Editing Exact functor

{{Short description|Functor that preserves short exact sequences}}
{{otheruses4|exact functors in homological algebra|exact functors between regular categories|regular category}}
In [[mathematics]], particularly [[homological algebra]], an '''exact functor''' is a [[functor]] that preserves [[short exact sequence]]s. 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 {{nowrap|1=''F'': '''P'''→'''Q'''}} be a [[covariant functor|covariant]] [[additive functor]] (so that, in particular, ''F''(0) = 0).  We say that ''F'' is an '''exact functor''' if whenever 

:<math>0 \to A\ \stackrel{f}{\to} \ B\ \stackrel{g}{\to} \ C \to 0</math> 

is a [[short exact sequence]] in '''P''' then 

:<math>0 \to F(A) \ \stackrel{F(f)}{\longrightarrow} \ F(B)\ \stackrel{F(g)}{\longrightarrow} \ F(C) \to 0</math> 

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 functor|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").

== Examples ==

Every [[equivalence of categories|equivalence or duality]] of abelian categories is exact.

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

If ''k'' is a [[field (mathematics)|field]] and ''V'' is a [[vector space]] over ''k'', we write ''V''&thinsp;* = Hom<sub>''k''</sub>(''V'',''k'') (this is commonly known as the [[dual space]]). This yields a contravariant exact functor from the [[category of vector spaces|category of ''k''-vector spaces]] to itself. (Exactness follows from the above: ''k'' is an [[injective module|injective]] ''k''-[[module (mathematics)|module]]. Alternatively, one can argue that every short exact sequence of ''k''-vector spaces [[split exact sequence|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 [[sheaf (mathematics)|sheaves]] of [[abelian group]]s 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 (mathematics)|ring]] and ''T'' is a right ''R''-[[module (mathematics)|module]], we can define a functor ''H''<sub>''T''</sub> from the abelian [[Category of modules|category of all left ''R''-modules]] to '''Ab''' by using the [[tensor product]] over ''R'': ''H''<sub>''T''</sub>(''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''<sub>''T''</sub> is exact if and only if ''T'' is [[flat module|flat]]. For example, <math>\mathbb{Q}</math> is a flat <math>\mathbb{Z}</math>-module. Therefore, tensoring with <math>\mathbb{Q}</math> as a <math>\mathbb{Z}</math>-module is an exact functor. '''Proof:''' It suffices to show that if ''i'' is an [[injective map]] of <math>\mathbb{Z}</math>-modules <math>i:M\to N</math>, then the corresponding map between the tensor products <math>M \otimes \mathbb{Q} \to N\otimes \mathbb{Q}</math> is injective. One can show that <math>m \otimes q = 0</math> if and only if <math>m</math> is a torsion element or <math>q = 0</math>. The given tensor products only have pure tensors. Therefore, it suffices to show that if a pure tensor <math>m \otimes q </math> is in the [[kernel (algebra)|kernel]], then it is zero. Suppose that <math>m \otimes q </math> is an element of the kernel. Then, <math>i(m)</math> is torsion. Since <math>i</math> is injective, <math>m</math> is torsion. Therefore, <math>m \otimes q  = 0</math>. Therefore, <math> M \otimes \mathbb{Q} \to N\otimes \mathbb{Q} </math> 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 <math>\mathbf{Z}</math>-modules <math>5\mathbf{Z} \hookrightarrow \mathbf{Z} \twoheadrightarrow \mathbf{Z}/5\mathbf{Z}</math>. Tensoring over <math>\mathbf{Z}</math> with <math>\mathbf{Z}/5\mathbf{Z}</math> gives a sequence that is no longer exact, since <math>\mathbf{Z}/5\mathbf{Z}</math> is not torsion-free and thus not flat. 

If '''A''' is an abelian category and '''C''' is an arbitrary [[small category|small]] [[category (mathematics)|category]], we can consider the [[functor category]] '''A<sup>C</sup>''' 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''<sub>''X''</sub> from '''A'''<sup>'''C'''</sup> to '''A''' by evaluating functors at ''X''. This functor ''E''<sub>''X''</sub> 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 <math>A \ \stackrel{f}{\to} \ B\ \stackrel{g}{\to} \ C \to 0</math> be a [[short exact sequence]] of  ''R''-modules. Then 
:<math>A\otimes_{R} P \stackrel{f \otimes P}\to B\otimes_{R} P \stackrel{g \otimes P}\to C \otimes_{R} P \to 0</math>
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
:<math>f \otimes P(a \otimes p):=f(a) \otimes p, g \otimes P(b \otimes p):=g(b) \otimes p</math>. 

This has a useful [[corollary]]: If ''I'' is an [[ideal (ring theory)|ideal]] of ''R'' and ''P'' is as above, then <math>P \otimes_{R} (R/I) \cong P/IP</math>.

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

As another application, we show that for, <math>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 </math> where <math> k=m/2^n </math> and ''n'' is the highest [[power of 2]] dividing ''m''. We prove a special case: ''m''=12.

Proof: Consider a pure tensor <math>(12z)\otimes (a/2^k ) \in  
 (12\mathbf{Z} \otimes_{Z} P).(12z)\otimes (a/2^k ) = (3z)\otimes (a/2^{k-2}) </math>. Also, for <math>(3z)\otimes (a/2^k ) \in  
 (3\mathbf{Z} \otimes_{Z} P), (3z)\otimes (a/2^k ) = (12z)\otimes (a/2^{k+2}) </math>.
This shows that <math>(12\mathbf{Z} \otimes_{Z} P) = (3\mathbf{Z} \otimes_{Z} P)</math>. Letting <math>P= \mathbf{Z}[1/2], A = 12\mathbf{Z}, B= \mathbf{Z}, C = \mathbf{Z}/12\mathbf{Z} </math>,  ''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
<math>: \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 </math>. The last congruence follows by a similar argument to one in the proof of the corollary showing that <math> I \otimes_R P \cong IP </math>.

== 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 [[limit (category theory)|limit]]s into limits; a covariant functor is right exact if and only if it turns finite [[colimit]]s 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|right derived functors]]; the degree to which a right exact functor fails to be exact can be measured with its [[derived functor|left derived functor]]s.

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

== Generalizations ==

In [[Grothendieck's Séminaire de géométrie algébrique|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. injective) limits. Then a functor from ''C'' to another category ''C&prime;'' 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|exact categories]] generalize the exact functors between abelian categories discussed here.

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

== Notes ==
<references/>

== References ==
{{refbegin}}
* {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | publisher=Dover| isbn = 978-0-486-47187-7}}
{{refend}}

{{Functors}}

{{DEFAULTSORT:Exact Functor}}
[[Category:Homological algebra]]
[[Category:Additive categories]]
[[Category:Functors]]