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 ar-
bitrary 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 *ε*:*P* //*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 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 ExtC* ^{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

*(−, G) the usual right satellite functor of the functor Hom*

_{K(Λ)}*(−, G).*

_{K(Λ)}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·*(λ* ^{0}*+

*λ*

*) =*

^{00}*λ·λ*

*+*

^{0}*λ·λ*

^{00}*,*(λ

*+*

^{0}*λ*

*)*

^{00}*·λ*=

*λ*

^{0}*·λ*+

*λ*

^{00}*·λ,*

*λ·*0 = 0

*·λ*= 0,

for all *λ, λ*^{0}*, λ*^{00}*∈* 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*+*a** ^{0}*) =

*λa*+

*λa*

^{0}*,*(λ+

*λ*

*)a=*

^{0}*λa*+

*λ*

^{0}*a,*(λ

*·λ*

*)a=*

^{0}*λ(λ*

^{0}*a),*1a=

*a,*0a= 0,

for all*λ, λ*^{0}*∈*Λ and *a, a*^{0}*∈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+*a** ^{0}*) =

*f*(a) +

*f*(a

*) and*

^{0}*f*(λa) =

*λf(a), for all*

*a, a*

^{0}*∈*

*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 *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* 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 [b1] = [b2] if and only if*a*1+*b*1=*a*2+*b*2 for some*a*1*, a*2*∈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*+*a** ^{0}* =

*a*+

*a*

*for*

^{00}*a, a*

^{0}*, a*

^{00}*∈*

*A*implies

*a*

*=*

^{0}*a*

*. Obviously,*

^{00}*A*is a cancellative Λ-semimodule if and only if

*A*is a cancellative

*C(Λ)-semimodule, whereC(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, 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 *g** ^{0}*:

*C*//

*D*

with *g*=*g*^{0}*u. 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 of*f*.

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

A *morphism* from *E*:*A* // * ^{λ}* //

*B*

*////*

^{τ}*C*to

*E*

*:*

^{0}*A*

*//*

^{0}

^{λ}*//*

^{0}*B*

^{0}

^{τ}*////*

^{0}*C*

*is a triple of Λ-homomorphisms (α, β, γ) such that*

^{0}*E*: *A*

*α*

²² // * ^{λ}* //

*B*

*////*

^{τ}*β*

²²

*C*

*γ*

²²*E** ^{0}*:

*A*

*//*

^{0}

^{λ}*//*

^{0}*B*

^{0}

^{τ}*////*

^{0}*C*

^{0}is a commutative diagram. For a morphism of the form
*E*: *A*

1_{A}

// //*B* ////

*β*

²²

*C*

1_{C}

*E** ^{0}*:

*A*// //

*B*

*////*

^{0}*C,*

we write*E> E−* * ^{0}*. If in addition

*β*is a Λ-isomorphism, we write

*E≡E*

*and say that*

^{0}*E*is

*equivalent*to

*E*

*.*

^{0}Next, suppose given a short exact sequence *E*:*A* // * ^{λ}* //

*B*

*////*

^{τ}*C*and a Λ-ho- momorphism

*γ*:

*C*

*//*

^{0}*C*. Then

*Eγ*:*A* // ^{λ}* ^{γ}* //

*B*

^{γ}

^{τ}*γ* ////*C*^{0}*,*

where *B** ^{γ}* = ©

(b, c* ^{0}*)

*∈*

*B⊕C*

^{0}*|τ(b) =*

*γ(c*

*)ª*

^{0}, *λ** ^{γ}*(a) = (λa,0),

*τ*

*(b, c*

^{γ}*) =*

^{0}*c*

*, is a short exact sequence of Λ-semimodules. Besides, if one defines a Λ-homomorphism*

^{0}*ξ*

*:*

^{γ}*B*

*//*

^{γ}*B*by

*ξ*

*(b, c*

^{γ}*) =*

^{0}*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^{0}*≡E(γγ** ^{0}*), (1.2)

*E≡E** ^{0}*=

*⇒Eγ≡E*

^{0}*γ,*(1.3)

*E> E−* * ^{0}* =

*⇒Eγ> E−*

^{0}*γ.*(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 denote

*S*by

*f◦E, andE*by

*S.*

A surjective Λ-homomorphism *τ*:*B* //*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 *τ*:*B* //*H* *with* *H* *a*
Λ-module is normal.

