# Exact functor

Last updated

In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.

## Definitions

Let P and Q be abelian categories, and let F: PQ be a covariant additive functor (so that, in particular, F(0)=0). We say that F is an exact functor if, whenever

${\displaystyle 0\to A{\stackrel {f}{\to }}B{\stackrel {g}{\to }}C\to 0}$

is a short exact sequence in P, then

${\displaystyle 0\to F(A){\stackrel {F(f)}{\to }}F(B){\stackrel {F(g)}{\to }}F(C)\to 0}$

is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0ABC0 is exact, then 0F(A)F(B)F(C)0 is also exact".)

Further, we say that F is

• left-exact if, whenever 0ABC0 is exact, then 0F(A)F(B)F(C) is exact;
• right-exact if, whenever 0ABC0 is exact, then F(A)F(B)F(C)0 is exact;
• half-exact if, whenever 0ABC0 is exact, then F(A)F(B)F(C) is exact. This is distinct from the notion of a topological half-exact functor.

If G is a contravariant additive functor from P to Q, we similarly define G to be

• exact if, whenever 0ABC0 is exact, then 0G(C)G(B)G(A)0 is exact;
• left-exact if, whenever 0ABC0 is exact, then 0G(C)G(B)G(A) is exact;
• right-exact if, whenever 0ABC0 is exact, then G(C)G(B)G(A)0 is exact;
• half-exact if, whenever 0ABC0 is exact, then G(C)G(B)G(A) is exact.

It is not always necessary to start with an entire short exact sequence 0ABC0 to have some exactness preserved. The following definitions are equivalent to the ones given above:

• F is exact if and only if ABC exact implies F(A)F(B)F(C) exact;
• F is left-exact if and only if 0ABC exact implies 0F(A)F(B)F(C) exact (i.e. if "F turns kernels into kernels");
• F is right-exact if and only if ABC0 exact implies F(A)F(B)F(C)0 exact (i.e. if "F turns cokernels into cokernels");
• G is left-exact if and only if ABC0 exact implies 0G(C)G(B)G(A) exact (i.e. if "G turns cokernels into kernels");
• G is right-exact if and only if 0ABC exact implies G(C)G(B)G(A)0 exact (i.e. if "G turns kernels into cokernels").

## Examples

Every equivalence or duality of abelian categories is exact.

The most basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. [1] The functor FA is exact if and only if A is projective. [2] The functor GA(X) = HomA(X,A) is a contravariant left-exact functor; [3] it is exact if and only if A is injective. [4]

If k is a field and V is a vector space over k, we write V* = Homk(V,k) (this is commonly known as the dual space). This yields a contravariant exact functor from the category of k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.)

If X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The covariant functor that associates to each sheaf F the group of global sections F(X) is left-exact.

If R is a ring and T is a right R-module, we can define a functor HT from the abelian category of all left R-modules to Ab by using the tensor product over R: HT(X) = TX. This is a covariant right exact functor; it is exact if and only if T is flat. In other words, given an exact sequence ABC0 of left R modules, the sequence of abelian groups T ⊗ AT ⊗ BT ⊗ C0 is exact.

For example, ${\displaystyle \mathbb {Q} }$ is a flat ${\displaystyle \mathbb {Z} }$-module. Therefore, tensoring with ${\displaystyle \mathbb {Q} }$ as a ${\displaystyle \mathbb {Z} }$-module is an exact functor. Proof: It suffices to show that if i is an injective map of ${\displaystyle \mathbb {Z} }$-modules ${\displaystyle i:M\to N}$, then the corresponding map between the tensor products ${\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }$ is injective. One can show that ${\displaystyle m\otimes q=0}$ if and only if ${\displaystyle m}$ is a torsion element or ${\displaystyle q=0}$. The given tensor products only have pure tensors. Therefore, it suffices to show that if a pure tensor ${\displaystyle m\otimes q}$ is in the kernel, then it is zero. Suppose that ${\displaystyle m\otimes q}$ is a nonzero element of the kernel. Then, ${\displaystyle i(m)}$ is torsion. Since ${\displaystyle i}$ is injective, ${\displaystyle m}$ is torsion. Therefore, ${\displaystyle m\otimes q=0}$, which is a contradiction. Therefore, ${\displaystyle M\otimes \mathbb {Q} \to N\otimes \mathbb {Q} }$ is also injective.

