Homology, Homotopy and Applications, vol.5(1), 2003, pp.387–406

Dedicated to Professor Hvedri Inassaridze on the occasion of his seventieth birthday

EXTENSIONS OF SEMIMODULES AND THE TAKAHASHI FUNCTOR Ext_{Λ}(C, A)

ALEX PATCHKORIA

(communicated by George Janelidze)

Abstract

Let Λ be a semiring with 1. By a Takahashi extension of a Λ-semimodule X by a Λ-semimodule Y we mean an extension of X by Y in the sense of M. Takahashi [10]. Let A be an arbitrary Λ-semimodule and C a Λ-semimodule which is normal in Takahashi’s sense, that is, there exist a projective Λ-semi-module P and a surjective Λ-homomorphism \xymatrix{ε : P \ar[r] & C} such that ε is a cokernel of the inclusion μ : Ker(ε) ↪ P. In [11], following the construction of the usual satellite functors, M. Takahashi defined Ext_{Λ}(C, A) by

Ext_{Λ}(C, A) = Coker(Hom_{Λ}(μ, A))

and used it to characterize Takahashi extensions of normal Λsemimodules by Λ-modules.

In this paper we relate Ext_{Λ}(C, A) with other known satellite functors of the functor Hom_{Λ}(−, A).

Section 1 is concerned with preliminaries. The purpose of Section 2 is to characterize Ext_{Λ}(C, A) in terms of Janelidze’s general Ext�^{n} -functors [5]. In Section 3 we show that Ext_{Λ}(C, A) with A cancellative can be described directly by Takahashi extensions of C by A. The last section is devoted to Ext_{Λ}(C, G) with G a Λ-module. We relate Ext_{Λ}(C, G) with Inassaridze extensions of C by G [4]. This allows to relate Ext_{Λ}(C, G) and S^{1} Hom_{K(Λ)}(−, G)(K(C)), where K(Λ) is the Grothendieck ring of Λ, K(C) the Grothendieck K(Λ)-module of C, and S^{1} Hom_{K(Λ)}(−, G) the usual right satellite functor of the functor Hom_{K(Λ)}(−, G).

Partially supported by INTAS grants 97-31961 and 00-566, and grant No 1.5 of the Georgian Academy of Sciences. Received July 7, 2003, revised September 26, 2003; published on October 1, 2003. 2000 Mathematics Subject Classification: 18G99, 18E25, 16Y60, 20M50. Key words and phrases: Semiring, semimodule, projective semimodule, normal semimodule, extension of semimodule, satellite functor. \begin{matrix}c \\ ◯ \end{matrix} 2003, Alex Patchkoria. Permission to copy for private use granted.

1. There are several concepts of semirings and semimodules (see for example, [2,3,9]). In this paper we use the following ones. A semiring Λ = (Λ, +, 0, ·, 1) is an algebraic structure in which (Λ, +, 0) is an abelian monoid, (Λ, ·, 1) a monoid, and

λ · (λ′ + λ′′) = λ · λ′ + λ · λ′′,
(λ′ + λ′′) · λ = λ′ · λ + λ′′ · λ,
λ · 0 = 0 · λ = 0,

for all λ, λ′, λ′′ ∈ Λ. An abelian monoid A = (A, +, 0) together with a map \xymatrix{Λ × A \ar[r] & A} , written as (λ, a) ⟼ λa, is called a (left) Λ-semimodule if

λ(a + a′) = λa + λa′,
(λ + λ′)a = λa + λ′a,
(λ · λ′)a = λ(λ′a),
1a = a, 0a = 0,

for all λ, λ′ ∈ Λ and a, a′ ∈ A. It immediately follows that λ0 = 0 for any λ ∈ Λ. Let us also recall: A map \xymatrix{f : A \ar[r] & B} between Λ-semimodules A and B is called a Λ-homo-

morphism if f(a + a′) = f(a) + f(a′) and f(λa) = λf(a), for all a, a′ ∈ A and λ ∈ Λ. It is obvious that any Λ-homomorphism carries 0 into 0. The abelian monoid of all Λ-homomorphisms from A to B is denoted by Hom_{Λ}(A, B). (Example: Let N be the semiring of non-negative integers. An N-semimodule A is simply an abelian monoid, and an N-homomorphism \xymatrix{f : A \ar[r] & B} is just a homomorphism of abelian monoids.)

A Λ-subsemimodule A of a Λ-semimodule B is a subsemigroup of (B, +) such that λa ∈ A for all a ∈ A and λ ∈ Λ. Clearly 0 ∈ A. The quotient Λ-semimodule B/A is defined as the quotient Λ-semimodule of B by the smallest congruence on the Λ-semimodule B some class of which contains A. Denote the congruence class of b ∈ B by [b]. Then [b_{1}] = [b_{2}] if and only if a_{1} + b_{1} = a_{2} + b_{2} for some a_{1}, a_{2} ∈ A. The Λ-homomorphism \xymatrix{p : B \ar[r] & B/A} that carries b ∈ B into [b] is called the canonical surjection.

A Λ-semimodule A is cancellative if a + a′ = a + a′′ for a, a′, a′′ ∈ A implies a′ = a′′. Obviously, A is a cancellative Λ-semimodule if and only if A is a cancellative C(Λ)-semimodule, where C(Λ) denotes the largest cancellative homomorphic image of Λ under addition. A Λ-semimodule A is called a Λ-module if A = (A, +, 0) is an abelian group. It is clear that A is a Λ-module if and only if A is a K(Λ)-module, where K(Λ) denotes the Grothendieck ring of Λ.

The categories of Λ-semimodules, cancellative Λ-semimodules, Λ-modules, abelian monoids, abelian groups, and sets are denoted by Λ-SMod, Λ-CSMod, Λ-Mod, Abm, Ab, and Set, respectively.

A cokernel of a Λ-homomorphism \xymatrix{f : A \ar[r] & B} is defined to be a Λ-homo-morphism \xymatrix{u:B \ar[r] & C} such that (i) uf = 0, and (ii) for any Λ-homomorphism \xymatrix{g : B \ar[r] & D} with gf = 0 there is a unique Λ-homomorphism \xymatrix{g′ : C \ar[r] & D} with g = g′u. One dually defines a kernel of f. Clearly, the canonical projection \xymatrix{p : B \ar[r] & B/f(A)} is a cokernel of f, and the inclusion Ker(f) ↪ A, where Ker(f) = {a ∈ A|f(a) = 0}, is a kernel of f.

A sequence [figure: E : A; λ; B; τ; C] of Λ-semimodules and Λ-homomorphisms is called a short exact sequence if λ is injective, τ is surjective, and λ(A) = Ker(τ) (cf. [9]). The following assertion is plain and well-known.

Proposition 1.1. If [figure: E : A; B; C] is a short exact sequence, then B is a Λ-module if and only if A and C are both Λ-modules.

A morphism from [figure: E : A; λ; B; τ; C] to [figure: E′ : A′; λ′; B′; τ′; C′] is a triple of Λ-homomorphisms (α, β, γ) such that

[figure: E :; A; α; λ; B; τ; β; C; γ; E′ :; A′; λ′; B′; τ′; C′]

is a commutative diagram. For a morphism of the form

[figure: E :; A; 1_{A}; B; β; C; 1_{C}; E′ :; A; B′; C,]

we write E − > E′. If in addition β is a Λ-isomorphism, we write E ≡ E′ and say that E is equivalent to E′.

