EXTENSIONS OF SEMIMODULES AND THE TAKAHASHI FUNCTOR Ext

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-semimoduleX by a Λ-semimoduleY we mean an extension ofX byY in the sense of M. Takahashi [10]. LetA be an ar- bitrary Λ-semimodule andCa Λ-semimodule which is normal in Takahashi’s sense, that is, there exist a projective Λ-semi- module P and a surjective Λ-homomorphism ε:P //C such thatε is a cokernel of the inclusionµ : 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 Λ- semimodules by Λ-modules.

In this paper we relate Ext_Λ(C, A) with other known satel- lite functors of the functor Hom_Λ(−, A).

Section 1 is concerned with preliminaries. The purpose of Section 2 is to charac- terize Ext_Λ(C, A) in terms of Janelidze’s general ExtCⁿ -functors [5]. In Section 3 we show that Ext_Λ(C, A) withA cancellative can be described directly by Takahashi extensions ofCbyA. The last section is devoted to Ext_Λ(C, G) withGa Λ-module.

We relate Ext_Λ(C, G) with Inassaridze extensions ofCbyG[4]. This allows to relate Ext_Λ(C, G) andS¹Hom_K(Λ)(−, G)(K(C)), whereK(Λ) is the Grothendieck ring of Λ, K(C) the Grothendieck K(Λ)-module of C, and S¹Hom_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, ex- tension of semimodule, satellite functor.

c

°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·(λ⁰+λ⁰⁰) =λ·λ⁰+λ·λ⁰⁰, (λ⁰+λ⁰⁰)·λ=λ⁰·λ+λ⁰⁰·λ, λ·0 = 0·λ= 0,

for all λ, λ⁰, λ⁰⁰ ∈ Λ. An abelian monoid A = (A,+,0) together with a map Λ×A //A, written as (λ, a)7−→λa, is called a (left) Λ-semimodule if

λ(a+a⁰) =λa+λa⁰, (λ+λ⁰)a=λa+λ⁰a, (λ·λ⁰)a=λ(λ⁰a), 1a=a, 0a= 0,

for allλ, λ⁰∈Λ and a, a⁰∈A. It immediately follows thatλ0 = 0 for anyλ∈Λ.

Let us also recall:

A map f :A //B between Λ-semimodules A and B is called a Λ-homo- morphism if f(a+a⁰) = f(a) +f(a⁰) and f(λa) = λf(a), for all a, a⁰ ∈ A and λ ∈ Λ. It is obvious that any Λ-homomorphism carries 0 into 0. The abelian monoid of all Λ-homomorphisms fromAtoBis denoted by Hom_Λ(A, B). (Example:

LetN be the semiring of non-negative integers. AnN-semimoduleA is simply an abelian monoid, and anN-homomorphism f :A //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 andλ∈Λ. Clearly 0 ∈A. The quotient Λ-semimodule B/Ais 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 ofb∈B by [b]. Then [b1] = [b2] if and only ifa1+b1=a2+b2 for somea1, a2∈A.

The Λ-homomorphism p:B //B/A that carries b ∈ B into [b] is called the canonical surjection.

A Λ-semimodule A is cancellative if a+a⁰ = a+a⁰⁰ for a, a⁰, a⁰⁰ ∈ A implies a⁰=a⁰⁰. Obviously,Ais a cancellative Λ-semimodule if and only ifAis a cancellative C(Λ)-semimodule, whereC(Λ) denotes the largest cancellative homomorphic image of Λ under addition. A Λ-semimoduleAis called a Λ-module ifA= (A,+,0) is an abelian group. It is clear that Ais a Λ-module if and only ifAis aK(Λ)-module, whereK(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, andSet, respectively.

A cokernel of a Λ-homomorphism f :A //B is defined to be a Λ-homo- morphism u:B //C such that (i) uf = 0, and (ii) for any Λ-homomorphism g:B //D with gf = 0 there is a unique Λ-homomorphism g⁰:C //D

with g=g⁰u. One dually defines a kernel of f. Clearly, the canonical projection p:B //B/f(A) is a cokernel of f, and the inclusion Ker(f) ,→ A, where Ker(f) =©

a∈A|f(a) = 0ª

, is a kernel off.

A sequence E:A // ^λ //B ^τ ////C of Λ-semimodules and Λ-homomorphisms is called ashort exact sequence ifλis injective,τ is surjective, andλ(A) = Ker(τ) (cf. [9]). The following assertion is plain and well-known.

Proposition 1.1. If E:A // //B ////C is a short exact sequence, thenB is aΛ-module if and only if AandC are both Λ-modules.

A morphism from E:A // ^λ //B ^τ ////C to E⁰ :A⁰ // ^λ0 //B^{0 τ0} ////C⁰ is a triple of Λ-homomorphisms (α, β, γ) such that

E: A

α

²² // ^λ //B ^τ ////

β

²²

C

γ

²²E⁰: A⁰ // ^λ0 //B^{0 τ0} ////C⁰

is a commutative diagram. For a morphism of the form E: A

1_A

// //B ////

β

²²

C

1_C

E⁰: A // //B⁰ ////C,

we writeE> E− ⁰. If in additionβis a Λ-isomorphism, we writeE≡E⁰and say that E isequivalenttoE⁰.

