ n) \xrightarrow {\alpha _ n} H^ p(B_ n) \xrightarrow {\beta _ n} H^ p(C_ n) \xrightarrow {\delta _ n} H^{p + 1}(A_ n) \xrightarrow {\epsilon _ n} H^{p + 1}(B_ n)
Namely, for every n we can find an m > n such that \mathop{\mathrm{Im}}(\beta _ m) maps to zero in H^ p(C_ n) because we may choose m such that H^ p(B_ m) \to H^ p(B_ n) factors through \alpha _ n : H^ p(A_ n) \to H^ p(B_ n). For a similar reason we may then choose k > m such that \mathop{\mathrm{Im}}(\delta _ k) maps to zero in H^{p + 1}(A_ m). Then H^ p(C_ k) \to H^ p(C_ n) is zero because H^ p(C_ k) \to H^ p(C_ m) maps into \mathop{\mathrm{Ker}}(\delta _ m) and H^ p(C_ m) \to H^ p(C_ n) annihilates \mathop{\mathrm{Ker}}(\delta _ m) = \mathop{\mathrm{Im}}(\beta _ m). \square
Comments (0)