Next, suppose given a short exact sequence [figure: E : A; λ; B; τ; C] and a Λ-ho-momorphism \xymatrix{γ : C′ \ar[r] & C} . Then

[figure: Eγ : A; λ^{γ}; B^{γ}; τ^{γ}; C′,]

where B^{γ} = {(b, c′) ∈ B ⊕ C′|τ(b) = γ(c′)}, λ^{γ} (a) = (λa, 0), τ^{γ} (b, c′) = c′, is a short exact sequence of Λ-semimodules. Besides, if one defines a Λ-homomorphism \xymatrix{ξ^{γ} : B^{γ} \ar[r] & B} by ξ^{γ} (b, c′) = b, then (1_{A}, ξ^{γ} , γ) is a morphism from Eγ to E. From the construction of Eγ it follows that

E ≡ E 1_{C} , (Eγ)γ′ ≡ E(γγ′),  (1.2)
E ≡ E′ ⟹ Eγ ≡ E′γ,  (1.3)
E − > E′ ⟹ Eγ − > E′γ.  (1.4)

We will also use sequences of the form [figure: S : A; X; f; Y; C] , where \xymatrix{f : X \ar[r] & A} is a Λ-homomorphism and [figure: E : X; Y; C] a short exact sequence of Λ-semimodules. It will be convenient to denote S by f∘E, and E by \overline{S}.

A surjective Λ-homomorphism \xymatrix{τ : B \ar[r] & C} is said to be a normal Λ-epimor-phism if it is a cokernel of the inclusion Ker(τ) ↪ B. One can easily see that τ is normal if and only if it is kernel-regular in the sense of [9]: if τ(b_{1}) = τ(b_{2}), then k_{1} + b_{1} = k_{2} + b_{2} for some k_{1}, k_{2} in Ker(τ).

Proposition 1.5. Any surjective Λ-homomorphism \xymatrix{τ : B \ar[r] & H} with H a Λ-module is normal.

Proof. Suppose τ(b_{1}) = τ(b_{2}). Take b ∈ B with τ(b) = −τ(b_{1}). Then (b_{2} + b), (b_{1} + b) ∈ Ker(τ) and (b_{2} + b) + b_{1} = (b_{1} + b) + b_{2}.

Note also that for any Λ-subsemimodule A of a Λ-semimodule B, the canonical projection \xymatrix{p : B \ar[r] & B/A} is normal.

Let A and C be Λ-semimodules. By a Takahashi (or normal) extension of C by A we mean an extension of C by A in the sense of [10], that is, a short exact se-

quence [figure: E : A; λ; B; τ; C] of Λ-semimodules with τ normal. Clearly, a short exact sequence of Λ-semimodules [figure: A; B; H] with H a Λ-module and a

sequence of the form Ker(p) ↪ \xymatrix{B \ar[r]^{p} & B/A} provide examples of Takahashi ex-

tensions. (Note that in general Ker(p) ≠ A.) Let� denote the class of all Takahashi extensions of Λ-semimodules. Then

E ∈� ⟹ Eγ ∈� ,  (1.6)
E, E′ ∈� ⟹ E ⊕ E′ ∈� .  (1.7)

Here E ⊕ E′ denotes [figure: A ⊕ A′; λ⊕λ′; B ⊕ B′; τ⊕τ′; C ⊕ C′] , the usual direct sum of E and E′. Two extensions E_{1} and E_{2} are equivalent if E_{1} ≡ E_{2}, i.e., if they are equivalent as short exact sequences. Following [10], we denote by E_{Λ}(C, A) the set of equivalence classes of Takahashi extensions of C by A. It contains at least the 0, the class of

[figure: 0 : A; i_{A}; A ⊕ C; π_{C}; C,]

where i_{A}(a) = (a, 0) and π_{C} (a, c) = c.

A Λ-semimodule P is projective if it satisfies the usual lifting property: Given a surjective Λ-homomorphism \xymatrix{τ : B \ar[r] & C} and a Λ-homomorphism \xymatrix{f : P \ar[r] & C},

there is a Λ-homomorphism \xymatrix{g : P \ar[r] & B} such that f = τg.

Proposition 1.8. Let \xymatrix{τ : B \ar[r] & C} be a normal Λ-epimorphism and let f_{1}, f_{2} : \xymatrix{P \ar[r] & B} be Λ-homomorphisms with P projective. If τf_{1} = τf_{2} then there exist Λ-homomorphisms \xymatrix{g_{1}, g_{2} : P \ar[r] & B} satisfying τg_{1} = 0 = τg_{2} and g_{1} + f_{1} = g_{2} + f_{2}. That is, the functor Hom_{Λ}(P, −) preserves normal epimorphisms.

This fact, proved in [1] (and first mentioned in [8]), implies

Proposition 1.9 (cf. [11]). Suppose given a diagram of Λ-semimodules and Λ-ho-momorphisms

[figure: A′; φ_{1}; φ_{2}; λ′; P; g_{2}; g_{1}; ψ_{1}; ψ_{2}; τ′; C′; γ; A; λ; B; τ; C,]

where the bottom row is a Takahashi extension, P is projective, τ′λ′ = 0, and τψ_{i} = γτ′, λφ_{i} = ψ_{i}λ′ for i = 1, 2. Then there are Λ-homomorphisms \xymatrix{g_{1}, g_{2} : P \ar[r] & A} such that g_{1}λ′ + φ_{1} = g_{2}λ′ + φ_{2}.

A Λ-semimodule C is called normal if there exist a projective Λ-semimodule P and a normal Λ-epimorphism \xymatrix{ε : P \ar[r] & C} [11]. In other words, C is normal if there is a Takahashi extension of Λ-semimodules [figure: R; P; C] with P projective, called a projective presentation of C. It follows from Proposition 1.5 that every Λ-module H is normal, since one has a free Λ-semimodule F and a surjective Λ-homomorphism \xymatrix{F \ar[r] & H} . Any quotient Λ-semimodule P/A of a projective Λ-semimodule P is also normal. Moreover, since the class of normal epimorphisms of Λ-semimodules is closed under composition, any quotient Λ-semimodule B/A of a normal Λ-semimodule B is normal [11]. We denote the category of normal Λ-semimodules and their Λ-homomorphisms by Λ-NSMod.

In [11] M. Takahashi has constructed Ext_{Λ}(C, A) as follows. Let (C, A) be an ob-

ject of (Λ-NSMod)^{op} × (Λ-SMod). Choose a projective presentation [figure: ℙ:R; μ]

\xymatrix{P \ar@{->>}[r]^{ε} & C} of C and define Ext_{Λ}(C, A) to be Coker(Hom_{Λ}(μ, A):[figure: Hom_{Λ}(P, A)] Hom_{Λ}(R, A)). That is,

Ext_{Λ}(C, A) = Hom_{Λ}(R, A)/ Hom_{Λ}(μ, A)(Hom_{Λ}(P, A)).

If \xymatrix{α : A \ar[r] & A′} is a homomorphism of Λ-semimodules, one defines Ext_{Λ}(C, α) :

Ext_{Λ}(C, A) [figure: Ext_{Λ}(C, A′)] by Ext_{Λ}(C, α)([φ])=[αφ]. Obviously, Ext_{Λ}(C, α) is

well defined. Next, any homomorphism \xymatrix{γ : C′ \ar[r] & C} of normal Λ-semimodules can be lifted to a morphism

[figure: ℙ′ :; R′; f; μ′; P′; ε′; g; C′; γ; ℙ :; R; μ; P; ε; C,]

and \xymatrix{Ext_{Λ}(γ, A) : Ext_{Λ}(C, A) \ar[r] & Ext_{Λ}(C′, A)} is defined by Ext_{Λ}(γ, A)([φ]) = [φf]. It follows from Proposition 1.9 that Ext_{Λ}(γ, A) is also well defined. Now one can easily see that Ext_{Λ}(C, A) is a functor from the category (Λ-NSMod)^{op} × (Λ-SMod) to the category Abm ([φ] ⟼ [αφ], [φ] ⟼ [φf]), additive in both its arguments. Further, Proposition 1.9 implies that a different choice of the projective presentations would yield a new functor Ẽxt_{Λ}(C, A) which is naturally isomorphic to the functor Ext_{Λ}(C, A).