Next, suppose given a short exact sequence E:A // ^λ //B ^τ ////C and a Λ-ho- momorphism γ:C⁰ //C. Then

Eγ:A // ^λγ //B^{γ τ}

γ ////C⁰,

where B^γ = ©

(b, c⁰) ∈ B⊕C⁰|τ(b) = γ(c⁰)ª

, λ^γ(a) = (λa,0), τ^γ(b, c⁰) = c⁰, is a short exact sequence of Λ-semimodules. Besides, if one defines a Λ-homomorphism ξ^γ:B^γ //B by ξ^γ(b, c⁰) = b, then (1_A, ξ^γ, γ) is a morphism from Eγ to E.

From the construction ofEγit follows that

E≡E1_C, (Eγ)γ⁰≡E(γγ⁰), (1.2)

E≡E⁰=⇒Eγ≡E⁰γ, (1.3)

E> E− ⁰ =⇒Eγ> E− ⁰γ. (1.4) We will also use sequences of the form S :A oo ^f X // //Y ////C , where f :X //A is a Λ-homomorphism and E:X // //Y ////C a short exact sequence of Λ-semimodules. It will be convenient to denoteS byf◦E, andEbyS.

A surjective Λ-homomorphism τ:B //C is said to be anormalΛ-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(b1) =τ(b2), then k1+b1=k2+b2 for somek1, k2 in Ker(τ).

Proposition 1.5. Any surjective Λ-homomorphism τ:B //H with H a Λ-module is normal.

Proof. Suppose τ(b1) = τ(b2). Take b ∈ B with τ(b) = −τ(b1). Then (b2+ b), (b1+ b)∈Ker(τ) and (b2+b) +b1= (b1+b) +b2.

Note also that for any Λ-subsemimoduleA of a Λ-semimoduleB, the canonical projection p:B //B/A is normal.

LetAand C be Λ-semimodules. By aTakahashi (ornormal)extensionofC by A we mean an extension of C byA in the sense of [10], that is, a short exact se- quence E:A //^λ //B ^τ ////C of Λ-semimodules with τ normal. Clearly, a short exact sequence of Λ-semimodules A // //B ////H with H a Λ-module and a sequence of the form Ker(p),→ B ^p //B/A provide examples of Takahashi ex- tensions. (Note that in general Ker(p)6=A.) Let

T

denote the class of all Takahashi extensions of Λ-semimodules. Then

E∈

T

^=⇒Eγ∈

T

^{, (1.6)}

E, E⁰∈

T

^{=⇒E⊕E0 ∈}

T

^{. (1.7)}

Here E⊕E⁰ denotes A⊕A⁰ //^λ⊕λ0//B⊕B^{0 τ⊕τ}

0////C⊕C⁰, the usual direct sum of E and E⁰. Two extensions E1 and E2 are equivalent if E1 ≡ E2, i.e., if they are equivalent as short exact sequences. Following [10], we denote byE_Λ(C, A) the set of equivalence classes of Takahashi extensions ofCbyA. It contains at least the 0, the class of

0 :A // ^iA //A⊕C ^πC ////C, wherei_A(a) = (a,0) and π_C(a, c) =c.

A Λ-semimoduleP is projectiveif it satisfies the usual lifting property: Given a surjective Λ-homomorphism τ :B //C and a Λ-homomorphismf :P //C, there is a Λ-homomorphism g:P //B such thatf =τ g.

Proposition 1.8. Let τ :B //C be a normalΛ-epimorphism and let f₁, f₂ : P //B be Λ-homomorphisms with P projective. If τ f₁ =τ f₂ then there exist Λ-homomorphisms g₁, g₂:P //B satisfying τ g₁ = 0 = τ g₂ and g₁ +f₁ = g₂+f₂. 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 ofΛ-semimodules andΛ-ho-

momorphisms

A⁰

ϕ₁

²²

ϕ₂

²²

λ⁰ //P

g₂

ÄÄÄÄÄÄÄÄÄÄ

g₁

ÄÄÄÄÄÄÄÄÄÄ

ψ₁

²²

ψ₂

²²

τ⁰ //C⁰

γ

²²

A // ^λ //B ^τ ////C,

where the bottom row is a Takahashi extension,P is projective,τ⁰λ⁰= 0, andτ ψ_i= γτ⁰,λϕ_i =ψ_iλ⁰ fori= 1,2. Then there areΛ-homomorphismsg₁, g₂ :P //A such that g₁λ⁰+ϕ₁ =g₂λ⁰+ϕ₂.