*Proof.* Suppose *τ*(b1) = *τ(b*2). Take *b* *∈* *B* with *τ(b) =* *−τ(b*1). Then (b2+ *b),*
(b1+ *b)∈*Ker(τ) and (b2+*b) +b*1= (b1+*b) +b*2.

Note also that for any Λ-subsemimodule*A* of a Λ-semimodule*B, the canonical*
projection *p*:*B* //*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 *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, E*^{0}*∈*

## T

^{=}

^{⇒}^{E}^{⊕}^{E}

^{0}

^{∈}## T

^{.}^{(1.7)}

Here *E⊕E** ^{0}* denotes

*A⊕A*

*//*

^{0}

^{λ⊕λ}*//*

^{0}*B⊕B*

^{0}

^{τ⊕τ}*0*////*C⊕C** ^{0}*, the usual direct sum of

*E*and

*E*

*. Two extensions*

^{0}*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

0 :*A* // ^{i}* ^{A}* //

*A⊕C*

^{π}*////*

^{C}*C,*where

*i*

*(a) = (a,0) and*

_{A}*π*

*(a, c) =*

_{C}*c.*

A Λ-semimodule*P* is *projective*if it satisfies the usual lifting property: Given a
surjective Λ-homomorphism *τ* :*B* //*C* and a Λ-homomorphism*f* :*P* //*C*,
there is a Λ-homomorphism *g*:*P* //*B* such that*f* =*τ g.*

Proposition 1.8. *Let* *τ* :*B* //*C* *be a normal*Λ-epimorphism and let *f*_{1}*, f*_{2} :
*P* //*B* *be* Λ-homomorphisms with *P* *projective. If* *τ f*_{1} =*τ f*_{2} *then there exist*
Λ-homomorphisms *g*_{1}*, g*_{2}:*P* //*B* *satisfying* *τ g*_{1} = 0 = *τ g*_{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 andΛ-ho-

*momorphisms*

*A*^{0}

*ϕ*_{1}

²²

*ϕ*_{2}

²²

*λ** ^{0}* //

*P*

*g*_{2}

ÄÄÄÄÄÄÄÄÄÄ

*g*_{1}

ÄÄÄÄÄÄÄÄÄÄ

*ψ*_{1}

²²

*ψ*_{2}

²²

*τ** ^{0}* //

*C*

^{0}*γ*

²²

*A* // * ^{λ}* //

*B*

*////*

^{τ}*C,*

*where the bottom row is a Takahashi extension,P* *is projective,τ*^{0}*λ** ^{0}*= 0, and

*τ ψ*

*=*

_{i}*γτ*

^{0}*,λϕ*

*=*

_{i}*ψ*

_{i}*λ*

^{0}*fori*= 1,2. Then there areΛ-homomorphisms

*g*

_{1}

*, g*

_{2}:

*P*//

*A*

*such that*

*g*

_{1}

*λ*

*+*

^{0}*ϕ*

_{1}=

*g*

_{2}

*λ*

*+*

^{0}*ϕ*

_{2}

*.*

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 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 *F* //*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 P:*R* //* ^{µ}* //

*P* * ^{ε}* ////

*C*of

*C*and 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* //*A** ^{0}* is a homomorphism of Λ-semimodules, one defines Ext

_{Λ}(C, α) : Ext

_{Λ}(C, A) //Ext

_{Λ}(C, A

*) by Ext*

^{0}_{Λ}(C, α)([ϕ]) = [αϕ]. Obviously, Ext

_{Λ}(C, α) is well defined. Next, any homomorphism

*γ*:

*C*

*//*

^{0}*C*of normal Λ-semimodules can be lifted to a morphism

P* ^{0}*:

*R*

^{0}*f*

²² // ^{µ}* ^{0}* //

*P*

^{0}

^{ε}*////*

^{0}*g*

²²

*C*^{0}

*γ*

²²

P: *R* // * ^{µ}* //

*P*

*////*

^{ε}*C,*

and Ext_{Λ}(γ, A) : Ext_{Λ}(C, A) //Ext_{Λ}(C^{0}*, 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 if*T*: Λ-Mod // Abmis 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 *X* // //*Y* ////*M* with *M* a Λ-module into a short exact sequence
*K(X)* // //*K(Y*) ////*M* of*K(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-*
module*A, one has a natural homomorphism*

Ext_{Λ}(H, A) * ^{K(H,A)}* //

*S*

^{1}Hom

*(*

_{K(Λ)}*−, K(A))(H), K*(H, A)([ϕ]) = [K(ϕ)], (1.10) where

*S*

^{1}Hom

*(−, K(A)) denotes the usual right satellite functor of the functor Hom*

_{K(Λ)}*(−, K(A)). A straightforward verification shows that*

_{K(Λ)}*K(H, A) is one-to-one*whenever

*A*is cancellative. Furthermore, it immediately follows from the universal property of

*K*that

*K(H, G) : Ext*_{Λ}(H, G) //*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 ExtC* ^{n}* -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 ExtT^{1}(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 ExtT

^{1}(C, A) to the category (Λ-NSMod)

^{op}

*×*(Λ-SMod) is naturally isomorphic to the functor Ext

_{Λ}(C, A).