It is evident that if T:Λ-Mod [figure]Abm is an additive functor, i.e., T (f +g) = T (f) + T (g) and T (0) = 0, then T (G) is an abelian group for any Λ-module G. Therefore, Ext_{Λ}(C, A) is an abelian group whenever either A or C is a Λ-module.

Next, the Grothendieck functor K carries any short exact sequence of Λ-se-mimodules [figure: X; Y; M] with M a Λ-module into a short exact sequence [figure: K(X); K(Y); M] of K(Λ)-modules. Therefore, if [figure: ℚ : V; Q; H]

is a Λ-projective presentation of a Λ-module H, then [figure: K(ℚ) : K(V); K(Q)]

[figure: H] is a K(Λ)-projective presentation of H. Consequently, for any Λ-semi-module A, one has a natural homomorphism

\xymatrix{Ext_{Λ}(H, A) \ar[r]^{K(H,A)} & S^{1} Hom_{K(Λ)}(−, K(A))(H)} , K(H, A)([φ])=[K(φ)], (1.10)

where S^{1} Hom_{K(Λ)}(−, K(A)) denotes the usual right satellite functor of the functor Hom_{K(Λ)}(−, K(A)). A straightforward verification shows that K(H, A) is one-to-one whenever A is cancellative. Furthermore, it immediately follows from the universal property of K that

\xymatrix{K(H, G) : Ext_{Λ}(H, G) \ar[r] & S^{1} Hom_{K(Λ)}(−, G)(H)}  (1.11)

ia an isomorphism for any Λ-modules G and H.

2. In [5] G. Janelidze introduced and studied general Ext�^{n} -functors, where� is an arbitrary class of diagrams of the form \xymatrix{X \ar[r] & Y \ar[r] & Z} in an arbitrary category. Suppose n = 1 and � = � , the class of all Takahashi extensions of Λ-semimodules. Then Ext�^{1} (C, A) is a functor from the category (Λ-SMod)^{op} × (Λ-SMod) to the category Abm. In this section we prove that the restriction of Ext�^{1} (C, A) to the category (Λ-NSMod)^{op} × (Λ-SMod) is naturally isomorphic to the functor Ext_{Λ}(C, A).

The functor Ext�^{1} (C, A), denoted by Ext_{Λ} JT (C, A) in this paper, is defined as follows. Let Ext_{Λ}JT (C, A) be the category of sequences of the form S = f ∘ \overline{S} :

[figure: A; X; f; Y; C] , where f is a Λ-homomorphism and [figure: \overline{S} : X] \xymatrix{Y \ar@{->>}[r] & C} a Takahashi extension of Λ-semimodules. Define Ext_{Λ} JT (C, A) to be the set of connected components of Ext_{Λ}JT (C, A), that is,

Ext_{Λ} JT (C, A) = Ext_{Λ}JT (C, A)/ ∼,

where ∼ is the smallest equivalence relation under which [figure: S : A; X; Y] [figure: C] is equivalent to [figure: S′ : A; X′; Y′; C] whenever there exists a commutative diagram of the form

[figure: S :; A; X; φ; Y; ψ; C; S′ :; A; X′; Y′; C.]

Further, for any Λ-homomorphisms \xymatrix{α : A \ar[r] & A′} and \xymatrix{γ : C′ \ar[r] & C} and any

object [figure: S : A; X; f; Y; C] of Ext_{Λ}JT (C, A), define αS and Sγ by

αS = αf ∘ \overline{S} and Sγ = f ∘ \overline{S}γ,  (2.1)

respectively. From (1.2), (1.6) and the fact that the above commutative diagram induces the morphism (φ, ψ^{γ}, 1_{C′}) : \overline{S}γ ⟶ \overline{S}′γ, ψ^{γ}(y, c′) = (ψ(y), c′), it follows that these operations make Ext_{Λ} JT (C, A) a functor from (Λ-SMod)^{op} × (Λ-SMod) to Set. Next, ( [figure: 0; Z; Z] ) ∈ � for every Λ-semimodule Z. This together with (1.7) implies that Ext_{Λ} JT (C,−) preserves all finite products [5]. Therefore Ext_{Λ} JT (C, A) is actually an abelian monoid-valued functor, additive in both its arguments. Observe that the addition in Ext_{Λ} JT (C, A) can be described by

cl(S) + cl(S′) = cl(▽_{A}(S ⊕ S′)△_{C}),

where \xymatrix{▽_{A} : A ⊕ A \ar[r] & A} and \xymatrix{△_{C} : C \ar[r] & C ⊕ C} are the codiagonal and diag-

onal maps, respectively. The class of [figure: 0=1_{A} ∘ 0 : A; A; i_{A}; A⊕C; π_{C}; C] coincides with the class of [figure: A; 0; C; C] and serves as a neutral element, i.e., cl(S) + cl(1_{A} ∘ 0) = cl(S).

For every Takahashi extension [figure: F : K; L; M] and every Λ-semimodu-le A, one has a natural connecting homomorphism of abelian monoids

\xymatrix{δ(F, A) : Hom_{Λ}(K, A) \ar[r] & Ext_{Λ} JT (M, A)}

defined by

[figure: δ(F, A)(f : K; A) = cl(A; K; f; L; M)] .

If L is a projective Λ-semimodule, than δ(F, A) is surjective [6]. Indeed, in this case

any object [figure: S : A; X; g; Y; M] of Ext_{Λ}JT (M, A) admits a commutative diagram

[figure: gφ ∘ F :; A; K; gφ; φ; L; ψ; M; S :; A; X; g; Y; M,]

i.e., δ(F, A)(gφ) = cl(S).

Proposition 2.2. Let C be a normal Λ-semimodule and [figure: ℙ : R; μ; P; ε; C] a projective presentation of C. Then \xymatrix{δ(ℙ, A) : Hom_{Λ}(R, A) \ar[r] & Ext_{Λ} JT (C, A)} is a

cokernel of \xymatrix{μ^{∗} = Hom_{Λ}(μ, A) : Hom_{Λ}(P, A) \ar[r] & Hom_{Λ}(R, A)} for every Λ-semi-module A.

Proof. Consider a diagram

\xymatrix{ &  & W \\ Hom_{Λ}(P, A) \ar[r]^{μ^{∗}} & Hom_{Λ}(R, A) \ar[ur]^{ω} \ar[rr]^{δ(ℙ,A)} &  & Ext_{Λ} JT (C, A), \ar@{-->}[ul]_{ω′}}

where ω is a homomorphism of abelian monoids such that ωμ^{∗} = 0. Take cl(S :

[figure: A; X; f; Y; C] ) ∈ Ext_{Λ} JT (C, A). There is a morphism (φ, ψ, 1_{C}) : ℙ ⟶ \overline{S}. Define ω′(cl(S)) = ω(fφ). If (φ′, ψ′, 1_{C}) : ℙ ⟶ \overline{S} is another morphism it follows from Proposition 1.9 that φ + βμ = φ′ + β′μ for some Λ-homomorphisms

\xymatrix{β, β′ : P \ar[r] & X} . Whence

ω(fφ′) = ω(fφ′) + ωμ^{∗}(fβ′) = ω(fφ′ + fβ′μ)
= ω(fφ + fβμ) = ω(fφ) + ωμ^{∗}(fβ) = ω(fφ),

i.e., ω(fφ′) = ω(fφ). On the other hand, a commutative diagram

[figure: S :; A; X; f; g; Y; h; C; S′ :; A; X′; f′; Y′; C,]

where S′ ∈ Ext_{Λ}JT (C, A), yields the morphism (gφ, hψ, 1_{C}) : ℙ ⟶ \overline{S}′. Therefore ω′(cl(S′)) = ω(f′gφ) = ω(fφ). Thus ω′ is well defined. Clearly ω = ω′δ(ℙ, A). This completes the proof since δ(ℙ, A) is surjective.