A Λ-semimodule C is called normal if there exist a projective Λ-semimodule P and a normal Λ-epimorphism ε:P //C [11]. In other words, C is normal if there is a Takahashi extension of Λ-semimodules R // //P ////C with P projective, called aprojective presentationofC. It follows from Proposition 1.5 that every Λ-moduleH is normal, since one has a free Λ-semimoduleF and a surjective Λ-homomorphism F //H . Any quotient Λ-semimodule P/A of a projective Λ-semimoduleP 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 P:R //^µ //

P ^ε ////C ofCand define Ext_Λ(C, A) to be Coker(Hom_Λ(µ, A) : 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 α:A //A⁰ is a homomorphism of Λ-semimodules, one defines Ext_Λ(C, α) : Ext_Λ(C, A) //Ext_Λ(C, A⁰) by Ext_Λ(C, α)([ϕ]) = [αϕ]. Obviously, Ext_Λ(C, α) is well defined. Next, any homomorphism γ:C⁰ //C of normal Λ-semimodules can be lifted to a morphism

P⁰: R⁰

f

²² // ^µ0 //P^{0 ε0} ////

g

²²

C⁰

γ

²²

P: R // ^µ //P ^ε ////C,

and Ext_Λ(γ, A) : Ext_Λ(C, A) //Ext_Λ(C⁰, 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 categoryAbm([ϕ]7−→[αϕ], [ϕ]7−→[ϕf]), additive in both its arguments. Further, Proposition 1.9 implies that a different choice of the projective

presentations would yield a new functorExtg_Λ(C, A) which is naturally isomorphic to the functor Ext_Λ(C, A).

It is evident that ifT: Λ-Mod // Abmis an additive functor, i.e.,T(f+g)

=T(f) +T(g) andT(0) = 0, thenT(G) is an abelian group for any Λ-moduleG.

Therefore, Ext_Λ(C, A) is an abelian group whenever eitherAorC is a Λ-module.

Next, the Grothendieck functor K carries any short exact sequence of Λ-se- mimodules X // //Y ////M with M a Λ-module into a short exact sequence K(X) // //K(Y) ////M ofK(Λ)-modules. Therefore, if Q:V // //Q ////H is a Λ-projective presentation of a Λ-module H, then K(Q) :K(V) // //K(Q)

////H is a K(Λ)-projective presentation of H. Consequently, for any Λ-semi- moduleA, one has a natural homomorphism

Ext_Λ(H, A) ^K(H,A) //S¹Hom_K(Λ)(−, K(A))(H), K(H, A)([ϕ]) = [K(ϕ)], (1.10) whereS¹Hom_K(Λ)(−, K(A)) denotes the usual right satellite functor of the functor Hom_K(Λ)(−, K(A)). A straightforward verification shows thatK(H, A) is one-to-one wheneverAis cancellative. Furthermore, it immediately follows from the universal property ofK that

K(H, G) : Ext_Λ(H, G) //S¹Hom_K(Λ)(−, G)(H) (1.11) ia an isomorphism for any Λ-modulesGandH.

2.In [5] G. Janelidze introduced and studied general ExtCⁿ -functors, where

C

is an arbitrary class of diagrams of the form X //Y //Z in an arbitrary category. Suppose n = 1 and

C

⁼

T

, the class of all Takahashi extensions of Λ-semimodules. Then ExtT¹ (C, A) is a functor from the category (Λ-SMod)^op× (Λ-SMod) to the categoryAbm. In this section we prove that the restriction of ExtT¹ (C, A) to the category (Λ-NSMod)^op×(Λ-SMod) is naturally isomorphic to the functor Ext_Λ(C, A).

The functor ExtT¹ (C, A), denoted by Ext_ΛJT(C, A) in this paper, is defined as follows. Let Ext_ΛJT(C, A) be the category of sequences of the form S = f ◦S : Aoo ^f X // //Y ////C, where f is a Λ-homomorphism and S: X // //

Y ////C a Takahashi extension of Λ-semimodules. Define Ext_ΛJT(C, A) to be the set of connected components ofExt_ΛJT(C, A), that is,

Ext_ΛJT(C, A) =Ext_ΛJT(C, A)/∼,

where ∼is the smallest equivalence relation under which S:Aoo X // //Y ////C is equivalent to S⁰:A X⁰oo // //Y⁰ ////C whenever there exists

a commutative diagram of the form S: Aoo X

ϕ

²² // //Y ////

ψ

²²

C

S⁰ : Aoo X⁰ // //Y⁰ ////C.

Further, for any Λ-homomorphisms α:A //A⁰ and γ:C⁰ //C and any object S:Aoo ^f X // //Y ////C ofExt_ΛJT(C, A), defineαS andSγby

αS=αf◦S and Sγ=f◦Sγ, (2.1)

respectively. From (1.2), (1.6) and the fact that the above commutative diagram induces the morphism (ϕ, ψ^γ,1_C0) : Sγ −→ S⁰γ, ψ^γ(y, c⁰) = (ψ(y), c⁰), it fol- lows that these operations make Ext_ΛJT(C, A) a functor from (Λ-SMod)^op× (Λ-SMod) to Set. Next, ( 0 //Z Z) ∈

T

for every Λ-semimodule Z.

This together with (1.7) implies that Ext_ΛJT(C,−) preserves all finite products [5]. Therefore Ext_ΛJT(C, A) is actually an abelian monoid-valued functor, additive in both its arguments. Observe that the addition in Ext_ΛJT(C, A) can be described by

cl(S) +cl(S⁰) =cl(5_A(S⊕S⁰)4_C),

where 5_A :A⊕A //A and 4_C :C //C⊕C are the codiagonal and diag- onal maps, respectively. The class of 0 = 1A◦0 :A A // ^iA //A⊕C ^πC ////C coincides with the class of Aoo 0 //C C and serves as a neutral ele- ment, i.e.,cl(S) +cl(1_A◦0) =cl(S).

