Abstract Nonsense

Crushing one theorem at a time

Left Exact, Right Exact, and Exact Functors


Point of Post: In this post we discuss the notion of left exact, right exact, and exact functors between abelian categories–giving several equivalent definitions.

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Motivation

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We finally get back to the homological algebra side of things. What we would now like to discuss is just a thinly veiled continuation of our last post on continuous and cocontinuous functors. Namely, we are going to ask when a functor preserves/partially preserves the exactness of a given short exact sequence in some abelian category. This is of absolutely fundamental importance in homological algebra because as we have previously stated the degree to which a sequence fails to be exact is the reason we study homological algebra and thus it behooves us to figure out when applying a functor to an exact sequence introduces no new obstruction. In fact, a huge part of the homological algebra to come will be measuring how badly a certain type of functor fails to be exact.

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Left Exact, Right Exact, and Exact Functors

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Let us get the definitions out of the way.

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Let \mathscr{A} and \mathscr{B} be abelian categories. We call a functor F:\mathscr{A}\to\mathscr{B} left exact if for every exact sequence

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0\to X\xrightarrow{\alpha} Y\xrightarrow{\beta} Z

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in \mathscr{A} the sequence

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0\to F(X)\xrightarrow{F(\alpha)}F(Y)\xrightarrow{F(\beta)}F(Z)

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is exact in \mathscr{B}. We say that F is right exact if every exact sequence X\xrightarrow{\alpha}Y\xrightarrow{\beta}Z\to0 in \mathscr{A} goes to an exact sequence

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F(X)\xrightarrow{F(\alpha)}F(Y)\xrightarrow{F(\beta)}F(Z)\to0

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in \mathscr{B}. We call a functor F exact if it is both left and right exact.

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For example, we have shown that \otimes_R A is a right exact functor R\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod} for commutative rings R and we have also shown that \text{Hom}_R(A,\bullet) is a left exact exact functor R\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod} with the same conditions. As a different example we have shown that the direct limit functor \varinjlim:\mathbf{DS}_\mathcal{A}(R\text{-}\mathbf{Mod})\to R\text{-}\mathbf{Mod} is exact when \mathcal{A} is directed–or said using our more recently language, if \mathcal{A} is a directed set then the colimit functor \text{colim }:(R\text{-}\mathbf{Mod})^\mathcal{A}\to R\text{-}\mathbf{Mod} is exact.

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Now, there are some nice equivalent definitions of left/right exactness if we assume our functors are additive. Namely:

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Theorem: Let F:\mathscr{A}\to\mathscr{B} be an additive functor. Then, the following are equivalent:

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\begin{aligned}&\mathbf{(1)}\quad F\text{ is left exact}\\&\mathbf{(2)}\quad F\text{ commutes with kernels}\\ &\mathbf{(3)}\quad F\text{ is finitely continuous}\\ &\mathbf{(4)}\quad F\text{ preserves left short exact sequences}\end{aligned}

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Proof: 

\mathbf{(1)}\implies\mathbf{(2)}: Let X\xrightarrow{\alpha}Y be an arbitrary arrow. We then have the exact sequence \ker \alpha\xrightarrow{k} X\xrightarrow{\alpha}Y which applying F gives the exact sequence 0\to F(\ker \alpha)\xrightarrow{F(k)} F(X)\xrightarrow{F(\alpha)}F(Y). By exactness we have a natural isomorphism \text{im }F(k)\xrightarrow{\approx}\ker F(\alpha), but since F(k) is a monomorphism it’s trivial \text{im }F(k) is naturally isomorphic to F(\ker \alpha). The rest follows from the fact that the category \bullet\overset{\displaystyle \longrightarrow}{\longrightarrow}\bullet (which is the category that equalizers, and thus kernels are limits of) is connected and applying theorem 2.2 of this paper.

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\mathbf{(2)}\implies\mathbf{(3)}: Since F is additive we know that it commutes with products, and so if it also commutes with kernels, and since all equalizers are just kernels in abelian categories we know that F commutes with all finite limits (this follows from previous comment) and thus is finitely cocontinuous.

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\mathbf{(3)}\implies\mathbf{(4)}: Let 0\to X\xrightarrow{\alpha}Y\xrightarrow{\beta}Z\to0 be a short exact sequence in \mathscr{A}. We need to verify that F(\alpha) is a monomorphism and that \text{im }F(\alpha)\cong\ker F(\beta). Now, to see that F(\alpha) is a monomorphism we merely note that by assumption \ker F(\alpha)\cong F(\ker\alpha)\cong F(0)=0. One can proceed similarly for the other point of exactness.

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\mathbf{(4)}\implies\mathbf{(1)}: This one is trivial.  \blacksquare

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Of course, there is a dualized version of this for right exact functors where we “co” everything.

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This tells us that, in particular, things like the tensor functor \otimes_R N where R is commutative is right exact, since it is left adjoint to the Hom functor and so cocontinuous and so right exact. This also allows us to finally complete a long-standing promise. Namely, a long time ago we claimed that the inverse limit functor was left exact. This now follows immediately since this is just a limit functor which we know is a right adjoint to the diagonal functor, and so continuous, and so finitely continuous and so left exact. Cool, huh?

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.

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April 24, 2012 - Posted by | Algebra, Category Theory, Homological Algebra | , , ,

3 Comments »

  1. […] postpone the proof of this, so that we may give a cool proof of it using abstract nonsense concerning adjoint […]

    Pingback by Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. IV) « Abstract Nonsense | April 24, 2012 | Reply

  2. […] that we can dualize the other properties of projective modules to get other characterizations, is exact, […]

    Pingback by Injective Modules (Pt. I) « Abstract Nonsense | April 28, 2012 | Reply

  3. […] We have so far discussed two kinds of modules that make certain half-exact (i.e. right exact or left exact) functors exact. Namely, we have discussed projective modules as the modules that make the Hom functor exact, and we have discussed injective modules as the modules that make the contravariant Hom functor exact. Both of these are very important since projective and injective modules are the “approximating” modules used to define powerful notions like derived functors. That said, they aren’t the most “functional” modules that make certain functors exact. In particular, in basic module theory there are three main functors, two of which are the covariant and contravariant Hom functor. The last, of course, is the tensor functor . The tensor functor shows up everywhere in mathematics and is, dare I say it, “more important” (whatever that really means) than the two Hom functors [note: this is just my limited opinion on the matter]. Thus, it would seem to behoove us to understand what kind of modules make the functor exact (it is already right exact). […]

    Pingback by Flat Modules (Pt. I) « Abstract Nonsense | May 4, 2012 | Reply


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