In general, if T is not flat, then tensor product is not left exact. For example, consider the short exact sequence of ${\displaystyle \mathbf {Z} }$-modules ${\displaystyle 5\mathbf {Z} \;\;{\hookrightarrow }\;\;\mathbf {Z} \twoheadrightarrow \mathbf {Z} /5\mathbf {Z} }$. Tensoring over ${\displaystyle \mathbf {Z} }$ with ${\displaystyle \mathbf {Z} /5\mathbf {Z} }$ gives a sequence that is no longer exact, since ${\displaystyle \mathbf {Z} /5\mathbf {Z} }$ is not torsion-free and thus not flat.

If A is an abelian category and C is an arbitrary small category, we can consider the functor category AC consisting of all functors from C to A; it is abelian. If X is a given object of C, then we get a functor EX from AC to A by evaluating functors at X. This functor EX is exact.

While tensoring may not be left exact, it can be shown that tensoring is a right exact functor:

Theorem: Let A,B,C and P be R modules for a commutative ring R having multiplicative identity. Let ${\displaystyle A{\stackrel {f}{\to }}B{\stackrel {g}{\to }}C\to 0}$

be a short exact sequence of R modules, then

${\displaystyle A\otimes _{R}P{\stackrel {f\otimes P}{\to }}B\otimes _{R}P{\stackrel {g\otimes P}{\to }}C\otimes _{R}P\to 0}$ is also a short exact sequence of R modules. (Since R is commutative, this sequence is a sequence of R modules and not merely of abelian groups). Here, we define :${\displaystyle f\otimes P(a\otimes p):=f(a)\otimes p,g\otimes P(b\otimes p):=g(b)\otimes p}$.

This has a useful corollary: If I is an ideal of R and P is as above, then ${\displaystyle P\otimes _{R}(R/I)\cong P/IP}$

Proof: :${\displaystyle I{\stackrel {f}{\to }}R{\stackrel {g}{\to }}R/I\to 0}$ , where f is the inclusion and g is the projection, is a exact sequence of R modules. By the above we get that :${\displaystyle I\otimes _{R}P{\stackrel {f\otimes P}{\to }}R\otimes _{R}P{\stackrel {g\otimes P}{\to }}R/I\otimes _{R}P\to 0}$ is also a short exact sequence of R modules. By exactness, ${\displaystyle R/I\otimes _{R}P\cong (R\otimes _{R}P)/Image(f\otimes P)=(R\otimes _{R}P)/(I\otimes _{R}P)}$, since f is the inclusion. Now, consider the R module homomorphism from ${\displaystyle R\otimes _{R}P\rightarrow P}$ given by R linearly extending the map defined on pure tensors: ${\displaystyle r\otimes p\mapsto rp.rp=0}$ implies that ${\displaystyle 0=rp\otimes 1=r\otimes p}$. So, the kernel of this map cannot contain any non zero pure tensors. ${\displaystyle R\otimes _{R}P}$ is composed only of pure tensors: For ${\displaystyle x_{i}\in R,\sum _{i}x_{i}(r_{i}\otimes p_{i})=\sum _{i}1\otimes (r_{i}x_{i}p_{i})=1\otimes (\sum _{i}r_{i}x_{i}p_{i})}$. So, this map is injective. It is clearly onto. So, ${\displaystyle R\otimes _{R}P\cong P}$. Similarly, ${\displaystyle I\otimes _{R}P\cong IP}$. This proves the corollary.

