4.2 Derived functors

Derived functors are a powerful tool in sheaf theory, extending the concept of functors to handle complexes. They measure how functors fail to be exact, revealing deeper relationships between algebraic and geometric objects.

Left and right derived functors use projective and injective resolutions respectively. Their existence depends on the availability of suitable resolutions in the source category. Derived functors play a crucial role in sheaf cohomology and have wide-ranging applications in algebraic geometry and topology.

Derived functors overview

• Derived functors generalize the notion of functors between categories to handle complexes and provide a way to measure the failure of a functor to be exact
• Play a crucial role in homological algebra and sheaf theory, allowing for the study of cohomology and the computation of invariants associated with sheaves
• Connect different algebraic and geometric objects, revealing deeper relationships and structures

Left vs right derived functors

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• Left derived functors $\mathbf{L}F$ are obtained by applying a functor $F$ to a projective resolution of an object and taking homology
  • Measure the extent to which $F$ fails to be right exact
• Right derived functors $\mathbf{R}F$ are obtained by applying a functor $F$ to an injective resolution of an object and taking homology
  • Measure the extent to which $F$ fails to be left exact
• The choice between left and right derived functors depends on the properties of the functor and the category under consideration

Existence of derived functors

• The existence of derived functors relies on the availability of suitable resolutions (projective or injective) in the source category
• In abelian categories with enough projectives or enough injectives, the existence of derived functors is guaranteed
• The derived functors are independent of the choice of resolution up to quasi-isomorphism, ensuring their well-definedness

Universal property of derived functors

• Derived functors satisfy a universal property that characterizes them uniquely up to isomorphism
• For a functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories, the left derived functor $\mathbf{L}F$ is the right adjoint to the derived functor of the inclusion functor from the subcategory of projective objects in $\mathcal{A}$ to $\mathcal{A}$
• Similarly, the right derived functor $\mathbf{R}F$ is the left adjoint to the derived functor of the inclusion functor from the subcategory of injective objects in $\mathcal{A}$ to $\mathcal{A}$

Derived functors of sheaf cohomology

• Sheaf cohomology is a powerful tool in algebraic geometry and topology that measures the global behavior of a sheaf
• Derived functors play a central role in the construction and computation of sheaf cohomology groups

Global sections functor

• The global sections functor $\Gamma: \mathsf{Sh}(X) \to \mathsf{Ab}$ assigns to each sheaf $\mathcal{F}$ on a topological space $X$ the abelian group of global sections $\Gamma(X, \mathcal{F})$
• The right derived functors of $\Gamma$ are the sheaf cohomology functors $H^i(X, -)$, which measure the obstruction to the existence of global sections
• The sheaf cohomology groups $H^i(X, \mathcal{F})$ provide information about the global properties of the sheaf $\mathcal{F}$ and the space $X$

Higher direct image functors

• For a continuous map $f: X \to Y$ between topological spaces, the direct image functor $f_*: \mathsf{Sh}(X) \to \mathsf{Sh}(Y)$ pushes forward sheaves from $X$ to $Y$
• The right derived functors of $f_*$ are the higher direct image functors $R^if_*$, which measure the failure of $f_*$ to be exact
• The higher direct image sheaves $R^if_*\mathcal{F}$ on $Y$ capture information about the cohomology of the sheaf $\mathcal{F}$ along the fibers of $f$

Čech cohomology and derived functors

• Čech cohomology is an alternative approach to sheaf cohomology that is based on open covers of the space $X$
• The Čech cohomology groups $\check{H}^i(X, \mathcal{F})$ can be computed using the Čech complex associated with an open cover
• Under suitable conditions (e.g., for paracompact Hausdorff spaces and fine sheaves), the Čech cohomology groups coincide with the derived functor cohomology groups

Derived functors in abelian categories

• Derived functors can be defined and studied in the general setting of abelian categories, which include categories of modules, sheaves, and more
• The properties and behavior of derived functors in abelian categories provide a unifying framework for homological algebra