For every Takahashi extension F :K // //L ////M and every Λ-semimodu- leA, one has a natural connecting homomorphism of abelian monoids

δ(F, A) : Hom_Λ(K, A) //Ext_ΛJT(M, A) defined by

δ(F, A)(f :K //A) =cl(Aoo ^f K // //L ////M).

IfLis a projective Λ-semimodule, thanδ(F, A) is surjective [6]. Indeed, in this case any object S:Aoo ^g X // //Y ////M of Ext_ΛJT(M, A) admits a commuta- tive diagram

gϕ◦F : Aoo ^gϕ K

ϕ

²² // //L ////

ψ

²²

M

S: Aoo ^g X // //Y ////M, i.e.,δ(F, A)(gϕ) =cl(S).

Proposition 2.2. LetCbe a normalΛ-semimodule and P:R //^µ //P ^ε ////C a

projective presentation ofC. Then δ(P, A) : Hom_Λ(R, A) //Ext_ΛJT(C, A) is a cokernel of µ^∗= Hom_Λ(µ, A) : Hom_Λ(P, A) //Hom_Λ(R, A) for every Λ-semi- module A.

Proof. Consider a diagram

W

Hom_Λ(P, A) ^µ

∗ //Hom_Λ(R, A)

ω

99s

ss ss ss ss

s _δ(P,A)

//Ext_ΛJT(C, A),

ω⁰

ffM M

M MM M

where ω is a homomorphism of abelian monoids such that ωµ^∗ = 0. Take cl(S : Aoo ^f X // //Y ////C)∈ Ext_ΛJT(C, A). There is a morphism (ϕ, ψ,1_C) : P−→S. Defineω⁰(cl(S)) =ω(f ϕ). If (ϕ⁰, ψ⁰,1_C) :P−→S is another morphism it follows from Proposition 1.9 that ϕ+βµ =ϕ⁰+β⁰µfor some Λ-homomorphisms

β, β⁰:P //X . Whence

ω(f ϕ⁰) =ω(f ϕ⁰) +ωµ^∗(f β⁰) =ω(f ϕ⁰+f β⁰µ)

=ω(f ϕ+f βµ) =ω(f ϕ) +ωµ^∗(f β) =ω(f ϕ), i.e.,ω(f ϕ⁰) =ω(f ϕ). On the other hand, a commutative diagram

S: A oo ^f X

g

²² // //Y ////

h

²²

C

S⁰ : A ^f X⁰

oo 0 // //Y⁰ ////C,

where S⁰ ∈ Ext_ΛJT(C, A), yields the morphism (gϕ, hψ,1_C) :P−→S⁰. Therefore ω⁰(cl(S⁰)) =ω(f⁰gϕ) =ω(f ϕ). Thusω⁰ is well defined. Clearlyω=ω⁰δ(P, A). This completes the proof sinceδ(P, A) is surjective.

As a consequence, we obtain

Theorem 2.3. Assume A is aΛ-semimodule andC a normal Λ-semimodule. Let P:R // ^µ //P ^ε ////C be the chosen projective presentation of C. Then

θ(C, A) : Ext_Λ(C, A) //Ext_ΛJT(C, A)

defined byθ(C, A)([ϕ]) =cl(ϕ◦P) =cl(Aoo ^ϕ R // ^µ //P ^ε ////C )is a natural isomorphism of abelian monoids.

Proof. Since δ(P, A) : Hom_Λ(R, A) //Ext_ΛJT(C, A) and the canonical pro- jection p: Hom_Λ(R, A) //Ext_Λ(C, A) are both cokernels of µ^∗= Hom_Λ(µ, A) :

Hom_Λ(P, A) //Hom_Λ(R, A) , we only note that θ(C, A)([ϕ]) =δ(P, A)(ϕ) and p(ϕ) = [ϕ], and thatδ andpare natural in (C, A).

3. In this section we concentrate on Ext_Λ(C, A) withA cancellative. It will be shown that this additional condition enables one to give a direct description of Ext_Λ(C, A) by Takahashi extensions ofC byA (cf. Theorem 2.3).

Let E:A //^λ //B ^τ ////C be a short exact sequence of Λ-semimodules and α:A //A⁰ a Λ-homomorphism. Following [10], denote by Bα the Λ-semimo- dule A⁰ ⊕B modulo the following congruence relation: (a⁰₁, b1)ρα(a⁰₂, b2) if there are a1, a2 ∈ A such that λ(a1) +b1 = λ(a2) +b2 and α(a2) +a⁰₁ = α(a1) +a⁰₂; and define λα:A⁰ //B_α, τ_α:B_α //C and ξα:B //B_α byλα(a⁰) = [a⁰,0],τα([a⁰, b]) =τ(b) andξα(b) = [0, b], respectively.

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

τ ////C ofΛ-semimodules and a Λ-homomorphism α:A //A⁰ withA⁰ can- cellative. Then

αE:A^{0 λα} //Bα

τα //C

is a short exact sequence of Λ-semimodules and(α, ξα,1_C) a morphism from E to αE. Furthermore, ifE∈

T

^thenαE∈

T

^.

