Dedicated to Professor Hvedri Inassaridze on the occasion of his seventieth birthday
EXTENSIONS OF SEMIMODULES AND THE TAKAHASHI FUNCTOR ExtΛ
(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 . 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 , following the construction of the usual satellite functors, M. Takahashi defined ExtΛ(C, A) by
ExtΛ(C, A) = Coker(HomΛ(µ, A))
and used it to characterize Takahashi extensions of normal Λ- semimodules by Λ-modules.
In this paper we relate ExtΛ(C, A) with other known satel- lite functors of the functor HomΛ(−, A).
Section 1 is concerned with preliminaries. The purpose of Section 2 is to charac- terize ExtΛ(C, A) in terms of Janelidze’s general ExtCn -functors . In Section 3 we show that ExtΛ(C, A) withA cancellative can be described directly by Takahashi extensions ofCbyA. The last section is devoted to ExtΛ(C, G) withGa Λ-module.
We relate ExtΛ(C, G) with Inassaridze extensions ofCbyG. This allows to relate ExtΛ(C, G) andS1HomK(Λ)(−, G)(K(C)), whereK(Λ) is the Grothendieck ring of Λ, K(C) the Grothendieck K(Λ)-module of C, and S1HomK(Λ)(−, G) the usual right satellite functor of the functor HomK(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.
°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 HomΛ(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. ). 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 //B0 τ0 ////C0 is a triple of Λ-homomorphisms (α, β, γ) such that
²² // λ //B τ ////
²²E0: A0 // λ0 //B0 τ0 ////C0
is a commutative diagram. For a morphism of the form E: A
// //B ////
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 //Bγ τ
where Bγ = ©
(b, c0) ∈ B⊕C0|τ(b) = γ(c0)ª
, λγ(a) = (λa,0), τγ(b, c0) = c0, is a short exact sequence of Λ-semimodules. Besides, if one defines a Λ-homomorphism ξγ:Bγ //B by ξγ(b, c0) = b, then (1A, ξγ, γ) is a morphism from Eγ to E.
From the construction ofEγit follows that
E≡E1C, (Eγ)γ0≡E(γγ0), (1.2)
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 : 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 , 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
Tdenote the class of all Takahashi extensions of Λ-semimodules. Then
Here E⊕E0 denotes A⊕A0 //λ⊕λ0//B⊕B0 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 , we denote byEΛ(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, whereiA(a) = (a,0) and πC(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 HomΛ(P,−)preserves normal epimorphisms.
This fact, proved in  (and first mentioned in ), implies
Proposition 1.9 (cf. ). Suppose given a diagram ofΛ-semimodules andΛ-ho-
A // λ //B τ ////C,
where the bottom row is a Takahashi extension,P is projective,τ0λ0= 0, andτ ψi= γτ0,λϕi =ψiλ0 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 . 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 . We denote the category of normal Λ-semimodules and their Λ-homomorphisms by Λ-NSMod.
In  M. Takahashi has constructed ExtΛ(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 ExtΛ(C, A) to be Coker(HomΛ(µ, A) : HomΛ(P, A) //
HomΛ(R, A)). That is,
ExtΛ(C, A) = HomΛ(R, A)/HomΛ(µ, A)(HomΛ(P, A)).
If α:A //A0 is a homomorphism of Λ-semimodules, one defines ExtΛ(C, α) : ExtΛ(C, A) //ExtΛ(C, A0) by ExtΛ(C, α)([ϕ]) = [αϕ]. Obviously, ExtΛ(C, α) is well defined. Next, any homomorphism γ:C0 //C of normal Λ-semimodules can be lifted to a morphism
²² // µ0 //P0 ε0 ////
P: R // µ //P ε ////C,
and ExtΛ(γ, A) : ExtΛ(C, A) //ExtΛ(C0, A) is defined by ExtΛ(γ, A)([ϕ]) = [ϕf]. It follows from Proposition 1.9 that ExtΛ(γ, A) is also well defined. Now one can easily see that ExtΛ(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 functorExtgΛ(C, A) which is naturally isomorphic to the functor ExtΛ(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, ExtΛ(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
ExtΛ(H, A) K(H,A) //S1HomK(Λ)(−, K(A))(H), K(H, A)([ϕ]) = [K(ϕ)], (1.10) whereS1HomK(Λ)(−, K(A)) denotes the usual right satellite functor of the functor HomK(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) : ExtΛ(H, G) //S1HomK(Λ)(−, G)(H) (1.11) ia an isomorphism for any Λ-modulesGandH.
