CONSTRUCTION OF HOMOTOPY EQUIVALENCE OF TRUNCATED COMPLEXES

For a ring R, given two truncated proper left C-resolutions of equal length for the same module, where C is a subcategory of R-modules, we obtain a pair of complexes of the same homotopy type and give some examples. Mathematics Subject Classification (2010): 16E05


Introduction
Throughout this paper, all rings are associative with identity and all modules are unitary modules.Let R be a ring.We denote by R-Mod the category of left R-modules.
Truncated projective resolutions are of interest in both algebraic geometry and algebraic topology.The final modules of two truncated projective resolutions of the same module may be stabilized to produce homotopy equivalent complexes.
In 2007, Mannan in [3] considered two truncated projective resolutions of equal length for the same module and obtained a pair of complexes of the same homotopy type.This paper generalizes projective resolutions to proper left C-resolutions and similar results are obtained, where C is a subcategory of R-modules.Moreover, some examples are given.

Main results
Let C be a subcategory of R-modules.Recall that a complex of modules Definition 2.2.Let C be a subcategory of R-modules and M a module in R-Mod.
be two complexes.Recall that a chain map f : X → Y is a family of morphisms {f i } such that the following diagram commutes: Recall that two chain maps f, g : X → Y are said to be chain homotopic, denoted by f ∼ g, if there exists a family of morphisms {s i } with each s i : The complexes X and Y are said to be chain homotopy equivalent, if there exist chain maps f : X → Y and g : Y → X, such that f g ∼ I Y and gf ∼ I X , where exact.Furthermore, the sequences F and G are chain homotopy equivalent.
Proof.For any C ∈ C, applying the functor Hom R (C, −) to G, we have the following commutative diagram The following result plays a crucial role in this paper.
Theorem 2.4.Let C be a subcategory of R-modules such that C is closed under finite direct sums, and M a module in R-Mod.Suppose that we have two proper left C-resolutions of M : Then the complexes are chain homotopy equivalent, where the modules T i , S i are defined inductively by Proof.We follow the proof of [3, Theorem 1.1].For each i = 1, 2, • • • , n, we have natural inclusions of summands: λ 0 : C 0 → S 0 both be the identity maps.We define ρ i : Also let C n denote chain complex Clearly, C 0 is the chain complex ( ).For r = 0, 1, • • • , n − 1, the chain complex Similarly, for r = 0, 1, • • • , n − 1, let D r denote the chain complex Again let D n denote chain complex Clearly, D 0 is the chain complex ( ).

By Lemma 2.3, for
) is chain homotopy equivalent to C r (respectively, D r ).Hence ( ) (respectively, ( )) is chain homotopy equivalent to To prove that ( ) and ( ) are chain homotopy equivalent, it suffices to show that C n is chain isomorphic to D n , that is, there exist isomorphisms h i , k i making the following diagram commute: We proceed by induction on n.For n = 0, as the sequences T 0 σ / / M / / 0 and S 0 σ / / M / / 0 are Hom R (C, −)-exact, there exist f 0 , g 0 such that the following diagrams commute: Then h 0 k 0 = 1 and k 0 h 0 = 1.From commutativity of ( ), we deduce: Hence we get the following commutative diagrams: Now suppose that for some j < i ≤ n, we have defined h j : T j ⊕ S j → S j ⊕ T j and k j : S j ⊕ T j → T j ⊕ S j for j = 0, 1, • • • , i − 1, so that for each j, we have h j k j = 1 and k j h j = 1.
Since the sequences T i ρi / / Ker(ρ i−1 ⊕ 0) and S i ρ i / / Ker(ρ i−1 ⊕ 0) are Hom R (C, −)-exact, and T i , S i ∈ C, there exist f i , g i such that the following diagrams commute: Then From commutativity of ( ), we deduce: So C n is chain isomorphic to D n , ( ) and ( ) are chain homotopy equivalent.This completes the proof.
As applications of Theorem 2.4, we will give some examples.Firstly, the following result follows immediately from Theorem 2.4 since the class of projective modules is closed under direct sums.Suppose we have exact sequences: with the P i and Q i all projective modules in R-Mod.Then the complexes are chain homotopy equivalent, where the modules T i , S i are defined inductively by T 0 ∼ = P 0 , S 0 ∼ = Q 0 , and for i = 1, 2, • • • , n, T i ∼ = P i ⊕ S i−1 , S i ∼ = Q i ⊕ T i−1 .
Recall from [5] that a module M is called FP-injective if Ext 1 R (F, M ) = 0 for any finitely presented module F .Recently, Pinzon in [4] shows that every module in R-Mod has an FP-injective cover if R is a left coherent ring.So every module M in R-Mod has a proper left FP-injective resolution if R is coherent.
remains exact after applying the functor Hom R (C, −) (respectively, Hom R (−, C)) for any object C ∈ C.This work was partly supported by NSF of China (Grant No. 11561039), and NSF of Gansu Province of China (No. 145RJZA079).CONSTRUCTION OF HOMOTOPY EQUIVALENCE OF TRUNCATED COMPLEXES 111 Definition 2.1.(see [1]) Let C be a subcategory of R-modules and M a module in R-Mod.A homomorphism f : C → M with C ∈ C is called a C-precover of M if the abelian group homomorphism Hom R (C , f ) : Hom R (C , C) → Hom R (C , M ) is surjective for any object C ∈ C. If every R-module has a C-precover, we say that C is a precovering class.Dually, we have the definitions of a C-preenvelope and a preenveloping class.
ϕ and ψ are isomorphisms.It is easy to see that the lower sequence in the above diagram is exact, so is the upper sequence.Thus the sequence G is Hom R (C, −)-exact.That sequences F and G are chain homotopy equivalent is simple.This completes the proof.

Example 2 . 6 .
Let R be a left coherent ring and M a module in R-Mod.Suppose the complexes ( ) and ( ) in Theorem 2.4 are two proper left FP-injective resolutions of M , then the complexes ( ) and ( ) in Theorem 2.4 are chain homotopy equivalent.Proof.Note that the class of FP-injective modules in R-Mod is closed under direct sums.Then the result follows from Theorem 2.4.