The functor ExtT^{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* *◦S* :
*A*oo ^{f}*X* // //*Y* ////*C*, where *f* is a Λ-homomorphism and *S*: *X* // //

*Y* ////*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 *S*:*A*oo *X* // //*Y*
////*C* is equivalent to *S** ^{0}*:

*A*

*X*

*oo // //*

^{0}*Y*

*////*

^{0}*C*whenever there exists

a commutative diagram of the form
*S*: *A*oo *X*

*ϕ*

²² // //*Y* ////

*ψ*

²²

*C*

*S** ^{0}* :

*A*oo

*X*

*// //*

^{0}*Y*

*////*

^{0}*C.*

Further, for any Λ-homomorphisms *α*:*A* //*A** ^{0}* and

*γ*:

*C*

*//*

^{0}*C*and any object

*S*:

*A*oo

^{f}*X*// //

*Y*////

*C*of

*Ext*

_{Λ}

*JT*(C, A), define

*αS*and

*Sγ*by

*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_{C}*0*) : *Sγ* *−→* *S*^{0}*γ,* *ψ** ^{γ}*(y, c

*) = (ψ(y), c*

^{0}*), it fol- lows that these operations make Ext*

^{0}_{Λ}

*JT*(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_{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** ^{0}*) =

*cl(5*

*(S*

_{A}*⊕S*

*)4*

^{0}*),*

_{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 = 1

*A*

*◦*0 :

*A*

*A*//

^{i}*//*

^{A}*A⊕C*

^{π}*////*

^{C}*C*coincides with the class of

*A*oo 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-
le*A, one has a natural connecting homomorphism of abelian monoids*

*δ(F, A) : Hom*_{Λ}(K, A) //Ext_{Λ}*JT*(M, A)
defined by

*δ(F, A)(f* :*K* //*A) =cl(A*oo ^{f}*K* // //*L* ////*M*)*.*

If*L*is a projective Λ-semimodule, than*δ(F, A) is surjective [6]. Indeed, in this case*
any object *S*:*A*oo ^{g}*X* // //*Y* ////*M* of *Ext*_{Λ}*JT*(M, A) admits a commuta-
tive diagram

*gϕ◦F* : *A*oo ^{gϕ}*K*

*ϕ*

²² // //*L* ////

*ψ*

²²

*M*

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

Proposition 2.2. *LetCbe a normal*Λ-semimodule and P:*R* //* ^{µ}* //

*P*

*////*

^{ε}*C*

*a*

*projective presentation ofC. Then* *δ(P, A) : Hom*_{Λ}(R, A) //Ext_{Λ}*JT*(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_{Λ}*JT*(C, A),

*ω*^{0}

ffM M

M MM M

where *ω* is a homomorphism of abelian monoids such that *ωµ** ^{∗}* = 0. Take

*cl(S*:

*A*oo

^{f}*X*// //

*Y*////

*C*)

*∈*Ext

_{Λ}

*JT*(C, A). There is a morphism (ϕ, ψ,1

*) : P*

_{C}*−→S. Defineω*

*(cl(S)) =*

^{0}*ω(f ϕ). If (ϕ*

^{0}*, ψ*

^{0}*,*1

*) :P*

_{C}*−→S*is another morphism it follows from Proposition 1.9 that

*ϕ*+

*βµ*=

*ϕ*

*+*

^{0}*β*

^{0}*µ*for some Λ-homomorphisms

*β, β** ^{0}*:

*P*//

*X*. Whence

*ω(f ϕ** ^{0}*) =

*ω(f ϕ*

*) +*

^{0}*ωµ*

*(f β*

^{∗}*) =*

^{0}*ω(f ϕ*

*+*

^{0}*f β*

^{0}*µ)*

=*ω(f ϕ*+*f βµ) =ω(f ϕ) +ωµ** ^{∗}*(f β) =

*ω(f ϕ),*i.e.,

*ω(f ϕ*

*) =*

^{0}*ω(f ϕ). On the other hand, a commutative diagram*

*S*: *A* oo ^{f}*X*

*g*

²² // //*Y* ////

*h*

²²

*C*

*S** ^{0}* :