2.In  G. Janelidze introduced and studied general ExtCn -functors, where
is an arbitrary class of diagrams of the form X //Y //Z in an arbitrary category. Suppose n = 1 and
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 ExtΛ(C, A).
The functor ExtT1 (C, A), denoted by ExtΛJT(C, A) in this paper, is defined as follows. Let ExtΛJT(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 ExtΛJT(C, A) to be the set of connected components ofExtΛJT(C, A), that is,
ExtΛJT(C, A) =ExtΛJT(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
²² // //Y ////
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 ofExtΛJT(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, ψγ,1C0) : Sγ −→ S0γ, ψγ(y, c0) = (ψ(y), c0), it fol- lows that these operations make ExtΛJT(C, A) a functor from (Λ-SMod)op× (Λ-SMod) to Set. Next, ( 0 //Z Z) ∈
Tfor every Λ-semimodule Z.
This together with (1.7) implies that ExtΛJT(C,−) preserves all finite products . Therefore ExtΛJT(C, A) is actually an abelian monoid-valued functor, additive in both its arguments. Observe that the addition in ExtΛJT(C, A) can be described by
cl(S) +cl(S0) =cl(5A(S⊕S0)4C),
where 5A :A⊕A //A and 4C :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(1A◦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) : HomΛ(K, A) //ExtΛJT(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 . Indeed, in this case any object S:Aoo g X // //Y ////M of ExtΛJT(M, A) admits a commuta- tive diagram
gϕ◦F : Aoo gϕ K
²² // //L ////
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) : HomΛ(R, A) //ExtΛJT(C, A) is a cokernel of µ∗= HomΛ(µ, A) : HomΛ(P, A) //HomΛ(R, A) for every Λ-semi- module A.
Proof. Consider a diagram
HomΛ(P, A) µ
∗ //HomΛ(R, A)
ss ss ss ss
M MM M
where ω is a homomorphism of abelian monoids such that ωµ∗ = 0. Take cl(S : Aoo f X // //Y ////C)∈ ExtΛJT(C, A). There is a morphism (ϕ, ψ,1C) : P−→S. Defineω0(cl(S)) =ω(f ϕ). If (ϕ0, ψ0,1C) :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
²² // //Y ////
S0 : A f X0
oo 0 // //Y0 ////C,
where S0 ∈ ExtΛJT(C, A), yields the morphism (gϕ, hψ,1C) :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) : ExtΛ(C, A) //ExtΛJT(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) : HomΛ(R, A) //ExtΛJT(C, A) and the canonical pro- jection p: HomΛ(R, A) //ExtΛ(C, A) are both cokernels of µ∗= HomΛ(µ, A) :
HomΛ(P, A) //HomΛ(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 ExtΛ(C, A) withA cancellative. It will be shown that this additional condition enables one to give a direct description of ExtΛ(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 , 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 //Bα, τα:Bα //C and ξα:B //Bα byλα(a0) = [a0,0],τα([a0, b]) =τ(b) andξα(b) = [0, b], respectively.
Proposition 3.1 (). 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:A0 λα //Bα
is a short exact sequence of Λ-semimodules and(α, ξα,1C) a morphism from E to αE. Furthermore, ifE∈
Also note that
E> E− 0 =⇒αE> αE− 0. (3.3) It is directly verified in  that
(α0α)E≡α0(αE), 1GE≡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 thatEΛ(C, G) is a functor from (Λ-SMod)op×(Λ-Mod) toSet(E7−→αE, E 7−→Eγ) .