As a consequence, we obtain

Theorem 2.3. Assume A is a Λ-semimodule and C a normal Λ-semimodule. Let [figure: ℙ : R; μ; P; ε; C] be the chosen projective presentation of C. Then

\xymatrix{θ(C, A) : Ext_{Λ}(C, A) \ar[r] & Ext_{Λ} JT (C, A)}

defined by θ(C, A)([φ]) = cl(φ ∘ ℙ) = cl( [figure: A; R; φ; μ; P; ε; C] ) is a natural isomorphism of abelian monoids.

Proof. Since \xymatrix{δ(ℙ, A) : Hom_{Λ}(R, A) \ar[r] & Ext_{Λ} JT (C, A)} and the canonical pro-

jection \xymatrix{p:Hom_{Λ}(R, A) \ar[r] & Ext_{Λ}(C, A)} are both cokernels of μ^{∗} =Hom_{Λ}(μ, A):

\xymatrix{Hom_{Λ}(P, A) \ar[r] & Hom_{Λ}(R, A)} , we only note that θ(C, A)([φ]) = δ(ℙ, A)(φ) and p(φ) = [φ], and that δ and p are natural in (C, A).

3. In this section we concentrate on Ext_{Λ}(C, A) with A cancellative. It will be shown that this additional condition enables one to give a direct description of Ext_{Λ}(C, A) by Takahashi extensions of C by A (cf. Theorem 2.3).

Let [figure: E : A; λ; B; τ; C] be a short exact sequence of Λ-semimodules and \xymatrix{α : A \ar[r] & A′} a Λ-homomorphism. Following [10], denote by B_{α} the Λ-semimo-dule A′ ⊕ B modulo the following congruence relation: (a′_{1}, b_{1})ρ_{α}(a′_{2}, b_{2}) if there are a_{1}, a_{2} ∈ A such that λ(a_{1}) + b_{1} = λ(a_{2}) + b_{2} and α(a_{2}) + a′_{1} = α(a_{1}) + a′_{2};

and define \xymatrix{λ_{α} : A′ \ar[r] & B_{α}} , \xymatrix{τ_{α} : B_{α} \ar[r] & C} and \xymatrix{ξ_{α} : B \ar[r] & B_{α}} by λ_{α}(a′) =

[a′, 0], τ_{α}([a′, b]) = τ(b) and ξ_{α}(b) = [0, b], respectively.

Proposition 3.1 ([10]). Suppose given a short exact sequence [figure: E : A; λ; B]

[figure: τ; C] of Λ-semimodules and a Λ-homomorphism \xymatrix{α : A \ar[r] & A′} with A′ cancellative. Then

\xymatrix{αE : A′ \ar[r]^{λ_{α}} & B_{α} \ar[r]^{τ_{α}} & C}

is a short exact sequence of Λ-semimodules and (α, ξ_{α}, 1_{C}) a morphism from E to αE. Furthermore, if E ∈� then αE ∈� .

Also note that

E ≡ E′ ⟹ αE ≡ αE′,  (3.2)
E − > E′ ⟹ αE − > αE′.  (3.3)

It is directly verified in [10] that

(α′α)E ≡ α′(αE), 1_{G}E ≡ E and α(Eγ) ≡ (αE)γ,  (3.4)

where E : [figure: G; B; C] is a short exact sequence with G a Λ-module, γ : \xymatrix{C′ \ar[r] & C} a homomorphism of Λ-semimodules, and α : \xymatrix{G \ar[r] & G′} and α′ : \xymatrix{G′ \ar[r] & G′′} are homomorphisms of Λ-modules. These equivalences together with (1.2), (1.3) and (3.2) show that E_{Λ}(C, G) is a functor from (Λ-SMod)^{op} ×(Λ-Mod) to Set (E ⟼ αE, E ⟼ Eγ) [10].

Definition 3.5. We say that a short exact sequence [figure: E : A; λ; B; τ; C] of Λ-semimodules is proper if λ(a) + b_{1} = λ(a) + b_{2}, a ∈ A, b_{1}, b_{2} ∈ B implies b_{1} = b_{2}.

Note that it immediately follows from the definition that if [figure: E:A]\xymatrix{B \ar@{->>}[r] & C}

is proper, then A is a cancellative Λ-semimodule. [figure: 0 : A; i_{A}; A ⊕ C; π_{C}; C] is proper if and only if A is cancellative. Also observe that any short exact sequence [figure: G; B; C] with G a Λ-module is proper.

Proposition 3.6. For every short exact sequence [figure: E : A; λ; B; τ; C] of Λ-se-mimodules and every Λ-homomorphism \xymatrix{α : A \ar[r] & A′} with A′ cancellative, αE :

[figure: A′; λ_{α}; B_{α}; τ_{α}; C] is proper.

Proof. Assume λ_{α}(a′) + [a′_{1}, b_{1}] = λ_{α}(a′) + [a′_{2}, b_{2}], a′, a′_{1}, a′_{2} ∈ A′, b_{1}, b_{2} ∈ B, i.e., [a′ + a′_{1}, b_{1}] = [a′ + a′_{2}, b_{2}]. By definition of ρ_{α}, there are a_{1}, a_{2} ∈ A such that λ(a_{1}) + b_{1} = λ(a_{2}) + b_{2} and α(a_{2}) + a′ + a′_{1} = α(a_{1}) + a′ + a′_{2}. Whence, since A′ is cancellative, λ(a_{1}) + b_{1} = λ(a_{2}) + b_{2} and α(a_{2}) + a′_{1} = α(a_{1}) + a′_{2}. That is, [a′_{1}, b_{1}] = [a′_{2}, b_{2}].

Remark 3.7. For a short exact sequence [figure: E : A; B; C] and a Λ-homo-morphism \xymatrix{γ : C′ \ar[r] & C} , Eγ is proper if and only if λ(a) + b_{1} = λ(a) + b_{2}, a ∈ A, b_{1}, b_{2} ∈ τ^{−1}(γ(C′)) implies b_{1} = b_{2}. In particular, it follows that if E is proper, then Eγ is proper.

Lemma 3.8. Suppose given a short exact sequence [figure: E : A; λ; B; τ; C] of Λ-semimodules and a Λ-homomorphism \xymatrix{α : A \ar[r] & A′} with A′ cancellative. And

assume that [figure: E′ : A′; λ′; B′; τ′; C′] is a proper short exact sequence and that (α, β, γ) is a morphism from E to E′. Then there exists a unique Λ-homomorphism \xymatrix{β′ : B_{α} \ar[r] & B′} such that (1_{A′} , β′, γ) is a morphism from αE to E′ and β = β′ξ_{α}. In particular, if γ = 1_{C} then αE − > E′.