As another application, we show that for, ${\displaystyle P=\mathbf {Z} [1/2]:=\{a/2^{k}:a,k\in \mathbf {Z} \},P\otimes \mathbf {Z} /m\mathbf {Z} \cong P/k\mathbf {Z} P}$ where ${\displaystyle k=m/2^{n}}$ and n is the highest power of 2 dividing m. We prove a special case: m=12.

Proof: Consider a pure tensor ${\displaystyle (12z)\otimes (a/2^{k})\in (12\mathbf {Z} \otimes _{Z}P).(12z)\otimes (a/2^{k})=(3z)\otimes (a/2^{k-2})}$. Also, for ${\displaystyle (3z)\otimes (a/2^{k})\in (3\mathbf {Z} \otimes _{Z}P),(3z)\otimes (a/2^{k})=(12z)\otimes (a/2^{k+2})}$. This shows that ${\displaystyle (12\mathbf {Z} \otimes _{Z}P)=(3\mathbf {Z} \otimes _{Z}P)}$. Letting ${\displaystyle P=\mathbf {Z} [1/2],A=12\mathbf {Z} ,B=\mathbf {Z} ,C=\mathbf {Z} /12\mathbf {Z} }$, A,B,C,P are R=Z modules by the usual multiplication action and satisfy the conditions of the main theorem. By the exactness implied by the theorem and by the above note we obtain that ${\displaystyle$ :\mathbf {Z} /12\mathbf {Z} \otimes _{Z}P\cong (\mathbf {Z} \otimes _{Z}P)/(12\mathbf {Z} \otimes _{Z}P)=(\mathbf {Z} \otimes _{Z}P)/(3\mathbf {Z} \otimes _{Z}P)\cong \mathbf {Z} P/3\mathbf {Z} P}. The last congruence follows by a similar argument to one in the proof of the corollary showing that ${\displaystyle I\otimes _{R}P\cong IP}$.

## Properties and theorems

A functor is exact if and only if it is both left exact and right exact.

A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into limits; a covariant functor is right exact if and only if it turns finite colimits into colimits; a contravariant functor is left exact if and only if it turns finite colimits into limits; a contravariant functor is right exact if and only if it turns finite limits into colimits.

The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.

Left and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and G is left exact.

## Generalizations

In SGA4, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows:

Let C be a category with finite projective (resp. inductive) limits. Then a functor from C to another category C is left (resp. right) exact if it commutes with finite projective (resp. inductive) limits.

Despite its abstraction, this general definition has useful consequences. For example, in section 1.8, Grothendieck proves that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category C.

The exact functors between Quillen's exact categories generalize the exact functors between abelian categories discussed here.

The regular functors between regular categories are sometimes called exact functors and generalize the exact functors discussed here.

## Notes

1. Jacobson (2009), p. 98, Theorem 3.1.
2. Jacobson (2009), p. 149, Prop. 3.9.
3. Jacobson (2009), p. 99, Theorem 3.1.
4. Jacobson (2009), p. 156.

## Related Research Articles

In mathematics, the tensor productVW of two vector spaces V and W is a vector space, endowed with a bilinear map from the Cartesian product V × W to VW. This bilinear map is universal in the sense that, for every vector space X, the bilinear maps from V × W to X are in one to one correspondence with the linear maps from VW to X.

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication on the integers. Through this generalization, theorems from arithmetic are extended to number rings and to non-numerical objects such as polynomials, series, matrices and functions.

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-moduleM to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients.

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In homological algebra and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another.

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.

In mathematics, the Grothendieck group construction constructs an abelian group from a commutative monoid M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of F and G.

In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.

In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module together with a perverse sheaf such that the functor from the Riemann–Hilbert correspondence sends to . This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves.

In mathematics the cotangent complex is roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases.

## References

• Jacobson, Nathan (2009). Basic algebra. 2 (2nd ed.). Dover. ISBN   978-0-486-47187-7.