Also note that

E≡E⁰=⇒αE≡αE⁰, (3.2)

E> E− ⁰ =⇒αE> αE− ⁰. (3.3) It is directly verified in [10] that

(α⁰α)E≡α⁰(αE), 1_GE≡E and α(Eγ)≡(αE)γ, (3.4) where E : G // //B ////C is a short exact sequence with G a Λ-module, γ : C⁰ //C a homomorphism of Λ-semimodules, and α : G //G⁰ and α⁰ : G⁰ //G⁰⁰ are homomorphisms of Λ-modules. These equivalences together with (1.2), (1.3) and (3.2) show thatE_Λ(C, G) is a functor from (Λ-SMod)^op×(Λ-Mod) toSet(E7−→αE, E 7−→Eγ) [10].

Definition 3.5. We say that a short exact sequence E:A //^λ //B ^τ ////C of Λ-semimodules is proper if λ(a) +b1=λ(a) +b2,a∈A,b1, b2∈B impliesb1=b2. Note that it immediately follows from the definition that if E:A // // B ////C is proper, then A is a cancellative Λ-semimodule. 0 :A // ^iA //A⊕C ^πC ////C is proper if and only if Ais cancellative. Also observe that any short exact sequence

G // //B ////C withGa Λ-module is proper.

Proposition 3.6. For every short exact sequence E:A //^λ //B ^τ ////C ofΛ-se- mimodules and everyΛ-homomorphism α:A //A⁰ with A⁰ cancellative, αE:

A⁰ // ^λα //Bα

τα ////C is proper.

Proof. Assume λα(a⁰) + [a⁰₁, b1] =λα(a⁰) + [a⁰₂, b2], a⁰, a⁰₁, a⁰₂ ∈ A⁰, b1, b2 ∈B, i.e., [a⁰ +a⁰₁, b1] = [a⁰ +a⁰₂, b2]. By definition of ρα, there are a1, a2 ∈ A such that λ(a1) +b1 = λ(a2) +b2 and α(a2) +a⁰ +a⁰₁ = α(a1) +a⁰ +a⁰₂. Whence, since A⁰ is cancellative,λ(a1) +b1 = λ(a2) +b2 and α(a2) +a⁰₁ =α(a1) +a⁰₂. That is, [a⁰₁, b1] = [a⁰₂, b2].

Remark 3.7. For a short exact sequence E:A // //B ////C and a Λ-homo- morphism γ:C⁰ //C,Eγis proper if and only ifλ(a) +b1=λ(a) +b2,a∈A, b1, b2∈τ⁻¹(γ(C⁰)) impliesb1=b2. In particular, it follows that ifEis proper, then Eγis proper.

Lemma 3.8. Suppose given a short exact sequence E:A // ^λ //B ^τ ////C of Λ-semimodules and a Λ-homomorphism α:A //A⁰ withA⁰ cancellative. And assume that E⁰ :A⁰ // ^λ0 //B^{0 τ0} ////C⁰ is a proper short exact sequence and that (α, β, γ)is a morphism fromE toE⁰. Then there exists a uniqueΛ-homomorphism β⁰:Bα //B⁰ such that(1_A0, β⁰, γ)is a morphism fromαEtoE⁰ andβ=β⁰ξα. In particular, ifγ= 1_C thenαE> E− ⁰.

Proof. Define β⁰:Bα //B⁰ by β⁰([a⁰, b]) = λ⁰(a⁰) +β(b). Assume [a⁰₁, b1] = [a⁰₂, b2], i.e.,λ(a1) +b1=λ(a2) +b2 andα(a2) +a⁰₁=α(a1) +a⁰₂ for somea1, a2∈ A. Then, since (α, β, γ) : E −→ E⁰ is a morphism, we have λ⁰α(a1) +β(b1) = λ⁰α(a2) +β(b2) and λ⁰α(a2) + λ⁰(a⁰₁) = λ⁰α(a1) +λ⁰(a⁰₂). These equations give λ⁰α(a₂) +λ⁰(a⁰₁) +β(b₁) = λ⁰α(a₁) +λ⁰(a⁰₂) +β(b₁) = λ⁰α(a₂) +λ⁰(a⁰₂) +β(b₂).

Whenceλ⁰(a⁰₁) +β(b1) =λ⁰(a⁰₂) +β(b2) sinceE⁰ is proper. Henceβ⁰ is well defined.

Clearly, β⁰ is a Λ-homomorphism with β⁰λα = λ⁰, γτα = τ⁰β⁰ and β = β⁰ξα. If β⁰⁰:Bα //B⁰ is another Λ-homomorphism such thatβ=β⁰⁰ξαand (1_A0, β⁰⁰, γ) is a morphism from αE to E⁰, then β⁰⁰([a⁰, b]) =β⁰⁰([a⁰,0] + [0, b]) =β⁰⁰λα(a⁰) + β⁰⁰ξ_α(b) =λ⁰(a⁰) +β(b) =β⁰([a⁰, b]).