Proof. Define \xymatrix{β′ : B_{α} \ar[r] & B′} by β′([a′, b]) = λ′(a′) + β(b). Assume [a′_{1}, b_{1}] = [a′_{2}, b_{2}], i.e., λ(a_{1}) + b_{1} = λ(a_{2}) + b_{2} and α(a_{2}) + a′_{1} = α(a_{1}) + a′_{2} for some a_{1}, a_{2} ∈ A. Then, since (α, β, γ) : E ⟶ E′ is a morphism, we have λ′α(a_{1}) + β(b_{1}) = λ′α(a_{2}) + β(b_{2}) and λ′α(a_{2}) + λ′(a′_{1}) = λ′α(a_{1}) + λ′(a′_{2}). These equations give λ′α(a_{2}) + λ′(a′_{1}) + β(b_{1}) = λ′α(a_{1}) + λ′(a′_{2}) + β(b_{1}) = λ′α(a_{2}) + λ′(a′_{2}) + β(b_{2}). Whence λ′(a′_{1}) + β(b_{1}) = λ′(a′_{2}) + β(b_{2}) since E′ is proper. Hence β′ is well defined. Clearly, β′ is a Λ-homomorphism with β′λ_{α} = λ′, γτ_{α} = τ′β′ and β = β′ξ_{α}. If \xymatrix{β′′ : B_{α} \ar[r] & B′} is another Λ-homomorphism such that β = β′′ξ_{α} and (1_{A′} , β′′, γ) is a morphism from αE to E′, then β′′([a′, b]) = β′′([a′, 0] + [0, b]) = β′′λ_{α}(a′) + β′′ξ_{α}(b) = λ′(a′) + β(b) = β′([a′, b]).

Let [figure: E : A; λ; B; τ; C] be a short exact sequence with A cancellative, and let \xymatrix{α : A \ar[r] & A′} , \xymatrix{α′ : A′ \ar[r] & A′′} and \xymatrix{γ : C′ \ar[r] & C} be Λ-homomorphisms with A′ and A′′ cancellative. Then Propositions 3.1 and 3.6 and Lemma 3.8 immediately provide the morphisms

(1_{A′′} , ν, 1_{C}) : (α′α)E ⟶ α′(αE), (1_{A}, ξ_{1_{A}} , 1_{C}) : E ⟶ 1_{A}E
and (1_{A′} , ι, 1_{C′}) : α(Eγ) ⟶ (αE)γ,

where \xymatrix{ν:B_{α′α} \ar[r] & (B_{α})_{α′}} and \xymatrix{ι:(B^{γ})_{α} \ar[r] & (B_{α})^{γ}} are defined by ν([a′′, b])= [a′′, [0, b]] and ι([a′, (b, c′)]) = ([a′, b], c′), respectively. Thus

(α′α)E − > α′(αE), E − > 1_{A}E and α(Eγ) − > (αE)γ.  (3.9)

Remark 3.10. One can easily verify that ν and ι are in fact Λ-isomorphisms. Hence (α′α)E ≡ α′(αE), E − > 1_{A}E and α(Eγ) ≡ (αE)γ (cf. (3.4)). Furthermore, ξ_{1_{A}} is a Λ-isomorphism if and only if E is proper. (Indeed, assume that E is proper.

Let ξ_{1_{A}} (b_{1}) = ξ_{1_{A}} (b_{2}), i.e., [0, b_{1}] = [0, b_{2}]. Then λ(a_{1}) + b_{1} = λ(a_{2}) + b_{2} and 1_{A}(a_{2})+0 = 1_{A}(a_{1})+0 for some a_{1}, a_{2} ∈ A. Whence, since E is proper, b_{1} = b_{2}. On the other hand, ξ_{1_{A}} is always surjective ([a, b] = [0, λ(a)+b] = ξ_{1_{A}} (λ(a)+b)). Hence ξ_{1_{A}} is a Λ-isomorphism. The converse immediately follows from Proposition 3.6.) Therefore 1_{A}E ≡ E if and only if E is proper. Denote by E_{Λ}P (C, A) the set of ≡- equivalence classes of proper Takahashi extensions of C by A. Then, by Proposition 3.6 and Remark 3.7, E_{Λ}P (C, A) is a functor from (Λ-SMod)^{op}×(Λ-CSMod) to Set which canonically extends the functor E_{Λ}(−, −) : (Λ-SMod)^{op} ×(Λ-Mod) ⟶Set.

Now let Ext_{Λ}T (C, A) be the category of Takahashi extensions of a Λ-semimodule C by a cancellative Λ-semimodule A. Define

Ext_{Λ} T (C, A) = Ext_{Λ}T (C, A)/⟨ − > ⟩ ,

where ⟨ − > ⟩ is the smallest equivalence relation containing the relation − >. By (1.2), (1.4), (1.6), Proposition 3.1, (3.3) and (3.9), the rules E ⟼ Eγ and E ⟼ αE make Ext_{Λ} T (C, A) a functor from (Λ-SMod)^{op} × (Λ-CSMod) to Set.

Remark 3.11. It follows from Proposition 3.6 and Remark 3.7 that one can similarly introduce the functor

Ext_{Λ} P T (C, A) = Ext_{Λ}P T (C, A)/⟨ − > ⟩ ,

where Ext_{Λ}P T (C, A) denotes the category of proper Takahashi extensions of C by A. Obviously, the maps

\xymatrix{Ext_{Λ} P T (C, A) \ar[r]^{Γ(C,A)} & Ext_{Λ} T (C, A) \ar[l]^{Γ′(C,A)}} , Γ(cl(E))=cl(E), Γ′(cl(E))=cl(1_{A}E)

are natural, and Γ′Γ = 1 and ΓΓ′ = 1.

In order to prove the following theorem, note that

α ∘ E ∼ 1_{A′} ∘ αE  (3.12)

for any short exact sequence [figure: E : A; λ; B; τ; C] and any Λ-homomorphism \xymatrix{α : A \ar[r] & A′} with A′ cancellative. Indeed, the morphism (α, ξ_{α}, 1_{C}) : E ⟶ αE gives the commutative diagram

[figure: α ∘ E :; A′; A; α; λ; α; B; τ; ξ_{α}; C; 1_{A′} ∘ αE :; A′; A′; λ_{α}; B_{α}; τ_{α}; C.]

Theorem 3.13. Let C be a Λ-semimodule and A a cancellative Λ-semimodule. Then

\xymatrix{χ(C, A) : Ext_{Λ} T (C, A) \ar[r] & Ext_{Λ} JT (C, A)}

defined by

[figure: χ(C, A)(cl(E:A; B; C))=cl(1_{A} ∘ E: A; A; B; C)]

is a natural bijection.

Proof. Define

\xymatrix{χ′(C, A) : Ext_{Λ} JT (C, A) \ar[r] & Ext_{Λ} T (C, A)}

by

[figure: χ′(C, A)(cl(S : A; X; f; κ; Y; σ; C)) = cl(f\overline{S}).]

This definition is independent of the chosen representative sequence S. Indeed, suppose given a commutative diagram

[figure: S :; A; X; f; κ; φ; Y; σ; ψ; C; S′ :; A; X′; g; κ′; Y′; σ′; C]

with S′ ∈ Ext_{Λ}JT (C, A). This commutative diagram and the morphism (g, ξ_{g}, 1_{C}) : \overline{S}′ ⟶ g\overline{S}′ yield the following commutative diagram

[figure: \overline{S} :; X; κ; f; Y; σ; ξ_{g} ψ; C; g\overline{S}′ :; A; κ′_{g}; Y′_{g}; σ′_{g}; C.]

