Derived functors $RF$ and $LF$ via derived categories

Intuition Beginner

Some useful functors do not preserve exact sequences. Global sections of sheaves are the standard example: local data may exist, but fail to glue globally.

Classical derived functors measure this failure degree by degree. The derived-category version packages all degrees at once.

Instead of producing separate groups one at a time, the total derived functor sends a complex to another complex in the derived category. Taking cohomology of that output recovers the classical derived functor groups.

The practical method is familiar: replace the input by a good resolution, apply the original functor, and then read the resulting complex in the derived category.

Visual Beginner

The resolution step makes the non-exact functor safe to apply.

Worked example Beginner

For a sheaf $\mathcal{F}$ on a space $X$, choose an injective resolution $$ \mathcal F\to I^\bullet. $$ Apply the global-sections functor $\Gamma (X,-)$ to get a complex of abelian groups.

The cohomology of that complex is sheaf cohomology. In derived-category language, the whole complex is $R\Gamma (X,\mathcal{F})$, and its cohomology groups are $H^i(X,\mathcal{F})$.

What this tells us: total derived functors keep the entire cohomological package together.

Check your understanding Beginner

Formal definition Intermediate+

Let $$ F:\mathcal A\to\mathcal B $$ be a left-exact additive functor between abelian categories, with enough injectives or enough $F$-acyclic objects. The total right derived functor is a triangulated functor $$ RF^+(\mathcal A)\to D^+(\mathcal B) $$ constructed by replacing a bounded-below complex $C^{\bullet }$ by a quasi-isomorphic complex $I^{\bullet }$ of $F$-acyclic or injective objects and setting $$ RF(C^\bullet)=F(I^\bullet) $$ in the derived category.

The classical right derived functors are recovered by $$ R^iF(A)=H^i(RF(A)) $$ for $A$ viewed as a complex concentrated in degree $0$.

Dually, if $$ G:\mathcal A\to\mathcal B $$ is right exact and enough projective or $G$-acyclic replacements exist, the total left derived functor is $$ LG^-(\mathcal A)\to D^-(\mathcal B), $$ computed by projective or flat-type resolutions. Classical left derived functors are recovered as the homology/cohomology objects of $LG$ with the chosen grading convention.

Examples:

• $R\Gamma (X,-)$ packages sheaf cohomology.
• $R{f}_{\ast }$ packages higher direct images.
• $R\mathrm{Hom}$ packages Ext.
• $-{\otimes }_R^L-$ packages Tor.

Counterexamples to common slips

• $RF$ is not obtained by applying $F$ to an arbitrary complex term by term.
• An injective resolution is a computation model, not extra data in the final derived object.
• $R^{i}F$ is a cohomology object of $RF$, not a separate unrelated construction.

Key theorem with proof Intermediate+

Theorem (recovery of classical derived functors). For a left-exact additive functor $F$ with enough injectives, $$ H^i(RF(A))\cong R^iF(A) $$ for every object $A\in \mathcal{A}$.

Proof. View $A$ as a complex concentrated in degree $0$. Choose an injective resolution $$ 0\to A\to I^0\to I^1\to I^2\to\cdots. $$ In $D^{+}(\mathcal{A})$, the resolution $I^{\bullet }$ is isomorphic to $A$ because the map $A\to I^{\bullet }$ is a quasi-isomorphism.

By definition of the total right derived functor, $$ RF(A)=F(I^\bullet) $$ in $D^{+}(\mathcal{B})$.

The classical derived functor $R^{i}F(A)$ is defined as the $i$-th cohomology of the complex $F(I^{\bullet })$. Therefore $$ H^i(RF(A))=H^i(F(I^\bullet))=R^iF(A). $$ Independence of the injective resolution follows from the comparison theorem: two injective resolutions are homotopy equivalent, hence give isomorphic objects in the derived category. $\square$

Bridge. This unit lifts the classical construction in 04.03.06 into the derived category. It prepares Grothendieck spectral sequences 04.03.13, derived tensor products 04.03.17, and the six-functor formalism 04.03.16.

Exercises Intermediate+

Advanced results Master

The total derived functor is the derived-category object that represents the best possible extension of a non-exact functor. For a left-exact $F$, the ordinary functor does not usually send quasi-isomorphic complexes to quasi-isomorphic complexes. Replacing inputs by $F$-acyclic objects repairs this.

The universal property can be phrased as follows: $RF$ is a functor on the derived category equipped with a comparison from $F$ on ordinary objects, and any other functor with the same quasi-isomorphism-invariance property factors through it in the appropriate derived sense.

Injective, projective, flat, and acyclic resolutions are not competing definitions. They are computation models for the same derived-category object when the comparison theorems apply. This is why sheaf cohomology can be computed by injective, flabby, soft, Godement, or Cech-type resolutions under the right hypotheses.

Total derived functors are triangulated functors. They send distinguished triangles to distinguished triangles, and therefore produce long exact cohomology sequences after applying $H^i$. This packages the classical long exact sequence of derived functors into one triangulated statement.

