13.41 Postnikov systems
A reference for this section is [Orlov-K3]. Let \mathcal{D} be a triangulated category. Let
X_ n \to X_{n - 1} \to \ldots \to X_0
be a complex in \mathcal{D}. In this section we consider the problem of constructing a “totalization” of this complex.
Definition 13.41.1. Let \mathcal{D} be a triangulated category. Let
X_ n \to X_{n - 1} \to \ldots \to X_0
be a complex in \mathcal{D}. A Postnikov system is defined inductively as follows.
If n = 0, then it is an isomorphism Y_0 \to X_0.
If n = 1, then it is a choice of an isomorphism Y_0 \to X_0 and a choice of a distinguished triangle
Y_1 \to X_1 \to Y_0 \to Y_1[1]
where X_1 \to Y_0 composed with Y_0 \to X_0 is the given morphism X_1 \to X_0.
If n > 1, then it is a choice of a Postnikov system for X_{n - 1} \to \ldots \to X_0 and a choice of a distinguished triangle
Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1]
where the morphism X_ n \to Y_{n - 1} composed with Y_{n - 1} \to X_{n - 1} is the given morphism X_ n \to X_{n - 1}.
Given a morphism
13.41.1.1
\begin{equation} \label{derived-equation-map-complexes} \vcenter { \xymatrix{ X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_0 \ar[d] \\ X'_ n \ar[r] & X'_{n - 1} \ar[r] & \ldots \ar[r] & X'_0 } } \end{equation}
between complexes of the same length in \mathcal{D} there is an obvious notion of a morphism of Postnikov systems.
Here is a key example.
Example 13.41.2. Let \mathcal{A} be an abelian category. Let \ldots \to A_2 \to A_1 \to A_0 be a chain complex in \mathcal{A}. Then we can consider the objects
X_ n = A_ n \quad \text{and}\quad Y_ n = (A_ n \to A_{n - 1} \to \ldots \to A_0)[-n]
of D(\mathcal{A}). With the evident canonical maps Y_ n \to X_ n and Y_0 \to Y_1[1] \to Y_2[2] \to \ldots the distinguished triangles Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1] define a Postnikov system as in Definition 13.41.1 for \ldots \to X_2 \to X_1 \to X_0. Here we are using the obvious extension of Postnikov systems for an infinite complex of D(\mathcal{A}). Finally, if colimits over \mathbf{N} exist and are exact in \mathcal{A} then
\text{hocolim} Y_ n[n] = (\ldots \to A_2 \to A_1 \to A_0 \to 0 \to \ldots )
in D(\mathcal{A}). This follows immediately from Lemma 13.33.7.
Given a complex X_ n \to X_{n - 1} \to \ldots \to X_0 and a Postnikov system as in Definition 13.41.1 we can consider the maps
Y_0 \to Y_1[1] \to \ldots \to Y_ n[n]
These maps fit together in certain distinguished triangles and fit with the given maps between the X_ i. Here is a picture for n = 3:
\xymatrix{ Y_0 \ar[rr] & & Y_1[1] \ar[dl] \ar[rr] & & Y_2[2] \ar[dl] \ar[rr] & & Y_3[3] \ar[dl] \\ & X_1[1] \ar[lu]_{+1} & & X_2[2] \ar[ll]_{+1} \ar[lu]_{+1} & & X_3[3] \ar[ll]_{+1} \ar[lu]_{+1} }
We encourage the reader to think of Y_ n[n] as obtained from X_0, X_1[1], \ldots , X_ n[n]; for example if the maps X_ i \to X_{i - 1} are zero, then we can take Y_ n[n] = \bigoplus _{i = 0, \ldots , n} X_ i[i]. Postnikov systems do not always exist. Here is a simple lemma for low n.
Lemma 13.41.3. Let \mathcal{D} be a triangulated category. Consider Postnikov systems for complexes of length n.
For n = 0 Postnikov systems always exist and any morphism (13.41.1.1) of complexes extends to a unique morphism of Postnikov systems.
For n = 1 Postnikov systems always exist and any morphism (13.41.1.1) of complexes extends to a (nonunique) morphism of Postnikov systems.
For n = 2 Postnikov systems always exist but morphisms (13.41.1.1) of complexes in general do not extend to morphisms of Postnikov systems.
For n > 2 Postnikov systems do not always exist.
Proof.