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 //B0 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(1A0, β0, γ)is a morphism fromαEtoE0 andβ=β0ξα. In particular, ifγ= 1C 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 (1A0, β00, γ) is a morphism from αE to E0, then β00([a0, b]) =β00([a0,0] + [0, b]) =β00λα(a0) + β00ξα(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
(1A00, ν,1C) : (α0α)E−→α0(αE), (1A, ξ1
A,1C) :E−→1AE and (1A0, ι,1C0) :α(Eγ)−→(αE)γ,
where ν:Bα0α //(Bα)α0 and ι: (Bγ)α //(Bα)γ are defined byν([a00, b]) = [a00,[0, b]] andι¡
= ([a0, b], c0), respectively. Thus
(α0α)E> α− 0(αE), E>−1AE 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>−1AE 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.
A(b1) = ξ1
A(b2), i.e., [0, b1] = [0, b2]. Then λ(a1) +b1 = λ(a2) + b2 and 1A(a2) + 0 = 1A(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 1AE≡E if and only ifE is proper. Denote byEΛP(C, A) the set of≡- equivalence classes of proper Takahashi extensions ofCbyA. Then, by Proposition 3.6 and Remark 3.7,EΛP(C, A) is a functor from (Λ-SMod)op×(Λ-CSMod) toSet which canonically extends the functorEΛ(−,−) : (Λ-SMod)op×(Λ-Mod)−→Set.
Now letExtΛT(C, A) be the category of Takahashi extensions of a Λ-semimodule C by a cancellative Λ-semimoduleA. Define
ExtΛT(C, A) =ExtΛT(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 ExtΛT(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
ExtΛP T(C, A) =ExtΛP T(C, A)/h −>i,
whereExtΛP T(C, A) denotes the category of proper Takahashi extensions ofC by A. Obviously, the maps
ExtΛP T(C, A) Γ(C,A)// ExtΛT(C, A)
oo , Γ(cl(E)) =cl(E), Γ0(cl(E)) =cl(1AE) are natural, and Γ0Γ = 1 and ΓΓ0 = 1.
In order to prove the following theorem, note that
α◦E∼1A0 ◦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,1C) :E−→αE gives the commutative diagram
α◦E: A0 oo α A // λ //
B τ ////
1A0 ◦αE: A0 A0 // λα //Bα
Theorem 3.13. Let C be a Λ-semimodule and A a cancellative Λ-semimodule.
χ(C, A) : ExtΛT(C, A) //ExtΛJT(C, A)
defined by χ(C, A)¡
cl(E:A // //B ////C)¢
1A◦E: A A // //B ////C¢ is a natural bijection.
χ0(C, A) : ExtΛJT(C, A) //ExtΛT(C, A) by
cl(S:Aoo f X // κ //Y σ ////C)¢
This definition is independent of the chosen representative sequence S. Indeed, suppose given a commutative diagram
S: Aoo f X // κ //
Y σ ////
S0: Aoo g X0 κ0 //Y0 σ0 ////C
withS0 ∈ ExtΛJT(C, A). This commutative diagram and the morphism (g, ξg,1C) : S0−→gS0 yield the following commutative diagram
S : X // κ //
Y σ ////
gS0 : A κ
0g //Yg0 σ
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(1A◦f S) =cl(f◦S) =cl(S).
On the other hand,χ0χ(cl(E)) =χ0(cl(1A◦E)) =cl(1AE) =cl(E) sinceE>−1AE.
Thusχ(C, A) is a bijection. Finally, consider the diagram ExtΛT(C, A) χ(C,A) //
ExtΛT(C0, A0) χ(C
0,A0) //ExtΛJT(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) ExtΛT(γ, α)(cl(E)) =cl(1A0 ◦αEγ) =cl(α◦Eγ)
= ExtΛJT(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 .
It is evident that Theorem 3.13 remains valid for any class
Cof short exact sequences E:A // //B ////C withAcancellative which satisfies the following conditions: (0 //Z Z)∈
Cfor every Λ-semimoduleZ; ifE∈
Theorem 3.13 shows that ExtΛT(C, A) is in fact an abelian monoid-valued func- tor, additive in each of its arguments; the addition in ExtΛT(C, A) obviously coin- cides with the Bear addition:
cl(E) +cl(E0) =cl¡
5A(E⊕E0)4C ¢ . 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.
w(C, A) : ExtΛ(C, A) //ExtΛT(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
ExtΛT(H, G) = Ext1K(Λ)(H, G) =EΛ(H, G),
where Ext1K(Λ) is the usual Ext functor. From this and Theorem 3.13 one has Corollary 3.16. Let GandH be Λ-modules. The map
Ext1K(Λ)(H, G) //ExtΛJT(H, G),
cl(G // //B ////H)7−→cl(G G // //B ////H) is a natural isomorphism of abelian groups.
