Bull. Ac. Sc. Georgian SSR,124, 205-208 (1986)

CLOSED MODEL CATEGORY STRUCTURES ON ABELIAN CATEGORIES

By T. PIRASHVILI

We establish conditions when homotopy theory of modules introduced by Eckmann and Hilton [1]-[3] is a closed model category in the sense of Quillen [4].

Let A be an abelian category. Moreover we let P be a class of objects in A and we let E be class of morphisms of A. We put

E(P) := {f | Hom_{A}(P, f) is surjective for all P ∈ P},
P(E) := {P | Hom_{A}(P, f) is surjective for all f ∈ E}.

Elements of E(P) are called P-epimorphisms, while elements of P(E) are called Eprojective objects.

Definition 1 A class of objects P is called a projective structure if P(E(P)) = P and for any object A there is a P-epimorphism P → A with P ∈ P.

By duality one can talk also on injective structures [5] in A as well as on I-monomorphisms for any injective structure I.

Proposition 2 a) Let (C, F, W) be a closed model category structure on an abelian category A (i.e. C is the class of all cofibrations, F is the class of all fibrations and W is the class of all weak equivalences). Suppose any object of A is fibrant and cofibrant. Then

P := {A ∈ Ob(A) | (A → 0) ∈ W}

form both injective and projective structures.

b) Let P be a class of objects in A forming both projective and injective structures. Let C be the class of P-monomorphisms, F be class of P-epimorphisms and let W be the class of morphisms f : A → B for which there exists a commutative diagram

\xymatrix{A ⊕ P_{1} \ar[r]^{≅}_{g} & B ⊕ P_{2} \ar[d]^{p_{1}} \\ A \ar[u]^{i_{1}} \ar[r]_{f} & B}

where P_{1}, P_{2} ∈ P, i_{1}(a) = (a, 0), p_{1}(a, b) = b and g is an isomorphism. Then (C, F, W) is a closed model category structure on the category A and every object in A is fibrant and cofibrant.

Definition 3 A Frobenius structure on an abelian category A is a class of objects of A forming both an injective and projective structure on A.

Definition 4 A closed model category is called very strict if every object is fibrant and cofibrant.

Corollary 5 There exists a one-to-one correspondence between structures of very strict model categories on an abelian category A and Frobenius structures on A. This correspondence is defined by Proposition 2.

Example 6 Let R be an associative ring with unit and let A be the category of left R-modules. Let P be the class of all projective R-modules and let I be the class of all injective R-modules. It is clear that P-epimorphisms are usual epimorphisms, P-monomorphisms are p-cofibrations in the sense of Eckmann and Hilton see [1]–[3]. It is also clear that I-monomorphisms are monomorphisms and I-epimorphisms are iepimorphisms [1]–[3]. Moreover weak equivalences defined by P and I are respectively p and i-equivalences respectively in the sense of Eckmann and Hilton ( see [1]–[3]).

Proposition 7 a) The class I is the Frobenius structure on A if and only if R is left Noetherian.

b) The class P is the Frobenius structure on A if and only if R is left perfect and right coherent.

Proof. It is clear that a Frobenius structure is closed under products and coproducts. Thus the if part follows from Theorem of Matlis [6] and from Theorem Chase [7].

a) Assume that R is left Noetherian. Let Spec(A) be the set of isomorphism classes of indecomposable injective R-modules. By [6] coproduct of injective objects is injective and every injective R-module is a coproduct of indecomposable injective objects. It follows that for any A ∈ A the homomorphism

∑_{Q∈Spec(A)} ∑_{f:Q→A} ⟶ A

is an I-epimorphism.

b) By [7] products of projective R-modules are projective. Let A be an R-module. We put

α = max(ℵ_{0}, Card(A), Card(R)).

Consider the natural mapping

A → ∏_{Card(P)≤α} ∏_{f:A→P} P

where P is projective. This map is a P-monomorphism, because any morphism A → L to a projective object L decomposes as a composite A → L_{1} → L, where Card(L_{1}) ≤ α and L_{1} is a projective object.

Corollary 8 a) Monomorphisms, i-fibrations, i-homotopy equivalences [1]-[3] form a closed model category structure if and only if R is left Noetherian.

b) p-cofibtations, epimorphisms and p-homotopy equivalence [1]-[3] form a closed model category structure if and only if R is left perfect and right coherence.

Example 9 Let A be a category of (unbounded) chain complexes and let C be the class of contractible complexes.

Proposition 10 The class C form a Frobenius structure on A. Furthermore Ccofibrations are degreewise split monomorphisms, C-fibrations are degreewise split epimorphisms and the class of C-weak equivalences coincides with the class of homotopy equivalences. Hence the homotopy category associated with this model category structure is equivalent to the category of chain complexes and homotopy classes of chain maps.

Therefore the model category structure associated to the class of all contractible chain complex coincides with the structure defined in [8].

References

[1] P. Hilton. Homotopy theory and duality. Gordon and Breach Science Publishers, New York-London-Paris 1965 x+224 pp.

[2] B. Eckmann and H. Kleisli. Algebraic homotopy groups and Frobenius algebras. Illinois J. Math. 6 1962 533–552.

[3] H. Kleisli. Homotopy theory in abelian categories. Canad. J. Math. 14 1962 139–169.

[4] D. Quillen. Homotopical algebra. Springer Lect. Notes Math. 41(1967)

[5] J. M. Maranda. Injective structures. Trans. Amer. Math. Soc. 110 1964 98–135.

[6] E. Matlis. Injective modules over Noetherian rings. Pacific J. Math. 8 1958 511– 528.

[7] S. U. Chase. Direct products of modules. Trans. Amer. Math. Soc. 97 1960 457– 473.

[8] M. Golasinski and G. Gromadzki. The homotopy category of chain complexes is a homotopy category. Colloq. Math. 47 (1982), no. 2, 173–178 (1983).