Injective vs projective resolutions

• Injective resolutions are used to compute right derived functors, while projective resolutions are used for left derived functors
• In an abelian category $\mathcal{A}$, an injective resolution of an object $A$ is a long exact sequence $0 \to A \to I^0 \to I^1 \to \cdots$ with each $I^i$ being an injective object
• Similarly, a projective resolution of $A$ is a long exact sequence $\cdots \to P_1 \to P_0 \to A \to 0$ with each $P_i$ being a projective object

Existence in abelian categories

• The existence of derived functors in an abelian category $\mathcal{A}$ depends on the availability of enough injectives or enough projectives
• If $\mathcal{A}$ has enough injectives, then every object admits an injective resolution, and right derived functors can be defined
• If $\mathcal{A}$ has enough projectives, then every object admits a projective resolution, and left derived functors can be defined

Long exact sequence of derived functors

• Derived functors preserve long exact sequences, leading to the long exact sequence of derived functors
• Given a short exact sequence $0 \to A \to B \to C \to 0$ in an abelian category and a left exact functor $F$, there is a long exact sequence of derived functors: $0 \to \mathbf{L}_0F(A) \to \mathbf{L}_0F(B) \to \mathbf{L}_0F(C) \to \mathbf{L}_1F(A) \to \mathbf{L}_1F(B) \to \cdots$
• The long exact sequence connects the derived functors of the objects in the short exact sequence and provides a powerful tool for computation and understanding their relationships

Applications of derived functors

• Derived functors have numerous applications in algebraic geometry, topology, and homological algebra
• They provide a way to study cohomology theories, compute invariants, and understand the global behavior of algebraic and geometric objects

Grothendieck spectral sequence

• The Grothendieck spectral sequence is a powerful tool that relates the derived functors of composed functors
• Given functors $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{C}$ between abelian categories, the Grothendieck spectral sequence relates the derived functors of the composition $G \circ F$ to the derived functors of $F$ and $G$
• The spectral sequence provides a systematic way to compute the derived functors of the composition and understand their structure

Derived functors in algebraic geometry

• Derived functors play a central role in algebraic geometry, particularly in the study of coherent sheaves and their cohomology
• The derived functors of the global sections functor and the pushforward functor provide important invariants of algebraic varieties and morphisms
• Derived categories and derived functors are used to formulate and solve problems in algebraic geometry, such as the construction of moduli spaces and the study of intersection theory

Derived categories and derived functors

• Derived categories are formed by localizing categories of complexes with respect to quasi-isomorphisms, allowing for a more flexible and invariant approach to homological algebra
• Derived functors can be defined and studied in the context of derived categories, providing a natural setting for their properties and applications
• Derived categories and derived functors have become essential tools in modern algebraic geometry, representation theory, and mathematical physics

Computation of derived functors

• Computing derived functors is a central problem in homological algebra and sheaf theory
• Various techniques and tools are available for the computation of derived functors, depending on the specific context and properties of the functors and categories involved

Acyclic resolutions and derived functors

• Acyclic resolutions provide a way to compute derived functors by replacing the original resolutions with simpler ones
• An object $A$ in an abelian category is called $F$-acyclic for a functor $F$ if the derived functors $\mathbf{L}_iF(A)$ or $\mathbf{R}^iF(A)$ vanish for all $i > 0$
• If a functor $F$ preserves acyclicity (i.e., maps $F$-acyclic objects to $F$-acyclic objects), then the derived functors can be computed using acyclic resolutions

Derived functors of representable functors

• Representable functors, which are functors of the form $\mathrm{Hom}(A, -)$ for an object $A$ in a category, have a special relationship with derived functors
• The derived functors of a representable functor $\mathrm{Hom}(A, -)$ are given by the ext functors $\mathrm{Ext}^i(A, -)$
• The ext functors measure the failure of the $\mathrm{Hom}$ functor to be exact and provide important cohomological information about the objects involved

Derived functors and satellites

• Satellites are a generalization of derived functors that allow for the construction of new functors from existing ones
• Given a functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories, the satellites of $F$ are functors $S_iF: \mathcal{A} \to \mathcal{B}$ that measure the deviation of $F$ from exactness
• The left and right derived functors of $F$ are particular examples of satellites, corresponding to the left and right satellites of $F$, respectively