Whence, by Proposition 3.6 and Lemma 3.8, f\overline{S} − > g\overline{S}′. Hence χ′(C, A) is well defined. Further, by (3.12), χχ′(cl(S)) = χ(cl(f\overline{S})) = cl(1_{A} ∘ f\overline{S}) = cl(f ∘ \overline{S}) = cl(S). On the other hand, χ′χ(cl(E)) = χ′(cl(1_{A} ∘ E)) = cl(1_{A}E) = cl(E) since E − > 1_{A}E. Thus χ(C, A) is a bijection. Finally, consider the diagram

\xymatrix{Ext_{Λ} T (C, A) \ar[r]^{χ(C,A)} \ar[d]_{Ext_{Λ} T (γ,α)} & Ext_{Λ} JT (C, A) \ar[d]^{Ext_{Λ} JT (γ,α)} \\ Ext_{Λ} T (C′, A′) \ar[r]^{χ(C′,A′)} & Ext_{Λ} JT (C′, A′),}

where \xymatrix{α : A \ar[r] & A′} is a homomorphism of cancellative Λ-semimodules and γ : \xymatrix{C′ \ar[r] & C} a homomorphism of Λ-semimodules. Using (3.12) and (2.1), we obtain

χ(C′, A′) Ext_{Λ} T (γ, α)(cl(E)) = cl(1_{A′} ∘ αEγ) = cl(α ∘ Eγ)
= cl((α ∘ E)γ) = cl(α(1_{A} ∘ E)γ) = Ext_{Λ} JT (γ, α) χ(C, A)(cl(E)),

i.e., the diagram is commutative. Thus χ(C, A) is natural.

Remark 3.14. More general results than Theorem 3.13 are discussed in [5,6]. However Theorem 3.13 is not merely a consequence of those since the span of Takahashi extensions is not regular in the sense of N. Yoneda [12].

It is evident that Theorem 3.13 remains valid for any class � of short exact sequences [figure: E : A; B; C] with A cancellative which satisfies the following

conditions: [figure: (0; Z; Z) ∈�] for every Λ-semimodule Z; if E ∈� then

αE, Eγ ∈� .

Theorem 3.13 shows that Ext_{Λ} T (C, A) is in fact an abelian monoid-valued functor, additive in each of its arguments; the addition in Ext_{Λ} T (C, A) obviously coincides with the Bear addition:

cl(E) + cl(E′) = cl( ▽_{A} (E ⊕ E′) △_{C} ).

As a corollary of Theorems 2.3 and 3.13 we have

Theorem 3.15. Assume A is a cancellative Λ-semimodule and C a normal Λ-semi-module. Let [figure: ℙ : R; ν; P; ε; C] be the chosen projective presentation of C. Then

\xymatrix{w(C, A) : Ext_{Λ}(C, A) \ar[r] & Ext_{Λ} T (C, A)}

defined by w(C, A)([φ]) = cl(φℙ) is a natural isomorphism of abelian monoids.

Proof. w(C, A) = χ′(C, A)θ(C, A).

Let G and H be Λ-modules, i.e., K(Λ)-modules. It immediately follows from Proposition 1.1 that

Ext_{Λ} T (H, G) = Ext1_{K(Λ)}(H, G) = E_{Λ}(H, G),

where Ext1_{K(Λ)} is the usual Ext functor. From this and Theorem 3.13 one has

Corollary 3.16. Let G and H be Λ-modules. The map

\xymatrix{Ext1_{K(Λ)}(H, G) \ar[r] & Ext_{Λ} JT (H, G)} ,

[figure: cl(G; B; H) ⟼ cl(G; G; B; H)]

is a natural isomorphism of abelian groups.

Note that in [6] Janelidze proved this for Λ = N, the semiring of non-negative integers.

Remark 3.17. Let Λ and Λ′ be additively cancellative semirings. In [8], for any contravariant additive functor T : (Λ-CSMod) [figure: (Λ′] -CSMod), we constructed

and studied right derived functors ℝ^{n}T : (Λ-CSMod) [figure: (Λ′] -CSMod), n = 0, 1, 2, . . .. In particular, we described ℝ^{n} Hom_{Λ}(−, A)(C) by means of certain nfold extensions of C by A. According to that description, for n = 1, H ∈ (Λ-Mod) and A ∈ (Λ-CSMod), one has a natural isomorphism of abelian groups

ℝ^{1} Hom_{Λ}(−, A)(H) ≅ Ext1_{Λ}(H, A)/⟨ − > ⟩,

where Ext1_{Λ}(H, A) denotes the category of short exact sequences [figure: A; B; H] of Λ-semimodules with B cancellative. (Note that R^{1} Hom_{Λ}(−, A)(H)=R^{1} Hom_{Λ}(−, U(A))(H) ≅ Ext1_{K(Λ)}(H, U(A)), where U(A) is the maximal Λ-submodule of A, and R^{1} Hom_{Λ}(−, A) denotes the usual right derived functor of the functor Hom_{Λ}(−, A) : (Λ-Mod) [figure]Ab.) On the other hand, if [figure: A; κ; X; σ; H] is a proper short exact sequence of Λ-semimodules with H ∈ (Λ-Mod), then X is cancellative. (To see this, suppose x + x_{1} = x + x_{2}, x, x_{1}, x_{2} ∈ X. Take x′ ∈ X such that σ(x′) = −σ(x). Then x′ + x = κ(a) for some a ∈ A; and we obtain κ(a) + x_{1} = x′ + x + x_{1} = x′ + x + x_{2} = κ(a) + x_{2}. Whence x_{1} = x_{2} since [figure: A; κ; X; σ; H] is proper.) Hence there is a natural isomorphism

ℝ^{1} Hom_{Λ}(−, A)(H) ≅ Ext_{Λ} P T (H, A).

Consequently, by Remark 3.11 and Theorems 3.13 and 3.15, each of the abelian groups Ext_{Λ} JT (H, A), Ext_{Λ} T (H, A) and Ext_{Λ}(H, A), where H is a Λ-module and A a cancellative Λ-semimodule, is naturally isomorphic to ℝ^{1} Hom_{Λ}(−, A)(H).

4. Let G be a Λ-module. In this section, continuing the investigation started in [11], we obtain some results relating Ext_{Λ}(C, G) and E_{Λ}(C, G). Besides, we relate Ext_{Λ}(C, G) with Inassaridze extensions of C by G [4], and also with S^{1} Hom_{K(Λ)}(−,G) (K(C)), where K(C) denotes the Grothendieck K(Λ)-module of C.

First of all, observe that the Baer addition of extensions, E+E′ = ▽_{G}(E⊕E′)△_{C}, makes E_{Λ}(C, G) an abelian monoid [4]. In addition the ≡-equivalence class of

[figure: 0 : G; i_{G}; G ⊕ C; π_{C}; C] serves as a neutral element. Furthermore, a straightforward verification shows that α(E + E′) ≡ αE + αE′, (E + E′)γ ≡ Eγ + E′γ, α0 ≡ 0 and 0γ ≡ 0. Thus E_{Λ}(C, G) is in fact a functor from (Λ-SMod)^{op} × (Λ-Mod) to Abm.