The examples form the backbone of modern algebraic geometry. $R\Gamma$ is sheaf cohomology; $R{f}_{\ast }$ is higher direct image; $R\mathcal{H}om$ is sheaf Ext; derived tensor product controls Tor and intersections; the six-functor formalism is built from derived versions of pullback, pushforward, tensor, Hom, and exceptional functors.

Synthesis. Total derived functors are the derived-category upgrade of classical derived functors. They replace object-by-object correction terms with functors between derived categories, making long exact sequences, spectral sequences, Ext, Tor, sheaf cohomology, and duality part of one formal system.

Full proof set Master

Proposition 1 (acyclic replacements give the same total derived functor). If two $F$-acyclic resolutions of $A$ are connected by a quasi-isomorphism over $A$, then applying $F$ gives quasi-isomorphic complexes.

Proof. The comparison theorem gives a chain map between the two resolutions lifting the identity of $A$, unique up to chain homotopy. Since the resolutions are $F$-acyclic, the cone of the comparison map is $F$-acyclic and acyclic. Applying $F$ preserves the relevant acyclicity, so the cone after applying $F$ is acyclic. Hence the induced map between the $F$-images is a quasi-isomorphism. $\square$

Proposition 2 (right derived functors are triangulated on the derived category). Under the hypotheses guaranteeing existence of $RF$, the functor $RF:D^{+}(\mathcal{A})\to D^{+}(\mathcal{B})$ sends distinguished triangles to distinguished triangles.

Proof. Represent a distinguished triangle in $D^{+}(\mathcal{A})$ by a cone triangle after choosing suitable acyclic replacements. Applying $F$ to those replacements sends the corresponding cone construction to a cone construction in $D^{+}(\mathcal{B})$ up to quasi-isomorphism. Since distinguished triangles in the derived category are images of cone triangles and localization preserves them, $RF$ is triangulated. $\square$

Proposition 3 (classical long exact sequence from a triangle). A short exact sequence in $\mathcal{A}$ gives the classical long exact sequence of right derived functors.

Proof. A short exact sequence $$ 0\to A\to B\to C\to 0 $$ defines a distinguished triangle $$ A\to B\to C\to A[1] $$ in $D^{+}(\mathcal{A})$. Applying $RF$ gives a distinguished triangle in $D^{+}(\mathcal{B})$. Applying the cohomology functors $H^i$ to that triangle gives a long exact sequence. By the recovery theorem, its terms are $R^{i}F(A)$, $R^{i}F(B)$, and $R^{i}F(C)$. $\square$

Connections Master

• Derived functors and Ext 04.03.06. This unit lifts the classical derived-functor construction into derived-category language.

• Derived category 04.03.11. Total derived functors are functors between derived categories.

• Triangulated categories 04.03.10. Total derived functors are triangulated and preserve distinguished triangles.

• Grothendieck spectral sequence 04.03.13. Composition of derived functors produces the spectral sequence.

• Derived tensor product and Tor 04.03.17. Derived tensor product is a central left-derived bifunctor.

• Direct and inverse image 04.01.04. $R{f}_{\ast }$ is the derived version of direct image.

Historical & philosophical context Master

Cartan and Eilenberg developed derived functors through projective and injective resolutions ^{[Cartan-Eilenberg]}. Grothendieck's Tohoku paper generalized the setting to abelian categories with enough injectives and made sheaf cohomology a derived-functor construction ^{[Grothendieck]}.

Verdier's derived categories changed the viewpoint. Instead of treating $R^{i}F$ as a list of separate functors, one treats $RF$ as one functor between derived categories ^[Verdier]. Gelfand-Manin and Weibel present this as the modern natural home of Ext, Tor, and sheaf cohomology ^{[Gelfand-Manin] [Weibel]}.

The philosophical shift is from correction terms to functorial geometry. Derived categories do not merely record how exactness fails; they provide a category where non-exact functors can be extended in a structurally coherent way.

Bibliography Master

@book{GelfandManinTotalDerivedFunctors,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  publisher = {Springer},
  year = {1996}
}

@book{WeibelTotalDerivedFunctors,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{CartanEilenbergDerivedFunctors,
  author = {Cartan, Henri and Eilenberg, Samuel},
  title = {Homological Algebra},
  publisher = {Princeton University Press},
  year = {1956}
}

@article{Grothendieck1957DerivedFunctors,
  author = {Grothendieck, Alexander},
  title = {Sur quelques points d'algebre homologique},
  journal = {Tohoku Mathematical Journal},
  volume = {9},
  pages = {119--221},
  year = {1957}
}

@book{VerdierTotalDerivedFunctors,
  author = {Verdier, Jean-Louis},
  title = {Des categories derivees des categories abeliennes},
  series = {Asterisque},
  volume = {239},
  publisher = {Societe Mathematique de France},
  year = {1996}
}