*A*

^{f}*X*

^{0}oo *0* // //*Y** ^{0}* ////

*C,*

where *S*^{0}*∈ Ext*_{Λ}*JT*(C, A), yields the morphism (gϕ, hψ,1* _{C}*) :P

*−→S*

*. Therefore*

^{0}*ω*

*(cl(S*

^{0}*)) =*

^{0}*ω(f*

^{0}*gϕ) =ω(f ϕ). Thusω*

*is well defined. Clearly*

^{0}*ω*=

*ω*

^{0}*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
P:*R* // * ^{µ}* //

*P*

*////*

^{ε}*C*

*be the chosen projective presentation of*

*C. Then*

*θ(C, A) : Ext*_{Λ}(C, A) //Ext_{Λ}*JT*(C, A)

*defined byθ(C, A)([ϕ]) =cl(ϕ◦*P) =*cl(A*oo ^{ϕ}*R* // * ^{µ}* //

*P*

*////*

^{ε}*C*)

*is a natural*

*isomorphism of abelian monoids.*

*Proof.* Since *δ(P, A) : Hom*_{Λ}(R, A) //Ext_{Λ}*JT*(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* 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 *E*:*A* //* ^{λ}* //

*B*

*////*

^{τ}*C*be a short exact sequence of Λ-semimodules and

*α*:

*A*//

*A*

*a Λ-homomorphism. Following [10], denote by*

^{0}*B*

*α*the Λ-semimo- dule

*A*

^{0}*⊕B*modulo the following congruence relation: (a

^{0}_{1}

*, b*1)ρ

*α*(a

^{0}_{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*

^{0}_{1}=

*α(a*1) +

*a*

^{0}_{2}; and define

*λ*

*α*:

*A*

*//*

^{0}*B*

*,*

_{α}*τ*

*:*

_{α}*B*

*//*

_{α}*C*and

*ξ*

*α*:

*B*//

*B*

*by*

_{α}*λ*

*α*(a

*) = [a*

^{0}

^{0}*,*0],

*τ*

*α*([a

^{0}*, b]) =τ(b) andξ*

*α*(b) = [0, b], respectively.

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

*B*

*τ* ////*C* *of*Λ-semimodules and a Λ-homomorphism *α*:*A* //*A*^{0}*withA*^{0}*can-*
*cellative. Then*

*αE*:*A*^{0}^{λ}* ^{α}* //

*B*

*α*

*τ**α* //*C*

*is a short exact sequence of* Λ-semimodules and(α, ξ*α**,*1* _{C}*)

*a morphism from*

*E*

*to*

*αE. Furthermore, ifE∈*

## T

^{then}^{αE}^{∈}## T

^{.}Also note that

*E≡E** ^{0}*=

*⇒αE≡αE*

^{0}*,*(3.2)

*E> E−* * ^{0}* =

*⇒αE> αE−*

^{0}*.*(3.3) It is directly verified in [10] that

(α^{0}*α)E≡α** ^{0}*(αE), 1

_{G}*E≡E*and

*α(Eγ)≡*(αE)γ, (3.4) where

*E*:

*G*// //

*B*////

*C*is a short exact sequence with

*G*a Λ-module,

*γ*:

*C*

*//*

^{0}*C*a homomorphism of Λ-semimodules, and

*α*:

*G*//

*G*

*and*

^{0}*α*

*:*

^{0}*G*

*//*

^{0}*G*

*are homomorphisms of Λ-modules. These equivalences together with (1.2), (1.3) and (3.2) show that*

^{00}*E*

_{Λ}(C, G) is a functor from (Λ-SMod)

^{op}

*×(Λ-Mod)*toSet(E

*7−→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) +b*1=

*λ(a) +b*2

*,a∈A,b*1

*, b*2

*∈B*

*impliesb*1=

*b*2

*.*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*//

^{i}*//*

^{A}*A⊕C*

^{π}*////*

^{C}*C*is proper if and only if

*A*is cancellative. Also observe that any short exact sequence

*G* // //*B* ////*C* with*G*a Λ-module is proper.