Note that in  Janelidze proved this for Λ =N, the semiring of non-negative integers.
Remark 3.17. Let Λ and Λ0be additively cancellative semirings. In , 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 RnHomΛ(−, 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
R1HomΛ(−, A)(H)∼=Ext1Λ(H, A)/h −>i,
whereExt1Λ(H, A) denotes the category of short exact sequences A // //B ////H of Λ-semimodules withBcancellative. (Note thatR1HomΛ(−, A)(H) =R1HomΛ(−, U(A))(H)∼= Ext1K(Λ)(H, U(A)), whereU(A) is the maximal Λ-submodule ofA, and R1HomΛ(−, A) denotes the usual right derived functor of the functor HomΛ(−, 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
R1HomΛ(−, A)(H)∼= ExtΛP T(H, A).
Consequently, by Remark 3.11 and Theorems 3.13 and 3.15, each of the abelian groups ExtΛJT(H, A), ExtΛT(H, A) and ExtΛ(H, A), where H is a Λ-module andAa cancellative Λ-semimodule, is naturally isomorphic toR1HomΛ(−, A)(H).
4. LetGbe a Λ-module. In this section, continuing the investigation started in , we obtain some results relating ExtΛ(C, G) andEΛ(C, G). Besides, we relate ExtΛ(C, G) with Inassaridze extensions ofCbyG, and also withS1HomK(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 =5G(E⊕E0)4C, makes EΛ(C, G) an abelian monoid . 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 EΛ(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 : whenever the equalityλ(g) +b=b holds for someg∈G,b∈B, theng= 0. Let ExtΛI(C, G) denote the set of≡-equivalence classes of Inassaridze extension ofCbyG. It is shown in  that ExtΛI(C, G) =U(EΛ(C, G)), the group of units of EΛ(C, G). This in particular means that ExtΛI(C, G) is an abelian group-valued subfunctor of EΛ(C, G). Moreover, ExtΛI(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 EΛS(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,EΛS(C, G) is another subfunctor of the functorEΛ(C, G).
We shall need the following four facts.
Proposition 4.1 (). Suppose given a morphism of the form E: G // λ //B τ ////
E0: G // λ0 //B0 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 kC :C //K(C) the canonicalΛ-homomorphism. Then the natural homomorphism
Ext1K(Λ)(K(C), G) = ExtΛI(K(C), G) ExtΛI(kC,G)//ExtΛI(C, G) is an isomorphism.
Proposition 4.3 (). 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
EΛ(C, G) η(C,G) //ExtΛ(C, G)
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) ∈ EΛ(C, G). There is a morphism (ϕ, ψ,1) :P−→E. And defineη(C, G)(cl(E)) = [ϕ]. It is shown in  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
EΛ(C, G) η(C,G) //
O ExtΛ(C, G)
wherem(C, G) is defined bym(C, G)(cl(E)) =clh−>i(E), andw(C, G) byw(C, G)([ϕ])
=clh−>i(ϕP) (see Theorem 3.15). One hasw(C, G)η(C, G) (cl(E)) =w(C, G)([ϕ]) = clh−>i(ϕP). But, by Lemma 3.8, ϕP> E, that is,− clh−>i(ϕP) = clh−>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)) =EΛS(C, G),
that is, the sequence
EΛS(C, G) j(C,G) // EΛ(C, G) η(C,G) // ExtΛ(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) ∈EΛS(C, G). If ν :C //B is a split- ting Λ-homomorphism, then the diagram
P: R // µ //
P ε ////
E: G // λ //B τ ////C
is commutative. Whence, by definition ofη(C, G), η(C, G)(cl(E)) =  = 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 //
P ε ////
0 : G // iG //G⊕C πC ////C,
where β =iGg+iCε, is commutative. By Lemma 3.8, this commutative diagram