THE SNAKE LEMMA IN AN ABELIAN CATEGORY WITH ENOUGH PROJECTIVES (INJECTIVES) David B. Surowski Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602 E-mail address: [email protected] Abstract. The Snake Lemma, fundamental in producing long exact sequences in homology or cohomology theories based on an abelian category, is shown to have a proof formally identical to the familiar one involving abelian groups when the category either has enough projectives or enough injectives. Easily one of the most important lemmas of homological algebra in an abelian category is the so-called Snake Lemma which is the key ingredient in showing that short exact sequences of (co)chain complexes generate natural long exact sequences in (co)homology. In turn, these long exact sequences are absolutely indispensable for explicit calculations. When the abelian category in question is the category of abelian groups (or R-modules) the construction of the connecting homomorphism is a very simple diagram chase. In the general case, however, proofs are based either on the deep Mitchell Embedding Theorem [M] or involve clever, but not so easily motivated “tricks,” such as the Two-Square Lemma of Fay, Hardie and Hilton [FHH]. Interestingly, and perhaps surprisingly (at least to the author!) is that if the given abelian category has enough projectives (or injectives), then the construc- tion of the connecting homomorphism is formally identical with that in the case of R-modules. Such an assumption on the category is quite reasonable, as most Communications in Algebra, Vol. 28, No. 1 (2000), 249–253. homology or cohomology theories (e.g., derived functors) are predicted on just such assumptions. Henceforth, we assume that A is an abelian category with enough projec- tives. (The “injectives” case can be handled with the obvious modifications.) Thus, given any object A ∈ A then there is a projective object P ∈ A and an epimorphism ε : P A. In what follows, arrows “” shall always denote epimorphisms and arrows “” shall denote monomorphisms. In the Snake lemma, one begins with the diagram with exact rows: µA εA A0 / A / / A00 f 0 f f 00 . µB εB B0 / / B / B00 If one enlarges this diagram so as to include kernels and cokernels of f 0, f, f 00, µK εK K0 / K / K00 µ0 µ µ00 µA εA A0 / A / / A00 f 0 f f 00 B0 / / B / B00 ε0 ε ε00 µC εC C0 / C / C00, where the morphisms µK , εK , µC , εC are the obvious induced ones, then there exists a connecting map δ : K00 −→ C0 so that the 6-term sequence µK εK δ µC εC K0 / K / K / C0 / C / C00 is exact. µ ε C P 00 The definition of δ proceeds as follows. Let KP P K be given, where P is projective, and where µP is the kernel of εP . Next, construct morphisms f1, f2 and f3 giving rise to a commutative diagram below: µ ε P P 00 KP / / P / / K Ð Ð 00 ÐÐf 1 µ ÐÐ Ð εA Ðf2 00 ÐÐ A / / A ÐÐ ÐÐ f f 00 Ð ÐÐ Ð Ö µB εB ÐÐ B0 / / B / B00. Ð?? ?? ?? ε0 f3 ? ? C00 00 f1: By projectivity of P , the morphism µ εP lifts to a morphism f1 : P −→ 00 A, satisfying εAf1 = µ εP . 00 00 00 f2: Since εBff1 = f εAf1 = f µ εP = 0, ff1 factors through µB = ker εB, 0 giving f2 : P −→ B , satisfying µBf2 = ff1. 0 0 f3: Just set f3 = ε f2 : P −→ C . Remark: The reader should have no difficulty in showing that f3 is independent of the original lift f1. We don’t need this fact, however. What remains in the construction is to factor f3 through εP , which is possible precisely when f3µP = 0, as εP = coker µP . To this end, we let µA = µA0 εA0 be an epi-mono factorization of µA: KA0 = = !C εA0 {{ CCµA0 {{ CC {{ CC { µA ! A0 / A. This fits into the larger diagram µ P 00 KP / / P / / K 0 00 f1 f1 µ ε ε 0 µ 0 0 0 A A 00 K0 / / A / / KA0 / / A / / A C CC C f 0 f 0 CC 2 f CC ! µB B0 / / B 0 where εσ = ker εA0 . In the above, f : KP −→ KA0 is induced by the universality of kernel. Next, note that 0 µBf ε0 = fµA0 εA0 ε0 = 0; 0 since µB is a monomorphism, we infer that f ε0 = 0. As εA0 = coker ε0, 0 0 f2 : KA0 −→ B exists as indicated above. 0 0 0 0 0 0 0 Therefore, f3µP = ε f2f1. Also ε f2εA0 = ε f = 0; since εA0 is an epimor- 0 0 phism, it follows that ε f2 = 0. Therefore f3µP = 0, and so there exists a map δ : K00 −→ C0 (the connecting map) as required. It remains to show that the resulting 6-term sequence is exact. This is best handled by the following modified version of the Yoneda Lemma: Lemma. Assume that the abelian category A has enough projectives (or injec- tives). Then the sequence β A −−→α B −−→ C is exact at B if and only if the sequence of abelian groups β Hom(P, A) −−−→α∗ Hom(P, B) −−→∗ Hom(P, C) is exact at Hom(P, B) for every projective object P ∈ A. (In the “enough injectives” case, make the corresponding statement involving the contravariant functor Hom(·,I), where I ∈ A is an arbitrary injective object.) Proof. Again, we shall be content to consider only the case in which A has enough projectives. The necessity of the above condition is clear. Therefore β assume that the condition holds for the sequence A −−→α B −−→ C. If the morphism α is given an epi-mono factorization K ? ? @ @ µ @@ @@ α @ β A / B / C, it suffices to prove that µ = ker β. Let ϕ : T −→ B be a morphism satisfying βϕ = 0; our objective is to factor ϕ through µ. To accomplish this, let ε : P T with P projective and let µP : KP −→ P be the kernel of ε: µ β K / / B / C ? O φP ~~ ~~ φ ~~ µP ~~ ε KP / / P / / T Here, ϕP = ϕ. By hypothesis Hom(P, A) −→ Hom(P, B) −→ Hom(P, C) is exact at Hom(P, B). As a result, µ β Hom(P, K) −−−→∗ Hom(P, B) −−→∗ Hom(P, C) 0 is also exact at Hom(P, B). But β∗(ϕP ) = βϕP = βϕ = 0 and so ϕP = µ∗(ϕP ) 0 for some ϕP ∈ Hom(P, K): µ β K / / B / C O ? O φ ~ 0 P ~~ φP ~ φ ~~ µP ~~ ε KP / / P / / T 0 0 However, as µϕP µP = ϕP µP = ϕεµP = 0, we infer that ϕP µP = 0. But 0 then, since ε = coker µP , we may factor ϕP through ε, giving a morphism 0 ϕT : T −→ K with ϕT ε = ϕP : µ K / / B O `A O A φT φ0 AA φ P AA ε A P / / T 0 Finally, µϕT ε = µϕP = ϕP = ϕε; since ε is an epimorphism, it follows that µϕT = ϕ, concluding the proof. References [FHH] T.H. Fay, K.A. Hardie, and P.J. Hilton, The two-square lemma, Publicacions Mathem´atiques 33 (1989), 133-137. [M] B. Mitchell, Theory of Categories, Academic Press, London, New York, 1965..