We call a Takahashi extension [figure: E : G; λ; B; τ; C] of a Λ-semimodule C by a Λ-module G an Inassaridze extension if E is an extension of C by G in the sense of H. Inassaridze [4]: whenever the equality λ(g) + b = b holds for some g ∈ G, b ∈ B, then g = 0. Let Ext_{Λ} I(C, G) denote the set of ≡-equivalence classes of Inassaridze extension of C by G. It is shown in [4] that Ext_{Λ} I(C, G) = U(E_{Λ}(C, G)), the group of units of E_{Λ}(C, G). This in particular means that Ext_{Λ} I(C, G) is an abelian group-valued subfunctor of E_{Λ}(C, G). Moreover, Ext_{Λ} I(C, G) is additive in both its arguments.

A short exact sequence of Λ-semimodules [figure: E : A; λ; B; τ; C] is said to be split if there exists a Λ-homomorphism \xymatrix{ν : C \ar[r] & B} such that τν = 1. Let E_{Λ}S(C, G) denote the set of ≡-equivalence classes of split Takahashi extensions of a Λ-semimodule C by a Λ-module G. It is easy to see that αE, Eγ and E ⊕ E′ are split whenever E and E′ are split. Consequently, E_{Λ}S(C, G) is another subfunctor of the functor E_{Λ}(C, G).

We shall need the following four facts.

Proposition 4.1 ([4]). Suppose given a morphism of the form

[figure: E :; G; λ; B; τ; β; C; E′ :; G; λ′; B′; τ′; C]

of Takahashi extensions of a Λ-semimodule C by a Λ-module G. If E′ is an Inassaridze extension, then β is an isomorphism.

Theorem 4.2 ([4,7]). Let K(C) be the Grothendieck K(Λ)-module of a Λ-semimo-dule C, and \xymatrix{k_{C} : C \ar[r] & K(C)} the canonical Λ-homomorphism. Then the natural homomorphism

\xymatrix{Ext1_{K(Λ)}(K(C), G) = Ext_{Λ} I(K(C), G) \ar[r]^{Ext_{Λ} I(k_{C} ,G)} & Ext_{Λ} I(C, G)}

is an isomorphism.

Proposition 4.3 ([7]). An Inassaridze extension [figure: E : G; B; C] is split

if and only if [figure: E ≡ 0 : G; i_{G}; G ⊕ C; π_{C}; C] .

Proposition 4.4. Let [figure: E : A; λ; B; τ; C] be a split short exact sequence of Λ-semimodules, and \xymatrix{ν : C \ar[r] & B} a splitting Λ-homomorphism, i.e., τν = 1. If C is a Λ-module, then the map

\xymatrix{q : A ⊕ C \ar[r] & B} , q(a, c) = λ(a) + ν(c)

is a Λ-isomorphism.

Proof. Define \xymatrix{q′ : B \ar[r] & A ⊕ C} by q′(b) = (λ^{−1}(b − ντ(b)), τ(b)). Then q′q = 1 and qq′ = 1.

For any Λ-module G and any normal Λ-semimodule C, M. Takahashi has defined a pair of maps

\xymatrix{E_{Λ}(C, G) \ar[r]^{η(C,G)} & Ext_{Λ}(C, G) \ar[l]^{ζ(C,G)}}

as follows. ζ(C, G)([f]) = cl(fℙ), where [figure: ℙ : R; μ; P; ε; C] is the chosen projective presentation of C. Next, take cl(E) ∈ E_{Λ}(C, G). There is a morphism (φ, ψ, 1) : ℙ ⟶ E. And define η(C, G)(cl(E)) = [φ]. It is shown in [11] that η(C, G) and ζ(C, G) are well defined, and η(C, G) is natural in each of its arguments, and η(C, G)ζ(C, G) = 1.

Observe that the surjection η(C, G) is in fact a homomorphism. In order to see this, consider the diagram

\xymatrix{E_{Λ}(C, G) \ar[rr]^{η(C,G)} \ar[dr]_{m(C,G)} &  & Ext_{Λ}(C, G) \ar[dl]^{w(C,G)} \\  & Ext_{Λ} T (C, G)}

where m(C, G) is defined by m(C, G)(cl(E))=cl_{⟨− >⟩}(E), and w(C, G) by w(C, G)([φ]) = cl_{⟨− >⟩}(φℙ) (see Theorem 3.15). One has w(C, G)η(C, G) (cl(E)) = w(C, G)([φ]) = cl_{⟨− >⟩}(φℙ). But, by Lemma 3.8, φℙ − > E, that is, cl_{⟨− >⟩}(φℙ) = cl_{⟨− >⟩}(E). Hence the diagram is commutative. Therefore, since m(C, G) is a homomorphism and w(C, G) an isomorphism, η(C, G) is a homomorphism.

Proposition 4.5. For any Λ-module G and any normal Λ-semimodule C,

Ker(μ(C, G)) = E_{Λ}S(C, G),

that is, the sequence

\xymatrix{E_{Λ}S(C, G) \ar[r]^{j(C,G)} & E_{Λ}(C, G) \ar[r]^{η(C,G)} & Ext_{Λ}(C, G)} ,  (4.6)

where j(C, G) denotes the inclusion, is a short exact sequence of abelian monoids.

Proof. Let cl( [figure: E : G; λ; B; τ; C] ) ∈ E_{Λ}S(C, G). If \xymatrix{ν : C \ar[r] & B} is a splitting Λ-homomorphism, then the diagram

[figure: ℙ :; R; μ; 0; P; ε; νε; C; E :; G; λ; B; τ; C]

is commutative. Whence, by definition of η(C, G), η(C, G)(cl(E)) = [0] = 0. Con-

versely, suppose cl( [figure: E : G; λ; B; τ; C] ) ∈ Ker(η(C, G)). Assume (φ, ψ, 1) is a morphism from ℙ to E. Then η(C, G)(cl(E)) = [φ] = 0, i.e., there exists a Λhomomorphism \xymatrix{g : P \ar[r] & G} such that φ = gμ. From this it follows that the diagram

[figure: ℙ :; R; μ; φ; P; ε; β; C; 0 :; G; i_{G}; G ⊕ C; π_{C}; C,]

where β = i_{G}g + i_{C} ε, is commutative. By Lemma 3.8, this commutative diagram and the morphism (φ, ψ, 1) : ℙ ⟶ E yield the commutative diagrams

[figure: φℙ :; G; μ_{φ}; P_{φ}; ε_{φ}; β_{1}; C; 0 :; G; i_{G}; G ⊕ C; π_{C}; C; and; φℙ :; G; μ_{φ}; P_{φ}; ε_{φ}; β_{2}; C; E :; G; λ; B; τ; C,]

respectively. Moreover, by Proposition 4.1, β_{1} is a Λ-isomorphism. Then one can

write τβ_{2}β_{1}^{−1}i_{C} = ε_{φ}β_{1}^{−1}i_{C} = π_{C} i_{C} = 1_{C}. Hence \xymatrix{β_{2}β_{1}^{−1}i_{C} : C \ar[r] & B} is a split-

ting Λ-homomorphism for E. Consequently, cl(E) ∈ E_{Λ}S(C, G). Corollary 4.7. The restriction of η(C, G) to Ext_{Λ} I(C, G), that is, the natural homomorphism

\xymatrix{η(C, G) : Ext_{Λ} I(C, G) \ar[r] & Ext_{Λ}(C, G)}

of abelian groups is one-to-one. Proof. Suppose cl(E) is contained in the kernel of this homomorphism. It then follows from Proposition 4.5 that E is split. Hence, by Proposition 4.3, cl(E)=0.