Proposition 3.6. *For every short exact sequence* *E*:*A* //* ^{λ}* //

*B*

*////*

^{τ}*C*

*of*Λ-se-

*mimodules and every*Λ-homomorphism

*α*:

*A*//

*A*

^{0}*with*

*A*

^{0}*cancellative,*

*αE*:

*A** ^{0}* //

^{λ}*//*

^{α}*B*

*α*

*τ**α* ////*C* *is proper.*

*Proof.* Assume *λ**α*(a* ^{0}*) + [a

^{0}_{1}

*, b*1] =

*λ*

*α*(a

*) + [a*

^{0}

^{0}_{2}

*, b*2],

*a*

^{0}*, a*

^{0}_{1}

*, a*

^{0}_{2}

*∈*

*A*

*,*

^{0}*b*1

*, b*2

*∈B, i.e.,*[a

*+*

^{0}*a*

^{0}_{1}

*, b*1] = [a

*+*

^{0}*a*

^{0}_{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*

*+*

^{0}*a*

^{0}_{1}=

*α(a*1) +

*a*

*+*

^{0}*a*

^{0}_{2}. Whence, since

*A*

*is cancellative,*

^{0}*λ(a*1) +

*b*1 =

*λ(a*2) +

*b*2 and

*α(a*2) +

*a*

^{0}_{1}=

*α(a*1) +

*a*

^{0}_{2}. That is, [a

^{0}_{1}

*, b*1] = [a

^{0}_{2}

*, b*2].

Remark 3.7. For a short exact sequence *E*:*A* // //*B* ////*C* and a Λ-homo-
morphism *γ*:*C** ^{0}* //

*C*,

*Eγ*is proper if and only if

*λ(a) +b*1=

*λ(a) +b*2,

*a∈A,*

*b*1

*, b*2

*∈τ*

*(γ(C*

^{−1}*)) implies*

^{0}*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* *E*:*A* // * ^{λ}* //

*B*

*////*

^{τ}*C*

*of*Λ-semimodules and a Λ-homomorphism

*α*:

*A*//

*A*

^{0}*withA*

^{0}*cancellative. And*

*assume that*

*E*

*:*

^{0}*A*

*//*

^{0}

^{λ}*//*

^{0}*B*

^{0}

^{τ}*////*

^{0}*C*

^{0}*is a proper short exact sequence and that*(α, β, γ)

*is a morphism fromE*

*toE*

^{0}*. Then there exists a unique*Λ-homomorphism

*β*

*:*

^{0}*B*

*α*//

*B*

^{0}*such that*(1

_{A}*0*

*, β*

^{0}*, γ)is a morphism fromαEtoE*

^{0}*andβ*=

*β*

^{0}*ξ*

*α*

*.*

*In particular, ifγ*= 1

_{C}*thenαE> E−*

^{0}*.*

*Proof.* Define *β** ^{0}*:

*B*

*α*//

*B*

*by*

^{0}*β*

*([a*

^{0}

^{0}*, b]) =*

*λ*

*(a*

^{0}*) +*

^{0}*β(b). Assume [a*

^{0}_{1}

*, b*1] = [a

^{0}_{2}

*, b*2], i.e.,

*λ(a*1) +

*b*1=

*λ(a*2) +

*b*2 and

*α(a*2) +

*a*

^{0}_{1}=

*α(a*1) +

*a*

^{0}_{2}for some

*a*1

*, a*2

*∈*

*A. Then, since (α, β, γ) :*

*E*

*−→*

*E*

*is a morphism, we have*

^{0}*λ*

^{0}*α(a*1) +

*β(b*1) =

*λ*

^{0}*α(a*2) +

*β(b*2) and

*λ*

^{0}*α(a*2) +

*λ*

*(a*

^{0}

^{0}_{1}) =

*λ*

^{0}*α(a*1) +

*λ*

*(a*

^{0}

^{0}_{2}). These equations give

*λ*

^{0}*α(a*

_{2}) +

*λ*

*(a*

^{0}

^{0}_{1}) +

*β(b*

_{1}) =

*λ*

^{0}*α(a*

_{1}) +

*λ*

*(a*

^{0}

^{0}_{2}) +

*β(b*

_{1}) =

*λ*

^{0}*α(a*

_{2}) +

*λ*

*(a*

^{0}

^{0}_{2}) +

*β(b*

_{2}).