The case n = 0 is immediate as isomorphisms are invertible. The case n = 1 follows immediately from TR1 (existence of triangles) and TR3 (extending morphisms to triangles). For the case n = 2 we argue as follows. Set Y_0 = X_0. By the case n = 1 we can choose a Postnikov system
Y_1 \to X_1 \to Y_0 \to Y_1[1]
Since the composition X_2 \to X_1 \to X_0 is zero, we can factor X_2 \to X_1 (nonuniquely) as X_2 \to Y_1 \to X_1 by Lemma 13.4.2. Then we simply fit the morphism X_2 \to Y_1 into a distinguished triangle
Y_2 \to X_2 \to Y_1 \to Y_2[1]
to get the Postnikov system for n = 2. For n > 2 we cannot argue similarly, as we do not know whether the composition X_ n \to X_{n - 1} \to Y_{n - 1} is zero in \mathcal{D}.
\square
Lemma 13.41.4. Let \mathcal{D} be a triangulated category. Given a map (13.41.1.1) consider the condition
13.41.4.1
\begin{equation} \label{derived-equation-P} \mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], X'_ j) = 0 \text{ for }i > j + 1 \end{equation}
Then
If we have a Postnikov system for X'_ n \to X'_{n - 1} \to \ldots \to X'_0 then property (13.41.4.1) implies that
\mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], Y'_ j) = 0 \text{ for }i > j + 1
If we are given Postnikov systems for both complexes and we have (13.41.4.1), then the map extends to a (nonunique) map of Postnikov systems.
Proof.
We first prove (1) by induction on j. For the base case j = 0 there is nothing to prove as Y'_0 \to X'_0 is an isomorphism. Say the result holds for j - 1. We consider the distinguished triangle
Y'_ j \to X'_ j \to Y'_{j - 1} \to Y'_ j[1]
The long exact sequence of Lemma 13.4.2 gives an exact sequence
\mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], Y'_{j - 1}[-1]) \to \mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], Y'_ j) \to \mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], X'_ j)
From the induction hypothesis and (13.41.4.1) we conclude the outer groups are zero and we win.
Proof of (2). For n = 1 the existence of morphisms has been established in Lemma 13.41.3. For n > 1 by induction, we may assume given the map of Postnikov systems of length n - 1. The problem is that we do not know whether the diagram
\xymatrix{ X_ n \ar[r] \ar[d] & Y_{n - 1} \ar[d] \\ X'_ n \ar[r] & Y'_{n - 1} }
is commutative. Denote \alpha : X_ n \to Y'_{n - 1} the difference. Then we do know that the composition of \alpha with Y'_{n - 1} \to X'_{n - 1} is zero (because of what it means to be a map of Postnikov systems of length n - 1). By the distinguished triangle Y'_{n - 1} \to X'_{n - 1} \to Y'_{n - 2} \to Y'_{n - 1}[1], this means that \alpha is the composition of Y'_{n - 2}[-1] \to Y'_{n - 1} with a map \alpha ' : X_ n \to Y'_{n - 2}[-1]. Then (13.41.4.1) guarantees \alpha ' is zero by part (1) of the lemma. Thus \alpha is zero. To finish the proof of existence, the commutativity guarantees we can choose the dotted arrow fitting into the diagram
\xymatrix{ Y_{n - 1}[-1] \ar[d] \ar[r] & Y_ n \ar[r] \ar@{..>}[d] & X_ n \ar[r] \ar[d] & Y_{n - 1} \ar[d] \\ Y'_{n - 1}[-1] \ar[r] & Y'_ n \ar[r] & X'_ n \ar[r] & Y'_{n - 1} }
by TR3.
\square
Lemma 13.41.5. Let \mathcal{D} be a triangulated category. Given a map (13.41.1.1) assume we are given Postnikov systems for both complexes. If
\mathop{\mathrm{Hom}}\nolimits (X_ i[i], Y'_ n[n]) = 0 for i = 1, \ldots , n, or
\mathop{\mathrm{Hom}}\nolimits (Y_ n[n], X'_{n - i}[n - i]) = 0 for i = 1, \ldots , n, or
\mathop{\mathrm{Hom}}\nolimits (X_{j - i}[-i + 1], X'_ j) = 0 and \mathop{\mathrm{Hom}}\nolimits (X_ j, X'_{j - i}[-i]) = 0 for j \geq i > 0,
then there exists at most one morphism between these Postnikov systems.
Proof.
Proof of (1). Look at the following diagram
\xymatrix{ Y_0 \ar[r] \ar[d] & Y_1[1] \ar[r] \ar[ld] & Y_2[2] \ar[r] \ar[lld] & \ldots \ar[r] & Y_ n[n] \ar[lllld] \\ Y'_ n[n] }
The arrows are the composition of the morphism Y_ n[n] \to Y'_ n[n] and the morphism Y_ i[i] \to Y_ n[n]. The arrow Y_0 \to Y'_ n[n] is determined as it is the composition Y_0 = X_0 \to X'_0 = Y'_0 \to Y'_ n[n]. Since we have the distinguished triangle Y_0 \to Y_1[1] \to X_1[1] we see that \mathop{\mathrm{Hom}}\nolimits (X_1[1], Y'_ n[n]) = 0 guarantees that the second vertical arrow is unique. Since we have the distinguished triangle Y_1[1] \to Y_2[2] \to X_2[2] we see that \mathop{\mathrm{Hom}}\nolimits (X_2[2], Y'_ n[n]) = 0 guarantees that the third vertical arrow is unique. And so on.
