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 satis ed:
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 ∈ A is a quotient of a projective object, meaning there exists a projective P ∈ 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 ∈ 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. √ √ 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]). De ne a map f : M → N in ChR to be: (i) a weak equivalence = f induces isomorphisms HkM → HkN (k ≥ 0), (ii) a co bration if ∀k ≥ 0, fk : Mk → Nk is a monomorphism + cokernel a projective R-module, (iii) a bration if ∀k > 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 co bration, 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 √ 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. Co brant = 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 co brant 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. √ 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 ∈ 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 co brations 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. De ne 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 co bration - 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 co bration followed by a bration. Since the con- struction P (−) will be functorial, the obtained factorization will be functorial as well. De ne 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 Co bration - 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.
5 Assume that in the following diagram
Mn Nn
∂ ∂
in−1 qn−1 Mn−1 / Qn−1 / Nn−1
∂ ∂ ∂
in−2 qn−2 Mn−2 / Qn−2 / Nn−2
. . . . . . i and q are a co bration+acyclic bration factorization of f up to (not including) dimension n. By the lemma, this means that the maps Qk → Zk−1Q × Nn−1 Zn−1N de ned as in (2.2) are surjective for all k < n. We wish to complete the diagram with the missing Qn. qn−1 restricts to a map on cycles. Because f is a chain map, the two maps in−1∂ : Mn → Zn−qQ and fn : Mn → Nn gives the following commutative diagram
Mn fn j
& Zk−1Q × Nn−1 / Nn Zn−1N in−1∂ ∂ ∂
) qn−1 Zn−1Q / Zn−1N
Because R-Mod has enough projectives, the canonical map j : Mn → Zk−1Q × Zn−1N Nn−1 can be factored as a monomorphism i with projective cokernel, followed by an epimorphism: i j⊕e Mn → Mn ⊕ P Zk−1Q × Nn−1 → Zk−1Q × Nn−1 Zn−1N Zn−1N
We de ne Qn to be the middle term in the last line. Because Qn → Zk−1Q × Zn−1N Nn−1 is a surjection, then by the induction hypothesis+the lemma we have that the map Qn → Nn is an acyclic bration (by abuse of notation). The inductive step is done. The base of the induction is left as an exercise.
6 3 Reminder - injectives
Problem. The model structure on ChR presented above relies heavily on the fact that R-Mod has enough projectives. We would like to generalize it to the category of sheaves of OX -modules, for instance. However, in general this cate- gory does not have enough projectives, so we cannot proceed in this direction. A hint for a solution is given by the fact that it does have enough injectives.
An R-module E is called injective if the following equivalent conditions are satis ed:
f • Given an injection 0 → A → B, any R-map A → E extends to a map from B: E O ` f 0 / A / B
• HomR (−,E) is exact . • Every short exact sequence 0 → E → B → C → 0 beginning with E splits.
f • (Baer Criterion) any R-map I → E, with I an ideal of R, can be extended to R. An abelian category A is said to have enough injectives if for every X ∈ A there exists an injective I such that X can be imbedded into I, meaning
0 → X → I is exact. Apart from Baer's Criterion, all of the de nitions above apply equally well to any abelian category. We will now present characterizations for injective objects in Ab and in R-Mod. See [Rotman, 7.70 .] for more details. • If R is a domain then F rac (R) is an injective R-module (by Baer's crite- rion).
• An R-module M is called divisible if any m ∈ M can be divided by any nonzero r ∈ R: ∀m ∈ M ∀r 6= 0 ∈ R, ∃m0 ∈ M s.t. m = rm0. Example 2. Let R be a domain. F rac (R) is divisible. Any direct sum of di- visible R-modules is divisible (⇒any vector space). Every quotient of a divisible R-modules is divisible.
• Let R be a domain. Then injective ⇒ divisible for an R-module. If R is in addition a PID, then divisible ⇔ injective.
7 Example 3. From example 2 it follows that the following abelian groups (Z- modules) are divisible, hence injective:
Q, R, C, Q/Z, R/Z By direct inspection, it is clear that S1, considered as the (multiplicative) unit circle in C, is divisible. • Let A be any abelian group. It is a quotient of a free abelian group, L L A = ( Z)/K ⊆ ( Q)/K with the latter injective. We have thus proved that any abelian group can be imbedded into an injective abelian group (=Ab has enough injectives). • This result can be extended to an R-module over an arbitrary ring: Theorem ([Rotman, 8.104]). For every ring R, every left R-module M can be imbedded as a sub-module of an injective left R-module. The proof is not di cult. The key is to imbed M, regarded as an abelian group, into a divisible abelian group D. This shows that R-Mod has enough injectives.
4 Injective model structure on ChR A nice thing about the axioms of model category is that they are self-dual. Meaning, they are left unchanged by inverting all the arrows and swapping between brations and co brations throughout. Since R-Mod has enough injec- tives, we have the following result for free7:
Theorem 4. De ne a map f : M → N in ChR to be: (i) a weak equivalence = f induces isomorphisms HkM → HkN (k ≥ 0), (ii) a co bration if ∀k > 0, fk : Mk → Nk is a monomorphism (iii) a bration if ∀k ≥ 0, fk : Mk → Nk is an epimorphism + kernel an injective R-module, Then with these choices ChR is a model category. In this injective model structure, it is clear either by direct inspection or by duality that every object is co brant, and that brant objects are precisely the chain complexes of injectives. In a similar manner to what we had before, for chain complexes concentrated in dimension 0, the brant replacement is an injective resolution
0 / X / 0 _ ∼ 0 / E0 / E1 / E2 / ...
7Note that monomorphisms and epimorphisms in R-Mod agree with their corresponding categorical counterparts.
8 which exists because R-Mod has enough injectives. The existence of a brant replacement for a general chain complex follow from MC5, which can now be proven thanks to the fact that R-Mod has enough injectives.
References
[DS95] W.G. Dwyer and J. Spalinski, Homotopy theories and model cate- gories. [Rotman] J. Rotman, Advanced Modern Algebra (1st edition, 2nd printing).
[Riehl] E. Riehl, Higher Dimensional Categories: Model Categories and Weak Factorisation Systems. Available on www.math.harvard. edu/~eriehl/essay.pdf
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