Whence*λ** ^{0}*(a

^{0}_{1}) +

*β(b*1) =

*λ*

*(a*

^{0}

^{0}_{2}) +

*β(b*2) since

*E*

*is proper. Hence*

^{0}*β*

*is well defined.*

^{0}Clearly, *β** ^{0}* is a Λ-homomorphism with

*β*

^{0}*λ*

*α*=

*λ*

*,*

^{0}*γτ*

*α*=

*τ*

^{0}*β*

*and*

^{0}*β*=

*β*

^{0}*ξ*

*α*. If

*β*

*:*

^{00}*B*

*α*//

*B*

*is another Λ-homomorphism such that*

^{0}*β*=

*β*

^{00}*ξ*

*α*and (1

_{A}*0*

*, β*

^{00}*, γ)*is a morphism from

*αE*to

*E*

*, then*

^{0}*β*

*([a*

^{00}

^{0}*, b]) =β*

*([a*

^{00}

^{0}*,*0] + [0, b]) =

*β*

^{00}*λ*

*α*(a

*) +*

^{0}*β*

^{00}*ξ*

*(b) =*

_{α}*λ*

*(a*

^{0}*) +*

^{0}*β(b) =β*

*([a*

^{0}

^{0}*, b]).*

Let *E*:*A* //* ^{λ}* //

*B*

*////*

^{τ}*C*be a short exact sequence with

*A*cancellative, and let

*α*:

*A*//

*A*

*,*

^{0}*α*

*:*

^{0}*A*

*//*

^{0}*A*

*and*

^{00}*γ*:

*C*

*//*

^{0}*C*be Λ-homomorphisms with

*A*

*and*

^{0}*A*

*cancellative. Then Propositions 3.1 and 3.6 and Lemma 3.8 imme- diately provide the morphisms*

^{00}(1_{A}*00**, ν,*1* _{C}*) : (α

^{0}*α)E−→α*

*(αE), (1*

^{0}

_{A}*, ξ*

_{1}

*A**,*1* _{C}*) :

*E−→*1

_{A}*E*and (1

_{A}*0*

*, ι,*1

_{C}*0*) :

*α(Eγ)−→*(αE)γ,

where *ν*:B*α*^{0}*α* //(B*α*)*α** ^{0}* and

*ι*: (B

*)*

^{γ}*α*//(B

*α*)

*are defined by*

^{γ}*ν([a*

^{00}*, b]) =*[a

^{00}*,*[0, b]] and

*ι*¡

[a^{0}*,*(b, c* ^{0}*)]¢

= ([a^{0}*, b], c** ^{0}*), respectively. Thus

(α^{0}*α)E> α−* * ^{0}*(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^{0}*α)E* *≡* *α** ^{0}*(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*(b1) = *ξ*_{1}

*A*(b2), i.e., [0, b1] = [0, b2]. Then *λ(a*1) +*b*1 = *λ(a*2) + *b*2 and
1* _{A}*(a2) + 0 = 1

*(a1) + 0 for some*

_{A}*a*1

*, a*2

*∈A. Whence, sinceE*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)/h −*>i,*

where*h −>i*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* *7−→* *Eγ* and *E* *7−→* *αE*
make Ext_{Λ}*T(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_{Λ}*P T*(C, A) =*Ext*_{Λ}*P T*(C, A)/h −*>i,*

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

Ext_{Λ}*P T*(C, A) ^{Γ(C,A)}// Ext_{Λ}*T*(C, A)

Γ* ^{0}*(C,A)

oo *,* Γ(cl(E)) =*cl(E),* Γ* ^{0}*(cl(E)) =cl(1

_{A}*E)*are natural, and Γ

*Γ = 1 and ΓΓ*

^{0}*= 1.*

^{0}In order to prove the following theorem, note that

*α◦E∼*1_{A}*0* *◦αE* (3.12)

for any short exact sequence *E*:*A* //* ^{λ}* //

*B*

*////*

^{τ}*C*and any Λ-homomorphism

*α*:

*A*//

*A*

*with*

^{0}*A*

*cancellative. Indeed, the morphism (α, ξ*

^{0}*α*

*,*1

*) :*

_{C}*E−→αE*gives the commutative diagram

*α◦E*: *A** ^{0}* oo

^{α}*A*//

*//*

^{λ}*α*

²²

*B* * ^{τ}* ////

*ξ**α*

²²

*C*

1_{A}*0* *◦αE*: *A*^{0}*A** ^{0}* //

^{λ}*//*

^{α}*B*

*α*

*τ**α* ////*C.*

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

*Then*

*χ(C, A) : Ext*_{Λ}*T(C, A)* //Ext_{Λ}*JT*(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

*χ** ^{0}*(C, A) : Ext

_{Λ}

*JT*(C, A) //Ext

_{Λ}

*T*(C, A) by

*χ** ^{0}*(C, A)¡

*cl(S*:*A*oo ^{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*: *A*oo ^{f}*X* // ^{κ} //