Proof of (2). The composition Y_ n[n] \to Y'_ n[n] \to X_ n[n] is the same as the composition Y_ n[n] \to X_ n[n] \to X'_ n[n] and hence is unique. Then using the distinguished triangle Y'_{n - 1}[n - 1] \to Y'_ n[n] \to X'_ n[n] we see that it suffices to show \mathop{\mathrm{Hom}}\nolimits (Y_ n[n], Y'_{n - 1}[n - 1]) = 0. Using the distinguished triangles
Y'_{n - i - 1}[n - i - 1] \to Y'_{n - i}[n - i] \to X'_{n - i}[n - i]
we get this vanishing from our assumption. Small details omitted.
Proof of (3). Looking at the proof of Lemma 13.41.4 and arguing by induction on n it suffices to show that the dotted arrow in the morphism of triangles
\xymatrix{ Y_{n - 1}[-1] \ar[d] \ar[r] & Y_ n \ar[r] \ar@{..>}[d] & X_ n \ar[r] \ar[d] & Y_{n - 1} \ar[d] \\ Y'_{n - 1}[-1] \ar[r] & Y'_ n \ar[r] & X'_ n \ar[r] & Y'_{n - 1} }
is unique. By Lemma 13.4.8 part (5) it suffices to show that \mathop{\mathrm{Hom}}\nolimits (Y_{n - 1}, X'_ n) = 0 and \mathop{\mathrm{Hom}}\nolimits (X_ n, Y'_{n - 1}[-1]) = 0. To prove the first vanishing we use the distinguished triangles Y_{n - i - 1}[-i] \to Y_{n - i}[-(i - 1)] \to X_{n - i}[-(i - 1)] for i > 0 and induction on i to see that the assumed vanishing of \mathop{\mathrm{Hom}}\nolimits (X_{n - i}[-i + 1], X'_ n) is enough. For the second we similarly use the distinguished triangles Y'_{n - i - 1}[-i - 1] \to Y'_{n - i}[-i] \to X'_{n - i}[-i] to see that the assumed vanishing of \mathop{\mathrm{Hom}}\nolimits (X_ n, X'_{n - i}[-i]) is enough as well.
\square
Lemma 13.41.6. Let \mathcal{D} be a triangulated category. Let X_ n \to X_{n - 1} \to \ldots \to X_0 be a complex in \mathcal{D}. If
\mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 2], X_ j) = 0 \text{ for }i > j + 2
then there exists a Postnikov system. If we have
\mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 1], X_ j) = 0 \text{ for }i > j + 1
then any two Postnikov systems are isomorphic.
Proof.
We argue by induction on n. The cases n = 0, 1, 2 follow from Lemma 13.41.3. Assume n > 2. Suppose given a Postnikov system for the complex X_{n - 1} \to X_{n - 2} \to \ldots \to X_0. The only obstruction to extending this to a Postnikov system of length n is that we have to find a morphism X_ n \to Y_{n - 1} such that the composition X_ n \to Y_{n - 1} \to X_{n - 1} is equal to the given map X_ n \to X_{n - 1}. Considering the distinguished triangle
Y_{n - 1} \to X_{n - 1} \to Y_{n - 2} \to Y_{n - 1}[1]
and the associated long exact sequence coming from this and the functor \mathop{\mathrm{Hom}}\nolimits (X_ n, -) (see Lemma 13.4.2) we find that it suffices to show that the composition X_ n \to X_{n - 1} \to Y_{n - 2} is zero. Since we know that X_ n \to X_{n - 1} \to X_{n - 2} is zero we can apply the distinguished triangle
Y_{n - 2} \to X_{n - 2} \to Y_{n - 3} \to Y_{n - 2}[1]
to see that it suffices if \mathop{\mathrm{Hom}}\nolimits (X_ n, Y_{n - 3}[-1]) = 0. Arguing exactly as in the proof of Lemma 13.41.4 part (1) the reader easily sees this follows from the condition stated in the lemma.
The statement on isomorphisms follows from the existence of a map between the Postnikov systems extending the identity on the complex proven in Lemma 13.41.4 part (2) and Lemma 13.4.3 to show all the maps are isomorphisms.
\square
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