Let E:A //^λ //B ^τ ////C be a short exact sequence withAcancellative, and let α:A //A⁰ , α⁰ :A⁰ //A⁰⁰ and γ:C⁰ //C be Λ-homomorphisms withA⁰ andA⁰⁰ cancellative. Then Propositions 3.1 and 3.6 and Lemma 3.8 imme- diately provide the morphisms

(1_A00, ν,1_C) : (α⁰α)E−→α⁰(αE), (1_A, ξ₁

A,1_C) :E−→1_AE and (1_A0, ι,1_C0) :α(Eγ)−→(αE)γ,

where ν:Bα⁰α //(Bα)α⁰ and ι: (B^γ)α //(Bα)^γ are defined byν([a⁰⁰, b]) = [a⁰⁰,[0, b]] andι¡

[a⁰,(b, c⁰)]¢

= ([a⁰, b], c⁰), respectively. Thus

(α⁰α)E> α− ⁰(αE), E>−1_AE and α(Eγ)>−(αE)γ. (3.9) Remark 3.10. One can easily verify that ν and ι are in fact Λ-isomorphisms.

Hence (α⁰α)E ≡ α⁰(αE), E>−1_AE and α(Eγ) ≡ (αE)γ (cf. (3.4)). Furthermore, ξ₁

A is a Λ-isomorphism if and only ifE is proper. (Indeed, assume thatEis proper.

Let ξ₁

A(b1) = ξ₁

A(b2), i.e., [0, b1] = [0, b2]. Then λ(a1) +b1 = λ(a2) + b2 and 1_A(a2) + 0 = 1_A(a1) + 0 for somea1, a2∈A. Whence, sinceEis proper,b1=b2. On the other hand,ξ₁

A is always surjective ([a, b] = [0, λ(a)+b] =ξ₁

A(λ(a)+b)). Hence ξ₁

A is a Λ-isomorphism. The converse immediately follows from Proposition 3.6.) Therefore 1_AE≡E if and only ifE is proper. Denote byE_ΛP(C, A) the set of≡- equivalence classes of proper Takahashi extensions ofCbyA. Then, by Proposition 3.6 and Remark 3.7,E_ΛP(C, A) is a functor from (Λ-SMod)^op×(Λ-CSMod) toSet which canonically extends the functorE_Λ(−,−) : (Λ-SMod)^op×(Λ-Mod)−→Set.

Now letExt_ΛT(C, A) be the category of Takahashi extensions of a Λ-semimodule C by a cancellative Λ-semimoduleA. Define

Ext_ΛT(C, A) =Ext_ΛT(C, A)/h −>i,

whereh −>iis 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 7−→ Eγ and E 7−→ αE make Ext_ΛT(C, A) a functor from (Λ-SMod)^op×(Λ-CSMod) toSet.

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

Ext_ΛP T(C, A) =Ext_ΛP T(C, A)/h −>i,

whereExt_ΛP T(C, A) denotes the category of proper Takahashi extensions ofC by A. Obviously, the maps

Ext_ΛP T(C, A) ^Γ(C,A)// Ext_ΛT(C, A)

Γ⁰(C,A)

oo , Γ(cl(E)) =cl(E), Γ⁰(cl(E)) =cl(1_AE) are natural, and Γ⁰Γ = 1 and ΓΓ⁰ = 1.

In order to prove the following theorem, note that

α◦E∼1_A0 ◦αE (3.12)

for any short exact sequence E:A //^λ //B ^τ ////C and any Λ-homomorphism α:A //A⁰ with A⁰ cancellative. Indeed, the morphism (α, ξα,1_C) :E−→αE gives the commutative diagram

α◦E: A⁰ oo ^α A // ^λ //

α

²²

B ^τ ////

ξα

²²

C

1_A0 ◦αE: A⁰ A⁰ // ^λα //Bα

τα ////C.

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

Then

χ(C, A) : Ext_ΛT(C, A) //Ext_ΛJT(C, A)

defined by χ(C, A)¡

cl(E:A // //B ////C)¢

=cl¡

1_A◦E: A A // //B ////C¢ is a natural bijection.

Proof. Define

χ⁰(C, A) : Ext_ΛJT(C, A) //Ext_ΛT(C, A) by

χ⁰(C, A)¡

cl(S:Aoo ^f X // ^κ //Y ^σ ////C)¢

=cl(f S).

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

S: Aoo ^f X // ^κ //

ϕ

²²

Y ^σ ////

ψ

²²

C

S⁰: Aoo ^g X^{0 κ0} //Y^{0 σ0} ////C

withS⁰ ∈ Ext_ΛJT(C, A). This commutative diagram and the morphism (g, ξg,1_C) : S⁰−→gS⁰ yield the following commutative diagram

S : X // ^κ //

f

²²

Y ^σ ////

ξ_gψ

²²

C

gS⁰ : A ^κ

0g //Y_g^{0 σ}

g0 ////C.

Whence, by Proposition 3.6 and Lemma 3.8,f S> gS− ⁰. Hence χ⁰(C, A) is well de- fined. Further, by (3.12),χχ⁰(cl(S)) =χ(cl(f S)) =cl(1_A◦f S) =cl(f◦S) =cl(S).

On the other hand,χ⁰χ(cl(E)) =χ⁰(cl(1_A◦E)) =cl(1_AE) =cl(E) sinceE>−1_AE.