*ϕ*

²²

*Y* * ^{σ}* ////

*ψ*

²²

*C*

*S** ^{0}*:

*A*oo

^{g}*X*

^{0}^{κ}

*//*

^{0}*Y*

^{0}

^{σ}*////*

^{0}*C*

with*S*^{0}*∈ Ext*_{Λ}*JT*(C, A). This commutative diagram and the morphism (g, ξ*g**,*1* _{C}*) :

*S*

^{0}*−→gS*

*yield the following commutative diagram*

^{0}*S* : *X* // ^{κ} //

*f*

²²

*Y* * ^{σ}* ////

*ξ*_{g}*ψ*

²²

*C*

*gS** ^{0}* :

*A*

^{κ}

*0**g* //*Y*_{g}^{0}^{σ}

*g**0* ////*C.*

Whence, by Proposition 3.6 and Lemma 3.8,*f S> gS−* * ^{0}*. Hence

*χ*

*(C, A) is well de- fined. Further, by (3.12),*

^{0}*χχ*

*(cl(S)) =*

^{0}*χ(cl(f S)) =cl(1*

_{A}*◦f S) =cl(f◦S) =cl(S).*

On the other hand,*χ*^{0}*χ(cl(E)) =χ** ^{0}*(cl(1

_{A}*◦E)) =cl(1*

_{A}*E) =cl(E) sinceE>−*1

_{A}*E.*

Thus*χ(C, A) is a bijection. Finally, consider the diagram*
Ext_{Λ}*T*(C, A) * ^{χ(C,A)}* //

Ext_{Λ}*T(γ,α)*

²²

Ext_{Λ}*JT*(C, A)

Ext_{Λ}*JT(γ,α)*

²²

Ext_{Λ}*T*(C^{0}*, A** ^{0}*)

^{χ(C}*0**,A** ^{0}*) //Ext

_{Λ}

*JT*(C

^{0}*, A*

*),*

^{0}where *α*:*A* //*A** ^{0}* is a homomorphism of cancellative Λ-semimodules and

*γ*:

*C*

*//*

^{0}*C*a homomorphism of Λ-semimodules. Using (3.12) and (2.1), we obtain

*χ(C*^{0}*, A** ^{0}*) Ext

_{Λ}

*T*(γ, α)(cl(E)) =

*cl(1*

_{A}*0*

*◦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]. 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*with

*A*cancellative which satisfies the following conditions: (0 //

*Z*

*Z)∈*

## C

for every Λ-semimodule*Z; ifE∈*

## C

^{then}

*αE,Eγ∈*

## C

^{.}

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

*cl(E) +cl(E** ^{0}*) =

*cl*¡

*5** _{A}*(E

*⊕E*

*)*

^{0}*4*

*¢*

_{C}*.*As a corollary of Theorems 2.3 and 3.13 we have

Theorem 3.15. *AssumeAis a cancellative*Λ-semimodule and*Ca normal*Λ-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_{Λ}*T*(C, A)

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

*Proof.* *w(C, A) =χ** ^{0}*(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) = Ext^{1}* _{K(Λ)}*(H, G) =

*E*

_{Λ}(H, G),

where Ext^{1}* _{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

Ext^{1}* _{K(Λ)}*(H, G) //Ext

_{Λ}

*JT*(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 Λ* ^{0}*be additively cancellative semirings. In [8], for any con-
travariant additive functor

*T*: (Λ-CSMod) //(Λ

*-CSMod), we constructed and studied right derived functors R*

^{0}

^{n}*T*: (Λ-CSMod) //(Λ

*-CSMod),*

^{0}*n*= 0,1,2, . . .. In particular, we described R

*Hom*

^{n}_{Λ}(−, A)(C) by means of certain

*n-*fold extensions of

*C*by

*A. According to that description, forn*= 1,

*H*

*∈*(Λ-Mod) and

*A∈*(Λ-CSMod), one has a natural isomorphism of abelian groups

R^{1}Hom_{Λ}(−, A)(H)*∼*=*Ext*^{1}_{Λ}(H, A)/h −*>i,*

where*Ext*^{1}_{Λ}(H, A) denotes the category of short exact sequences *A* // //*B* ////*H*
of Λ-semimodules with*B*cancellative. (Note that*R*^{1}Hom_{Λ}(−, A)(H) =*R*^{1}Hom_{Λ}(−,
*U*(A))(H)*∼*= Ext^{1}* _{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) // Ab.) On the other hand, if

