Exact functor

 ( Homological Algebra ) 

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.

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

$0\; \backslash to\; A\; \backslash \; \backslash stackrel\{f\}\{\backslash longrightarrow\}\; \backslash \; B\; \backslash \; \backslash stackrel\{g\}\{\backslash longrightarrow\}\; \backslash \; C\; \backslash to\; 0$

is a short exact sequence in P then

$0\; \backslash to\; F(A)\; \backslash \; \backslash stackrel\{F(f)\}\{\backslash longrightarrow\}\; \backslash \; F(B)\backslash \; \backslash stackrel\{F(g)\}\{\backslash longrightarrow\}\; \backslash \; F(C)\; \backslash to\; 0$

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

Further, we say that F is

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

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

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

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Examples Every equivalence or duality of abelian categories is exact.

The most basic examples of left exact functors are the : if A is an abelian category and A is an object of A, then F _A( X) = Hom _A( A, X) defines a covariant left-exact functor from A to the category Ab of abelian groups.Jacobson (2009), p. 98, Theorem 3.1. The functor F _A is exact if and only if A is projective.Jacobson (2009), p. 149, Prop. 3.9. The functor G _A( X) = Hom _A( X, A) is a contravariant left-exact functor;Jacobson (2009), p. 99, Theorem 3.1. it is exact if and only if A is injective module.Jacobson (2009), p. 156.

If k is a field and V is a vector space over k, we write V * = Hom _k( 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 module 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 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 H _T from the abelian category of all left R-modules to Ab by using the tensor product over R: H _T( X) = T ⊗ X. This is a covariant right exact functor; in other words, given an exact sequence A→ B→ C→0 of left R modules, the sequence of abelian groups T ⊗ A → T ⊗ B → T ⊗ C → 0 is exact.

The functor H _T is exact if and only if T is flat module. For example, $\backslash mathbb\{Q\}$ is a flat $\backslash mathbb\{Z\}$-module. Therefore, tensoring with $\backslash mathbb\{Q\}$ as a $\backslash mathbb\{Z\}$-module is an exact functor. Proof: It suffices to show that if i is an injective map of $\backslash mathbb\{Z\}$-modules $i:M\backslash to\; N$, then the corresponding map between the tensor products $M\; \backslash otimes\; \backslash mathbb\{Q\}\; \backslash to\; N\backslash otimes\; \backslash mathbb\{Q\}$ is injective. One can show that $m\; \backslash otimes\; q\; =\; 0$ if and only if $m$ is a torsion element or $q\; =\; 0$. The given tensor products only have pure tensors. Therefore, it suffices to show that if a pure tensor $m\; \backslash otimes\; q$ is in the kernel, then it is zero. Suppose that $m\; \backslash otimes\; q$ is an element of the kernel. Then, $i(m)$ is torsion. Since $i$ is injective, $m$ is torsion. Therefore, $m\; \backslash otimes\; q\; =\; 0$. Therefore, $M\; \backslash otimes\; \backslash mathbb\{Q\}\; \backslash to\; N\backslash otimes\; \backslash mathbb\{Q\}$ 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 $\backslash mathbf\{Z\}$-modules $5\backslash mathbf\{Z\}\; \backslash hookrightarrow\; \backslash mathbf\{Z\}\; \backslash twoheadrightarrow\; \backslash mathbf\{Z\}/5\backslash mathbf\{Z\}$. Tensoring over $\backslash mathbf\{Z\}$ with $\backslash mathbf\{Z\}/5\backslash mathbf\{Z\}$ gives a sequence that is no longer exact, since $\backslash mathbf\{Z\}/5\backslash mathbf\{Z\}$ is not torsion-free and thus not flat.

If A is an abelian category and C is an arbitrary small category category, we can consider the functor category A^C consisting of all functors from C to A; it is abelian. If X is a given object of C, then we get a functor E _X from A ^C to A by evaluating functors at X. This functor E _X 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 $A\; \backslash \; \backslash stackrel\{f\}\{\backslash to\}\; \backslash \; B\backslash \; \backslash stackrel\{g\}\{\backslash to\}\; \backslash \; C\; \backslash to\; 0$ be a short exact sequence of R-modules. Then

$A\backslash otimes\_\{R\}\; P\; \backslash stackrel\{f\; \backslash otimes\; P\}\backslash to\; B\backslash otimes\_\{R\}\; P\; \backslash stackrel\{g\; \backslash otimes\; P\}\backslash to\; C\; \backslash otimes\_\{R\}\; P\; \backslash to\; 0$

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

$f\; \backslash otimes\; P(a\; \backslash otimes\; p):=f(a)\; \backslash otimes\; p,\; g\; \backslash otimes\; P(b\; \backslash otimes\; p):=g(b)\; \backslash otimes\; p$.

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

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

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

Proof: Consider a pure tensor $(12z)\backslash otimes\; (a/2^k\; )\; \backslash in$

(12\mathbf{Z} \otimes_{Z} P).(12z)\otimes (a/2^k ) = (3z)\otimes (a/2^{k-2}) . Also, for $(3z)\backslash otimes\; (a/2^k\; )\; \backslash in\; (3\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P),\; (3z)\backslash otimes\; (a/2^k\; )\; =\; (12z)\backslash otimes\; (a/2^\{k+2\})$.
     

This shows that $(12\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)\; =\; (3\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)$. Letting $P=\; \backslash mathbf\{Z\}1/2,\; A\; =\; 12\backslash mathbf\{Z\},\; B=\; \backslash mathbf\{Z\},\; C\; =\; \backslash mathbf\{Z\}/12\backslash mathbf\{Z\}$, 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 $:\; \backslash mathbf\{Z\}/12\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P\; \backslash cong\; (\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)\; /\; (12\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)\; =\; (\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)\; /\; (3\backslash mathbf\{Z\}\; \backslash otimes\_\{Z\}\; P)\; \backslash cong\; \backslash mathbf\{Z\}P/3\backslash mathbf\{Z\}\; P$. The last congruence follows by a similar argument to one in the proof of the corollary showing that $I\; \backslash otimes\_R\; P\; \backslash cong\; IP$.

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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 into colimits; a contravariant functor is left exact iff it turns finite colimits into limits; a contravariant functor is right exact iff it turns finite limits into colimits.

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

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

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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 category generalize the exact functors between abelian categories discussed here.

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

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Notes

• Nathan Jacobson (2025). 9780486471877, Dover. ISBN 9780486471877

Categories: Homological Algebra, Additive Categories, Functors

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