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A partial morphism of delta functors extends through one dimension shift

Statement

Let $S$ and $T$ be delta functors on an abelian category. In the homological case fix $n > 0$ ; in the cohomological case fix $n \geq 0$ .

Homological case: suppose $S = (S_{i}, \partial^{S})$ and $T = (T_{i}, \partial^{T})$ are homological, suppose natural transformations $u_{i} : S_{i} \Rightarrow T_{i}, 0 \leq i < n,$ have already been chosen compatibly with the connecting maps in degrees $< n$ , and choose for an object $A$ a short exact sequence $0 \to K \to P \overset{p}{\to} A \to 0$ such that $P$ is projective and $T_{n} (p) = 0$ . Then there is a unique morphism $u_{n}^{(p)} (A) : S_{n} (A) \to T_{n} (A)$ such that $\partial_{n}^{T} u_{n}^{(p)} (A) = u_{n - 1} (K) \partial_{n}^{S} .$ If a morphism $f : A \to A^{'}$ is covered by a morphism between two such chosen short exact sequences, then the maps $u_{n}^{(p)} (A)$ and $u_{n}^{(p^{'})} (A^{'})$ are natural with respect to $f$ .

Cohomological case: suppose $S = (S^{i}, \partial_{S})$ and $T = (T^{i}, \partial_{T})$ are cohomological, suppose natural transformations $u^{i} : S^{i} \Rightarrow T^{i}, 0 \leq i \leq n,$ have already been chosen compatibly with the connecting maps in degrees $< n$ , and choose for an object $A$ a short exact sequence $0 \to A \overset{e}{\to} I \to C \to 0$ such that $I$ is injective and $S^{n + 1} (e) = 0$ . Then there is a unique morphism $u_{A}^{n + 1, (e)} : S^{n + 1} (A) \to T^{n + 1} (A) .$ More explicitly, let $q_{S} : coker (S^{n} (I) \to S^{n} (C)) \tilde{\to} S^{n + 1} (A)$ be induced by $\partial_{S}^{n}$ , let ${\overline{u}}^{n}$ be the map on cokernels induced by $u^{n}$ , and let $q_{T} : coker (T^{n} (I) \to T^{n} (C)) \to T^{n + 1} (A)$ be induced by $\partial_{T}^{n}$ . Then $u_{A}^{n + 1, (e)}$ is characterized by $u_{A}^{n + 1, (e)} q_{S} = q_{T} {\overline{u}}^{n} .$ If a morphism $f : A \to A^{'}$ is covered by a morphism between two such chosen short exact sequences, then these maps are natural with respect to $f$ .

Facts & Assumptions

Given: An object $A$ and a chosen effacement sequence as in the statement.

[L1]

Effaceability supplies the chosen projective or injective short exact sequence (Effaceable homological delta functor in positive degrees, Effaceable cohomological delta functor in positive degrees).

[L2]

In the homological case, the chosen effacement makes the connecting map $\partial_{n}^{T} : T_{n} (A) \to T_{n - 1} (K)$ monic; in the cohomological case, the chosen effacement makes $S^{n + 1} (A)$ the cokernel of $S^{n} (I) \to S^{n} (C)$ (Dimension shift for a homological delta functor effaced in the middle, Dimension shift for a cohomological delta functor effaced in the middle).

[L3]

A natural transformation is defined by commuting with the maps induced by the chosen morphisms (Natural transformation and its components).

Proof

technique · direct

1.1L1L2givenconstruct

In the homological case, exactness of the long sequences for the chosen short exact sequence gives $S_{n} (P) \to S_{n} (A) \overset{\partial_{n}^{S}}{\to} S_{n - 1} (K) \to S_{n - 1} (P)$ and $T_{n} (P) \to T_{n} (A) \overset{\partial_{n}^{T}}{\to} T_{n - 1} (K) \to T_{n - 1} (P) .$ Because the lower incoming map is zero by the chosen effacement, [L2] makes $\partial_{n}^{T}$ monic. The already defined map $u_{n - 1} (K)$ therefore determines at most one map $u_{n}^{(p)} (A)$ satisfying $\partial_{n}^{T} u_{n}^{(p)} (A) = u_{n - 1} (K) \partial_{n}^{S},$ and exactness shows that the right-hand side lands in $\ker (T_{n - 1} (K) \to T_{n - 1} (P)) = im (\partial_{n}^{T})$ , so this map exists.

2.1L2L3step 1.1algebra

If $f : A \to A^{'}$ is covered by a morphism of chosen effacement sequences, then the already defined degree- $(n - 1)$ maps are natural by [L3]. Applying the defining equation from step 1.1 on both objects and using naturality of the connecting morphisms inside the two long exact sequences shows that $\partial_{n}^{T}$ has the same composite with both $T_{n} (f) u_{n}^{(p)} (A)$ and $u_{n}^{(p^{'})} (A^{'}) S_{n} (f)$ . Since the target $\partial_{n}^{T}$ is monic by [L2], these two maps are equal.

3.1L2L3givenconstruct∎

In the cohomological case, [L2] makes $q_{S}$ an isomorphism. The known map $u^{n}$ induces ${\overline{u}}^{n}$ on the displayed cokernels, while exactness makes $\partial_{T}^{n}$ factor through the map $q_{T}$ from the target cokernel. Define $u_{A}^{n + 1, (e)} : = q_{T} {\overline{u}}^{n} q_{S}^{- 1} .$ This is the unique map satisfying $u_{A}^{n + 1, (e)} q_{S} = q_{T} {\overline{u}}^{n}$ . The same cokernel equation, together with naturality of the known degree- $n$ maps from [L3], gives naturality for morphisms covered by chosen effacement ladders.

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)
The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)