*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*

^{0}*∈X*such that

*σ(x*

*) =*

^{0}*−σ(x).*

Then *x** ^{0}*+

*x*= κ(a) for some

*a*

*∈*

*A; and we obtain*κ(a) +

*x*1 =

*x*

*+*

^{0}*x*+

*x*1 =

*x*

*+*

^{0}*x*+

*x*2 = κ(a) +

*x*2. Whence

*x*1 =

*x*2 since

*A*//

^{κ}//

*X*

*////*

^{σ}*H*is proper.) Hence there is a natural isomorphism

R^{1}Hom_{Λ}(−, A)(H)*∼*= ExtΛ*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 toR^{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 GrothendieckK(Λ)-module of*

*C.*

First of all, observe that the Baer addition of extensions,*E+E** ^{0}* =

*5*

*(E⊕E*

_{G}*)4*

^{0}*, makes*

_{C}*E*

_{Λ}(C, G) an abelian monoid [4]. In addition the

*≡-equivalence class of*0 :

*G*//

^{i}*//*

^{G}*G⊕C*

^{π}*////*

^{C}*C*serves as a neutral element. Furthermore, a straightfor- ward verification shows that

*α(E*+

*E*

*)*

^{0}*≡αE*+

*αE*

*, (E+*

^{0}*E*

*)γ*

^{0}*≡Eγ*+E

^{0}*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 *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 *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*

_{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 thatαE,Eγ*and

*E⊕E*

*are split whenever*

^{0}*E*and

*E*

*are split. Consequently,*

^{0}*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*
*E*: *G* // * ^{λ}* //

*B*

*////*

^{τ}*β*

²²

*C*

*E** ^{0}*:

*G*//

^{λ}*//*

^{0}*B*

^{0}

^{τ}*////*

^{0}*C*

*of Takahashi extensions of a*Λ-semimodule*C* *by a*Λ-module*G. IfE*^{0}*is an Inas-*
*saridze extension, then* *β* *is an isomorphism.*

Theorem 4.2 ([4,7]). *LetK(C)be the GrothendieckK(Λ)-module of a*Λ-semimo-
*duleC, and* *k** _{C}* :

*C*//

*K(C)*

*the canonical*Λ-homomorphism. Then the natural

*homomorphism*

Ext^{1}* _{K(Λ)}*(K(C), G) = Ext

_{Λ}

*I(K(C), G)*

^{Ext}

^{Λ}

^{I(k}

^{C}*//Ext*

^{,G)}_{Λ}

*I(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* // ^{i}* ^{G}* //

*G⊕C*

^{π}*////*

^{C}*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 a*Λ-module, then the map

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

*Proof.* Define *q** ^{0}*:

*B*//

*A⊕C*by

*q*

*(b) = (λ*

^{0}*(b*

^{−1}*−ντ(b)), τ*(b)). Then

*q*

^{0}*q*= 1 and

*= 1.*

^{0}For any Λ-module*G*and any normal Λ-semimodule*C, 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(f*P), 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_{Λ}*T*(C, G)

where*m(C, G) is defined bym(C, G)(cl(E)) =cl*_{h−}* _{>i}*(E), and

*w(C, G) byw(C, G)([ϕ])*

=*cl*_{h−}* _{>i}*(ϕP) (see Theorem 3.15). One has

*w(C, G)η(C, G) (cl(E)) =w(C, G)([ϕ]) =*

*cl*

_{h−}*(ϕP). But, by Lemma 3.8,*

_{>i}*ϕP> E, that is,−*

*cl*

_{h−}*(ϕP) =*

_{>i}*cl*

_{h−}*(E). Hence the diagram is commutative. Therefore, since*

_{>i}*m(C, G) is a homomorphism andw(C, G)*an isomorphism,

*η(C, G) is a homomorphism.*

Proposition 4.5. *For any* Λ-module*Gand any normal*Λ-semimodule*C,*
Ker(µ(C, G)) =*E*_{Λ}*S(C, G),*

*that is, the sequence*

*E*_{Λ}*S(C, G)* * ^{j(C,G)}* //

*E*

_{Λ}(C, G)

*// Ext*

^{η(C,G)}_{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*

_{Λ}

*S(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 of*η(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* // ^{i}* ^{G}* //

*G⊕C*

^{π}*////*

^{C}*C,*

where *β* =*i*_{G}*g*+*i*_{C}*ε, is commutative. By Lemma 3.8, this commutative diagram*