Thusχ(C, A) is a bijection. Finally, consider the diagram Ext_ΛT(C, A) ^χ(C,A) //

Ext_ΛT(γ,α)

²²

Ext_ΛJT(C, A)

Ext_ΛJT(γ,α)

²²

Ext_ΛT(C⁰, A⁰) ^χ(C

0,A⁰) //Ext_ΛJT(C⁰, A⁰),

where α:A //A⁰ is a homomorphism of cancellative Λ-semimodules and γ : C⁰ //C a homomorphism of Λ-semimodules. Using (3.12) and (2.1), we obtain

χ(C⁰, A⁰) Ext_ΛT(γ, α)(cl(E)) =cl(1_A0 ◦αEγ) =cl(α◦Eγ)

=cl¡

(α◦E)γ¢

=cl¡

α(1_A◦E)γ¢

= Ext_ΛJT(γ, α)χ(C, A)(cl(E)), i.e., the diagram is commutative. Thusχ(C, A) is natural.

Remark 3.14. More general results than Theorem 3.13 are discussed in [5,6]. How- ever 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

C

of short exact sequences E:A // //B ////C withAcancellative which satisfies the following conditions: (0 //Z Z)∈

C

for every Λ-semimoduleZ; ifE∈

C

^then

αE,Eγ∈

C

^.

Theorem 3.13 shows that Ext_ΛT(C, A) is in fact an abelian monoid-valued func- tor, additive in each of its arguments; the addition in Ext_ΛT(C, A) obviously coin- cides with the Bear addition:

cl(E) +cl(E⁰) =cl¡

5_A(E⊕E⁰)4_C ¢ . As a corollary of Theorems 2.3 and 3.13 we have

Theorem 3.15. AssumeAis a cancellativeΛ-semimodule andCa normalΛ-semi- module. Let P:R // ^ν //P ^ε ////C be the chosen projective presentation of C.

Then

w(C, A) : Ext_Λ(C, A) //Ext_ΛT(C, A)

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

Proof. w(C, A) =χ⁰(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_ΛT(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 GandH be Λ-modules. The map

Ext¹_K(Λ)(H, G) //Ext_ΛJT(H, G),

cl(G // //B ////H)7−→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 Λ⁰be additively cancellative semirings. In [8], for any con- travariant additive functor T : (Λ-CSMod) //(Λ⁰ -CSMod), we constructed and studied right derived functors RⁿT : (Λ-CSMod) //(Λ⁰ -CSMod), n= 0,1,2, . . .. In particular, we described RⁿHom_Λ(−, A)(C) by means of certain n- fold extensions ofC byA. According to that description, forn= 1,H ∈(Λ-Mod) andA∈(Λ-CSMod), one has a natural isomorphism of abelian groups

R¹Hom_Λ(−, A)(H)∼=Ext¹_Λ(H, A)/h −>i,

whereExt¹_Λ(H, A) denotes the category of short exact sequences A // //B ////H of Λ-semimodules withBcancellative. (Note thatR¹Hom_Λ(−, A)(H) =R¹Hom_Λ(−, U(A))(H)∼= Ext¹_K(Λ)(H, U(A)), whereU(A) is the maximal Λ-submodule ofA, and R¹Hom_Λ(−, A) denotes the usual right derived functor of the functor Hom_Λ(−, A) : (Λ-Mod) // Ab.) On the other hand, ifA // ^κ //X ^σ ////H is a proper short exact sequence of Λ-semimodules withH ∈(Λ-Mod), thenXis cancellative. (To see this, supposex+x1=x+x2,x, x1, x2∈X. Takex⁰∈X such thatσ(x⁰) =−σ(x).

Then x⁰+x= κ(a) for some a ∈ A; and we obtain κ(a) +x1 = x⁰+x+x1 = x⁰+x+x2 = κ(a) +x2. Whence x1 = x2 since A // ^κ //X ^σ ////H is proper.) Hence there is a natural isomorphism

R¹Hom_Λ(−, A)(H)∼= ExtΛP T(H, A).

Consequently, by Remark 3.11 and Theorems 3.13 and 3.15, each of the abelian groups Ext_ΛJT(H, A), Ext_ΛT(H, A) and Ext_Λ(H, A), where H is a Λ-module andAa cancellative Λ-semimodule, is naturally isomorphic toR¹Hom_Λ(−, A)(H).

4. LetGbe a Λ-module. In this section, continuing the investigation started in [11], we obtain some results relating Ext_Λ(C, G) andE_Λ(C, G). Besides, we relate Ext_Λ(C, G) with Inassaridze extensions ofCbyG[4], and also withS¹Hom_K(Λ)(−,G) (K(C)), whereK(C) denotes the GrothendieckK(Λ)-module of C.

