Derived Functors

edit_note

This post is being revised. The text below is the version as of 2026-07-30, so some statements may be out of date and references to this post from other posts may not line up.

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

\(\delta\)-functor

We previously proved that, given any short exact sequence in \(\Ch(\mathcal{A})\)

\[0\longrightarrow A_\bullet\longrightarrow B_\bullet\longrightarrow C_\bullet\longrightarrow 0\]

we can construct the long exact sequence

\[\cdots\rightarrow H_n(A)\rightarrow H_n(B)\rightarrow H_n(C)\rightarrow H_{n-1}(A)\rightarrow \cdots\]

The most essential part of this proof was the definition of the connecting map \(\delta\), and we generalize this process as follows.

Definition 1 Let two abelian categories \(\mathcal{A},\mathcal{B}\) be given. Then a homological \(\delta\)-functor from \(\mathcal{A}\) to \(\mathcal{B}\) consists of a collection of additive functors \(T_n:\mathcal{A}\rightarrow\mathcal{B}\) (\(n\geq 0\)), together with morphisms \(\delta_n:T_n(C)\rightarrow T_{n-1}(A)\) defined for every short exact sequence

\[0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0\]

For \(n<0\), we regard all \(T_n\) as zero. These satisfy the following conditions.

1. (Long exact sequence) The sequence

   \[\cdots\longrightarrow T_{n+1}(C)\overset{\delta}{\longrightarrow}T_n(A)\longrightarrow T_n(B)\longrightarrow T_n(C)\overset{\delta}{\longrightarrow}T_{n-1}(A)\longrightarrow \cdots\]

   is exact.

2. (Naturality) Given a homomorphism between short exact sequences

   the following diagram

   commutes.

Rewriting the above definition with \(T^n\) and \(\delta^n:T^n(C)\rightarrow T^{n+1}(A)\), we obtain the definition of a cohomological \(\delta\)-functor. Since we have agreed to regard both \(T_n\) and \(T^n\) as zero when \(n<0\), the first condition for a homological \(\delta\)-functor means in particular that

\[\cdots\longrightarrow T_0(A)\longrightarrow T_0(B)\longrightarrow T_0(C)\longrightarrow0\longrightarrow 0\longrightarrow\cdots,\]

that is, \(T_0\) is a right exact functor. Similarly, the first condition for a cohomological \(\delta\)-functor makes \(T^0\) a left exact functor.

Also, the second condition of a \(\delta\)-functor, naturality, means that when we regard \(T_i(C)\) and \(T_{i-1}(A)\) as functors from the category of short exact sequences \(\mathbf{S}(\mathcal{A})\) to \(\mathcal{A}\), each \(\delta_i\) is a natural transformation between them.

As always, the cohomological case can be easily derived from the homological one, so we shall consider only homological \(\delta\)-functors from now on.

Definition 2 Let two \(\delta\)-functors \(S,T\) be given. Then a morphism \(S\rightarrow T\) is a collection of natural transformations from \(S_n\) to \(T_n\) that commute with \(\delta\).

In other words, it is a collection of natural transformations \(\alpha_n:S_n\Rightarrow T_n\) making the following diagram commute for every short exact sequence

\[0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0\]

Definition 3 A \(\delta\)-functor \(T\) is called a universal \(\delta\)-functor if, whenever a \(\delta\)-functor \(S\) and a natural transformation \(\alpha_0:S_0\rightarrow T_0\) are given, there exists a unique morphism \((\alpha_n:S_n\Rightarrow T_n)\) of \(\delta\)-functors extending it.

Derived Functors

Consider a right exact functor \(F: \mathcal{A}\rightarrow \mathcal{B}\) between two abelian categories \(\mathcal{A}\) and \(\mathcal{B}\). Then \(F\) does not preserve left exactness. For example, if \(F\) is covariant, then even when the short exact sequence

\[0 \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow 0\]

is given, only the exactness of the sequence

\[F(A_1) \rightarrow F(A_2) \rightarrow F(A_3)\rightarrow 0\]

is preserved. Likewise, a left exact functor does not preserve right exactness.

The philosophy of derived functors is to recover the information lost on one side by introducing infinitely many additional terms. Thus, for a right exact functor \(F\), our goal is to find a homological \(\delta\)-functor with \(T_0=F\), and similarly for a left exact functor, our goal is to find a cohomological \(\delta\)-functor.

