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Projective and Injective Model Structure on Chr Projective and injective model structure on ChR Shlomi Agmon Thursday, July 2nd, 2012 The material in sections 1 and 3 is taken mostly from [Rotman, 7.4]. The rest of the material is from [DS95, Section 7], unless mentioned otherwise. 1 Reminder - projectives Reminder: Both Hom functors are generally only left exact, while tensor prod- uct M ⊗ − is only right exact. An R-module P is called projective if the following equivalent conditions are satised: f • Given a surjection B ! C ! 0, any R-map P ! C lifts to B: P f B / / C / 0 • HomR (P; −) is exact. • Every short exact sequence 0 ! A ! B ! P ! 0 ending with P splits. • P is a direct summand of a free R module. An abelian category A is said to have enough projectives if every object B 2 A is a quotient of a projective object, meaning there exists a projective P 2 A such that P ! B ! 0 is exact. Note that all of the above could have actually been done in an arbitrary abelian category. For a general (not necessarily abelian) category C, P 2 C is called a projective object if HomC (P; −) sends epimorphisms (in C) to epimor- phisms (in Sets). In the following we specialize to R-modules. Example. A free module is projective. R-Mod has enough projectives because every module is a quotient of a free module. In particular, Ab = Z − Mod has enough projectives. 1 There are modules which are projective but not free1: Let . Then is a projective -module because it is a direct R = Z3 ⊕ Z2 Z3 R summand of the free R-module R, but it is not free. p p A more interesting example: R := Z −5 , then the ideal I = 3; 2 + −5 is a non-principal ideal, therefore cannot be free. However, I is a direct sum- mand of R2 .2 2 Projective model structure on ChR Let R be an associative ring with unit. ModR be left R-modules, ChR be (non- negatively graded) chain complexes of R-modules. Hk : ChR ! ModR the homology functors (k ≥ 0). We will prove portions of the following standard projective model structure on ChR: Theorem 1 ([DS95, 7.2]). Dene a map f : M ! N in ChR to be: (i) a weak equivalence = f induces isomorphisms HkM ! HkN (k ≥ 0), (ii) a cobration if 8k ≥ 0, fk : Mk ! Nk is a monomorphism + cokernel a projective R-module, (iii) a bration if 8k > 0, fk : Mk ! Nk is an epimorphism. Then with these choices ChR is a model category. 2.1 Reminder - basics of MC 3 classes that include identities, closed under composition, and: MC1 All small (co+)limits exist. MC2 2 out of 3 for w.e.. MC3 All 3 classes are closed under retracts. MC4 A lift B 99K X exists for A / X i p B / Y whenever i is a cobration, p a bration, and at least one of them acyclic. ∼ MC5 Any map f can be factored (functorially) in two ways: • ,!• • and ∼ • ,!• • Remarks: 1Over a PID R, projective , free. Because M is a projective R-module =) it is a direct summand of a free R-module. But over a PID, any submodule of a free module is free. u p 2Take the surjection R2 ! I; 7! u · 3 + v · 2 + −5 in one direction, the other is a v direct calculation 2 • Initial/terminal object, ;=∗ : the zero complex. ∼ c ∼ f • (co)brant; (co)brant replacement of X : ; ,! X X, X ,! X ∗. • In particular, Fibrant: any object is brant in this model structure. So we can take Xf = X. Cobrant = chain complexes of projectives, meaning chain complexes which are level-wise projective. 3 Warning: A projective object in ChR is a chain complex of projectives , but 4 a chain complex of projectives need not be a projective object in ChR . • Xc = a projective resolution (at least when X is concentrated in dim 0), ::: / P2 / P1 / P0 / 0 ∼ 0 / X0 / 0 For a chain complex concentrated in dim 0, the fact that R-Mod has enough projectives implies that a cobrant replacements exist. For a general chain complex this follow from MC5, which we will also prove using the same fact. 2.2 Proof of projective model structure - MC1 to MC4 Includes identities + closed under composition = easy5. MC1 Dimension-wise. p MC2 (2 out of 3) MC3 Technical and easy. 3Denote by K (A; n) the chain complex which is A at dimension n, and zero otherwise. We have the adjunction ∼ , where sends HomChR (M; K (A; n)) = HomR−Mod (Mn;A) M 7! Mn a chain complex M to the R-module at it's n-th dimension. Since the right adjoint K (−; n) preserves epomirohisms, we have that the left adjoint M 7! Mn preserves projectives. 