First of all, observe that the Baer addition of extensions,E+E⁰ =5_G(E⊕E⁰)4_C, makes E_Λ(C, G) an abelian monoid [4]. In addition the ≡-equivalence class of 0 :G // ^iG //G⊕C ^πC ////C serves as a neutral element. Furthermore, a straightfor- ward verification shows thatα(E+E⁰)≡αE+αE⁰, (E+E⁰)γ≡Eγ+E⁰γ,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 E:G //^λ //B ^τ ////C of a Λ-semimoduleCby a Λ-moduleGanInassaridze extensionifEis an extension ofCbyGin the sense of H. Inassaridze [4]: whenever the equalityλ(g) +b=b holds for someg∈G,b∈B, theng= 0. Let Ext_ΛI(C, G) denote the set of≡-equivalence classes of Inassaridze extension ofCbyG. It is shown in [4] that Ext_ΛI(C, G) =U(E_Λ(C, G)), the group of units of E_Λ(C, G). This in particular means that Ext_ΛI(C, G) is an abelian group-valued subfunctor of E_Λ(C, G). Moreover, Ext_ΛI(C, G) is additive in both its arguments.

A short exact sequence of Λ-semimodules E:A //^λ //B ^τ ////C is said to be split if there exists a Λ-homomorphism ν:C //B such that τ ν = 1. Let E_ΛS(C, G) denote the set of ≡-equivalence classes of split Takahashi extensions of a Λ-semimoduleC by a Λ-moduleG. It is easy to see thatαE,EγandE⊕E⁰ are split wheneverE andE⁰ are split. Consequently,E_ΛS(C, G) is another subfunctor of the functorE_Λ(C, G).

We shall need the following four facts.

Proposition 4.1 ([4]). Suppose given a morphism of the form E: G // ^λ //B ^τ ////

β

²²

C

E⁰: G // ^λ0 //B^{0 τ0} ////C

of Takahashi extensions of aΛ-semimoduleC by aΛ-moduleG. IfE⁰ is an Inas- saridze extension, then β is an isomorphism.

Theorem 4.2 ([4,7]). LetK(C)be the GrothendieckK(Λ)-module of aΛ-semimo- duleC, and k_C :C //K(C) the canonicalΛ-homomorphism. Then the natural homomorphism

Ext¹_K(Λ)(K(C), G) = Ext_ΛI(K(C), G) ^ExtΛI(kC,G)//Ext_ΛI(C, G) is an isomorphism.

Proposition 4.3 ([7]). An Inassaridze extension E:G // //B ////C is split if and only if E≡0 :G // ^iG //G⊕C ^πC ////C.

Proposition 4.4. Let E:A // ^λ //B ^τ ////C be a split short exact sequence of Λ-semimodules, and ν :C //B a splitting Λ-homomorphism, i.e., τ ν = 1. If C is aΛ-module, then the map

q:A⊕C //B , q(a, c) =λ(a) +ν(c) is aΛ-isomorphism.

Proof. Define q⁰:B //A⊕C byq⁰(b) = (λ⁻¹(b−ντ(b)), τ(b)). Then q⁰q= 1 andqq⁰ = 1.

For any Λ-moduleGand any normal Λ-semimoduleC, M. Takahashi has defined a pair of maps

E_Λ(C, G) ^η(C,G) //Ext_Λ(C, G)

ζ(C,G)

oo

as follows.ζ(C, G)([f]) =cl(fP), where P:R // ^µ //P ^ε ////C is the chosen pro- jective presentation of C. Next, take cl(E) ∈ E_Λ(C, G). There is a morphism (ϕ, ψ,1) :P−→E. And defineη(C, G)(cl(E)) = [ϕ]. It is shown in [11] thatη(C, G) and ζ(C, G) are well defined, andη(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

E_Λ(C, G) ^η(C,G) //

m(C,G)OOOOOOOOOO''

O Ext_Λ(C, G)

w(C,G)

wwnnnnnnnnnnnn

Ext_ΛT(C, G)

wherem(C, G) is defined bym(C, G)(cl(E)) =cl_h−>i(E), andw(C, G) byw(C, G)([ϕ])

=cl_h−>i(ϕP) (see Theorem 3.15). One hasw(C, G)η(C, G) (cl(E)) =w(C, G)([ϕ]) = cl_h−>i(ϕP). But, by Lemma 3.8, ϕP> E, that is,− cl_h−>i(ϕP) = cl_h−>i(E). Hence the diagram is commutative. Therefore, sincem(C, G) is a homomorphism andw(C, G) an isomorphism,η(C, G) is a homomorphism.

Proposition 4.5. For any Λ-moduleGand any normalΛ-semimoduleC, Ker(µ(C, G)) =E_ΛS(C, G),

that is, the sequence

E_ΛS(C, G) ^j(C,G) // E_Λ(C, G) ^η(C,G) // Ext_Λ(C, G), (4.6) wherej(C, G)denotes the inclusion, is a short exact sequence of abelian monoids.

Proof. Let cl(E:G // ^λ //B ^τ ////C) ∈E_ΛS(C, G). If ν :C //B is a split- ting Λ-homomorphism, then the diagram

P: R // ^µ //

0

²²

P ^ε ////

νε

²²

C

E: G // ^λ //B ^τ ////C

is commutative. Whence, by definition ofη(C, G), η(C, G)(cl(E)) = [0] = 0. Con- versely, suppose cl(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 Λ- homomorphism g:P //G such that ϕ = gµ. From this it follows that the diagram

P: R // ^µ //

ϕ

²²

P ^ε ////

β

²²

C

0 : G // ^iG //G⊕C ^πC ////C,

where β =i_Gg+i_Cε, is commutative. By Lemma 3.8, this commutative diagram