This corollary allows to relate S^{1} Hom_{K(Λ)}(−, G)(K(C)) and Ext_{Λ}(C, G) (cf. (1.10)) as follows. Let [figure: 𝔽 : L; F; K(C)] be the chosen K(Λ)-projective presentation of K(C). Define

\xymatrix{S(C, G) : S^{1} Hom_{K(Λ)}(−, G)(K(C)) \ar[r] & Ext_{Λ}(C, G)}

by S(C, G)([ \xymatrix{f : L \ar[r] & G} ]) = [fφ], where \xymatrix{φ : R \ar[r] & L} is any Λ-homomor-phism such that (φ, ψ, k_{C}) is a morphism from the Λ-projective presentation ℙ : [figure: R; μ; P; ε; C] to 𝔽. By Proposition 1.9, S(C, G) is well defined. Clearly S(C, G) = Ext_{Λ}(k_{C} , G)(K(K(C), G))^{−1} (see (1.11)). Corollary 4.8. For any Λ-module G and any normal Λ-semimodule C, S(C, G) is an injective natural homomorphism. Proof. Consider the diagram

[figure: S^{1} Hom_{K(Λ)}(−, G)(K(C)); S(C,G); r(K(C),G); Ext_{Λ}(C, G); Ext_{Λ} I(K(C), G) = Ext1_{K(Λ)}(K(C), G); Ext_{Λ} I(k_{C} ,G); Ext_{Λ} I(C, G),; η(C,G)]  (4.9)

where r(K(C), G), defined by r(K(C), G)([ \xymatrix{g : L \ar[r] & G} ]) = cl(g𝔽), is a wellknown natural isomorphism. Let \xymatrix{f : L \ar[r] & G} be a homomorphism of Λ-modules and (φ, ψ, k_{C}) a morphism from ℙ to 𝔽. By (3.4), (f𝔽)k_{C} ≡ f(𝔽k_{C}). Besides, it is easy to see that (φ, ψ, k_{C}) and f give the morphism (fφ, ψ′, 1_{C}) : ℙ ⟶ f(𝔽k_{C}), where \xymatrix{ψ′ : P \ar[r] & (F^{k_{C}})_{f}} is defined by ψ′(p) = [0, (ψ(p), ε(p))]. Consequently, one can write

η(C, G) Ext_{Λ} I(k_{C} , G) r(K(C), G)([ \xymatrix{f : L \ar[r] & G} ])

= η(C, G) Ext_{Λ} I(k_{C} , G)(cl(f𝔽)) = η(C, G)(cl((f𝔽)k_{C}))

= η(C, G)(cl((f(𝔽k_{C}))) = [fφ] = S(C, G)([ \xymatrix{f : L \ar[r] & G} ]).

That is, the diagram is commutative. It then follows from Theorem 4.2 and Corollary 4.7 that S(C, G) is an injective natural homomorphism.

Before discussing the following results, recall that a semiring Λ = (Λ, +, 0, ., 1) is called additively cancellative if (Λ, +, 0) is cancellative (e.g., N, the semiring of nonnegative integers). In this case every projective Λ-semimodule is obviously cancellative. Moreover, if Λ is additively cancellative, then every normal Λ-semimodule C is cancellative. (Indeed, let [figure: M; κ; Q; σ; C] be a projective presentation of C. Suppose c_{1} + c = c_{2} + c, c, c_{1}, c_{2} ∈ C. Take q, q_{1}, q_{2} ∈ Q so that σ(q) = c, σ(q_{1}) = c_{1} and σ(q_{2}) = c_{2}. Then σ(q_{1} + q) = σ(q_{2} + q). Hence κ(m_{1}) + q_{1} + q = κ(m_{2}) + q_{2} + q for some m_{1}, m_{2} ∈ M. Whence, since Q is cancellative, κ(m_{1}) + q_{1} = κ(m_{2}) + q_{2}. Therefore σ(q_{1}) = σ(q_{2}), i.e., c_{1} = c_{2}.)

Theorem 4.10. If Λ is an additively cancellative semiring, then the natural map

\xymatrix{η(C, G) : Ext_{Λ} I(C, G) \ar[r] & Ext_{Λ}(C, G)}

is an isomorphism for any Λ-module G and any normal Λ-semimodule C.

Proof. The chosen projective presentation [figure: ℙ : R; μ; P; ε; C] of C and a Λ-ho-

momorphism \xymatrix{φ:R \ar[r] & G} give the Takahashi extension [figure: φℙ:G; μ_{φ}]\xymatrix{P_{φ} \ar@{->>}[r]^{ε_{φ}} & C} and the morphism (φ, ξ_{φ}, 1_{C}) : ℙ ⟶ φℙ (see Proposition 3.1). Suppose μ_{φ}(g) + [h, p] = [h, p], g, h ∈ G, p ∈ P, i.e., [g + h, p] = [h, p]. Then μ(r_{1}) + p = μ(r_{2}) + p and φ(r_{2}) + g + h = φ(r_{1}) + h for some r_{1}, r_{2} ∈ R. These two equations imply g = 0 since P is cancellative. Hence φℙ is an Inassaridze extension. On the other hand, by definition of η(C, G), one has η(C, G)(cl(φℙ)) = [φ]. Thus \xymatrix{η(C, G) : Ext_{Λ} I(C, G) \ar[r] & Ext_{Λ}(C, G)} is surjective. This together with Corollary 4.7 gives the desired result.

Theorems 4.2 and 4.10 and the commutative diagram (4.9) (see the proof of Corollary 4.8) yield

Theorem 4.11. If Λ is an additively cancellative semiring, then the natural map

\xymatrix{S(C, G) : S^{1} Hom_{K(Λ)}(−, G)(K(C)) \ar[r] & Ext_{Λ}(C, G)}

is an isomorphism for any Λ-module G and any normal Λ-semimodule C.

We have already mentioned that \xymatrix{ζ(C, G) : Ext_{Λ}(C, G) \ar[r] & E_{Λ}(C, G)} is defined by ζ(C, G)([φ]) = cl(φℙ) and that η(C, G)ζ(C, G) = 1. On the other hand, the proof of Theorem 4.10 shows that if Λ is an additively cancellative semiring, then ζ(C, G) maps Ext_{Λ}(C, G) into Ext_{Λ} I(C, G). Hence, by Theorem 4.10, ζ(C, G) is the two-sided inverse for \xymatrix{η(C, G):Ext_{Λ} I(C, G) \ar[r] & Ext_{Λ}(C, G)} whenever Λ is additively cancellative. Consequently, under the given hypothesis, ζ(C, G) is a splitting homomorphism for the short exact sequence (4.6). Thus, by Proposition 4.4, one has

Theorem 4.12. Let Λ be an additively cancellative semiring. Suppose G is a Λ-module and C a normal Λ-semimodule. Then the map

\xymatrix{E_{Λ}S(C, G) ⊕ Ext_{Λ}(C, G) \ar[r] & E_{Λ}(C, G)} , (cl(E), [φ]) ⟼ cl(E) + cl(φℙ)

is a natural isomorphism of abelian monoids.

This theorem together with Theorems 4.10 and 4.2 gives

Corollary 4.13. Let Λ be an additively cancellative semiring. Assume G is a Λ-module and C a normal Λ-semimodule. Then the maps

\xymatrix{E_{Λ}S(C, G) ⊕ Ext_{Λ} I(C, G) \ar[r] & E_{Λ}(C, G)} , (cl(E), cl(E′)) ⟼ cl(E) + cl(E′)

and

E_{Λ}S(C, G) ⊕ Ext1_{K(Λ)}(K(C), G) ⟶ E_{Λ}(C, G), (cl(E), cl(T)) ↦ cl(E) + cl(T k_{C})

are natural isomorphisms of abelian monoids.

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Alex Patchkoria lexo@rmi.acnet.ge

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 0193 Georgia