4 The projective objects of ChR are precisely the acyclic chain complexes of projectives. ·r So ···! 0 ! R ! R ! 0 for r 2 R non-unit is an example of a chain complex of projectives which is not a projective chain complex. Projective unbounded chain complexes have a slightly dierent character- ization. See more details at: http://mathoverflow.net/questions/103584/ on-the-difference-between-a-projective-chain-complex-and-a-level-wise-projective. 5The only point which requires some eort is that a composition of cobrations is a co- bration. Suppose 0 ! L ! M ! N is a series of monomorphisms, each with projective (level-wise) cokernels. Then M N N is exact, with the rst and last terms 0 ! L ! L ! M ! 0 being projective, thus also N is projective, as a direct sum of projective modules. L 3 MC4 Technical. There are, however, some interesting results en-route: Id Let A be an R-module. Dene Dn (A) to be A ! A in dimensions n; n − 1, zero otherwise. We have an adjunction ∼= (2.1) HomChR (Dn (A) ;M) ! HomModR (A; Mn) sending f 7! fn. From this adjunction it follows immediately that if A is a projective R- 6 module then Dn (A) is an acyclic and projective object in ChR. Obviously, a direct sum of such projective disks is projective and acyclic. Denote by ZkM the k-cycles of a chain complex M. We have a converse: Lemma ([DS95, 7.10]). Suppose that P in ChR is an acyclic chain complex of projectives (=level-wise projective). Then each module ZkP (k ≥ 0) is projec- tive, and is isomorphic as a chain complex to L . P k≥1 Dk (Zk−1P ) n Denote for short by Dn the chain complex Dn (R). Set S to be K (R; n), the chain complex which is R in dimension n, zero otherwise. There is the n−1 n obvious inclusion jn : S ! D . We have the following characterization of brations, which is very resemblant to that of Serre brations: Proposition ([DS95, 7.19]). A map f : X ! Y in ChR is (i) a bration i it has the RLP w.r.s.t. the maps 0 ! Dn for all n ≥ 1, and n−1 n (ii) an acyclic bration i it has the RLP w.r.s.t. the maps jn : S ! D for all n ≥ 0. The proof is a nice exercise in diagram chasing. 2.3 Proof of MC5 (factorization) A useful argument in order to produce factorizations of maps with lifting prop- erties is the small object argument. We will not present it, however, and will instead pursue a more direct approach. The reader is urged to look the small object argument in the literature ([DS95, 7.12], for example). The proof below is taken from [Riehl]. 2.3.1 Acyclic cobration - bration factorization Let M be an R-module, and choose a set of generators. M can be written as a quotient of the projective R-module L R, where we have taken one copy of R for each generator of M. The problem with this construction is that it is not functorial. 6A fancier way to show the same thing is the following: In an adjoint pair, if the right adjoint preserves epimorphisms, then the left adjoint preserves projective objects (direct proof). The functor M 7! Mn obviously preserves epimorphisms. 4 Let f : M ! N be a morphism in ChR. If we could make a functorial construction in ChR similar to the above of an acyclic chain complex P (N) of projectives, then we could factorize f in the form f M N 9/ i f⊕e M % M ⊕ P (N) which is clearly an acyclic cobration followed by a bration. Since the con- struction P (−) will be functorial, the obtained factorization will be functorial as well. Dene now M P (N) := Dk maps Dk!N with the obvious projection e : P (N) ! N, sending the copy of Dk indexed by g : Dk ! N to it's image under g. It is a direct sum of acyclic chain complexes of projectives, hence an acyclic complex of projectives. 2.3.2 Cobration - acyclic bration factorization For a chain map f : M ! N, consider the commutative diagram fn Mn / Nn @ @ Zn−1M / Zn−1N fn−1 induced by f. It induces a map to the pullback (omitting Mn from the corner of the last diagram) Mn ! Zn−1M × Nn (2.2) Zn−1N The proof of this factorization relies on the following technical lemma, which can be proven by standard diagram chasing methods. Lemma ([Riehl, 3.19]). f is a trivial bration i the map in (2.2) is a surjection for all n ≥ 0. Now that we are properly equipped, we can turn to the heart of the proof.
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