Definition 4 Let a right exact functor \(F:\mathcal{A}\rightarrow \mathcal{B}\) be given, and suppose \(\mathcal{A}\) has enough projectives. Then the left derived functors \(L_iF\) of \(F\) are defined by the formula

\[(L_iF)(A)=H_i(F(P_\bullet)),\qquad\text{$P_\bullet$ a projective resolution of $A$}\]

For this definition to make sense, \(L_iF(A)\) must not depend on the choice of \(P_\bullet\) above.

Lemma 5 \(L_iF(A)\) does not depend on the choice of \(P_\bullet\) above.

Proof

Take two projective resolutions and apply §Resolutions, ⁋Theorem 6 to the identity map.

Now let us examine left derived functors in more detail. First, since \(F\) is right exact, we know that the sequence

\[F(P_1) \overset{Fd_1}{\longrightarrow} F(P_0) \overset{F\epsilon_0}{\longrightarrow} F(A) \longrightarrow 0\]

is exact. Therefore, we obtain

\[L_0F(A)=H_0(F(P))=\frac{F(P_0)}{\im Fd_1}=\frac{F(P_0)}{\ker F\epsilon_0}\cong F(A)\]

To show that the \(L_\bullet F\) form a homological \(\delta\)-functor, we must first show that they are additive functors, and then construct the connecting maps \(\delta\). We divide this into two steps.

Lemma 6 The \(L_iF\) are additive functors.

Proof

First, given any \(f: A' \rightarrow A\) and projective resolutions of \(A'\) and \(A\) respectively, we can apply §Resolutions, ⁋Theorem 6 to obtain \(L_nF(f)\). That this satisfies functoriality and additivity is obvious from the universal property.

Lemma 7 The \(L_iF\) form a homological \(\delta\)-functor.

Proof

First, suppose a short exact sequence

\[0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\]

is given. If projective resolutions \(P_\bullet\) of \(A\) and \(R_\bullet\) of \(C\) are given, then using §Resolutions, ⁋Lemma 7 we obtain a projective resolution \(Q_\bullet \rightarrow B\). On the other hand, since each \(R_n\) is projective, the sequence

\[0 \rightarrow P_n \rightarrow Q_n \rightarrow R_n \rightarrow 0\]

is split exact. From this,

\[0 \rightarrow F(P_\bullet) \rightarrow F(Q_\bullet) \rightarrow F(R_\bullet) \rightarrow 0\]

is also a short exact sequence ([Multilinear Algebra] §Hom and the Tensor Product, ⁋Proposition 1), and considering the homology sequence here, we obtain the desired connecting maps and the long exact sequence of left derived functors

\[\cdots\overset{\partial}{\longrightarrow}L_iF(A)\longrightarrow L_iF(B)\longrightarrow L_iF(C)\overset{\partial}{\longrightarrow}L_{i-1}F(A)\longrightarrow L_{i-1}F(B)\longrightarrow L_{i-1}F(C)\overset{\partial}{\longrightarrow}\cdots\]

That the information thus obtained satisfies the second condition of Definition 1 follows from §Resolutions, ⁋Theorem 6.

Moreover, they define a universal homological \(\delta\)-functor in the sense of Definition 3. We omit the proof of this.

Proposition 8 Consider an abelian category \(\mathcal{A}\) with enough projectives and any right exact functor \(F: \mathcal{A}\rightarrow \mathcal{B}\). Then the derived functors \(L_nF\) are universal \(\delta\)-functors.

Just as in the discussion above, we can also define right derived functors for a left exact functor. Its definition is the “dual” of Definition 4.

Definition 9 Let a left exact functor \(F:\mathcal{A}\rightarrow \mathcal{B}\) be given, and suppose \(\mathcal{A}\) has enough injectives. Then the right derived functors \(R^i F\) of \(F\) are defined by the formula

\[(R^iF)(A)=H_i(F(I^\bullet)),\qquad\text{$I^\bullet$ an injective resolution of $A$}\]

Then one can also show that these are universal cohomological \(\delta\)-functors. The reason we use superscripts, unlike in Definition 4, is that these are literally cohomological \(\delta\)-functors, and they arise mainly when dealing with matters related to cohomology.

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References

[Wei] C.A. Weibel. An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1995.