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Hom is left exact in each variable

Statement

Let $A$ be an abelian category and $X$ an object of $A$ .

If $0 \to A \to B \to C$ is exact, then $0 \to A (X, A) \to A (X, B) \to A (X, C)$ is exact in $A b$ .
If $A \to B \to C \to 0$ is exact, then $0 \to A (C, X) \to A (B, X) \to A (A, X)$ is exact in $A b$ .

Thus Hom is left exact in each variable.

Facts & Assumptions

Given: An abelian category $A$ and an object $X$ of $A$ .

[L1]

In an abelian category, exactness at the left end means that the first displayed map is a kernel of the second (Degenerate exactness criteria).

[L2]

Abelian categories have all finite limits, and representable functors preserve existing small limits (An abelian category has all finite limits and all finite colimits, Every covariantly representable functor to Set preserves all existing small limits).

[L3]

The covariant and contravariant Hom assignments are the representable functors $A (X, -)$ and $A^{o p} (X, -)$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, The opposite of an abelian category is abelian).

[L4]

The target category of these Hom functors is $A b$ , which is abelian (Abelian groups form an abelian category).

Proof

technique · direct

1.1L1L2L3L4assume-hyp

Assume $0 \to A \to B \to C$ is exact. By [L1], the map $A \to B$ is a kernel of $B \to C$ . By [L2] and [L3], the representable functor $A (X, -)$ preserves that kernel. Therefore $0 \to A (X, A) \to A (X, B) \to A (X, C)$ is exact in $A b$ by [L1] applied inside the abelian category [L4].

2.1L2L3L4step 1.1

Passing to the opposite category, the exact sequence $A \to B \to C \to 0$ becomes a left-exact sequence in $A^{o p}$ . Applying step 1.1 there to the representable functor $A^{o p} (X, -) = A (-, X)$ gives $0 \to A (C, X) \to A (B, X) \to A (A, X)$ exact in $A b$ .

3.1step 1.1step 2.1∎

Hence Hom is left exact in each variable.

Depends on

Used by

Hom is not exact Counterexample
L₀ of a non-right-exact functor need not recover the functor Counterexample
A contravariant functor derived via the opposite category Example
The right derived functors of Hom from a fixed object Example
Local-to-global Ext collapse for a regular immersion Lemma
An object is projective exactly when Hom out of it is exact Theorem

Dependency tree · two levels

29 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

The Stacks Project, Section 12.5, Lemma 12.5.8 (standard reference, not scraped)
David Mehrle, Category Theory, Part III, Definition 7.22 (standard reference, not scraped)