ESTUDIO DE IMPACTO AMBIENTAL

If the purpose of the splitting lemma is to understand duality, then the purpose of duality, for us, is to understand exchange. An exchange is an operation of form ω7→ω−α∪β, whereω and ω−α∪β are product (respectively, coproduct) structures, and

α⊆α∩ω, β∩ω⊆α. (6.4.1)

In practice the two conditions encoded in (6.4.1) may be safely ignored; we impose them only to avoid degenerate cases, e.g. where the sets (ω−α) and β intersect nontrivially, hence where β “adds” elements toω that are already present.

A pair (α, β) that meets all the given criteria as anexchange pair for ω. We refer to an exchange operation of form

ω7→ω− {a} ∪ {b}

as anelementary exchange, and to the pair (a, b) as anelementary exchange pair, or simply as an exchange pair, where context leaves no room for confusion.

Exchange operations are fundamental both to modern methods in matrix algebra and to matroid theory. One of the simplest questions surrounding this subject is the following: Givenω, when is (α, β) an exchange pair? This question reduces determining when (a, b) is an elementary exchange pair, since, for example, ifω is a coproduct thenω−α∪β is a coproduct iff

is a coproduct.

The question of determining exchange pairs is therefore answered exhaustively by Lemma 6.4.6, the Exchange Lemma. This result is mathematically equivalent to the splitting lemma, and retains much of its flavor. Like the splitting lemma, it has several variations, the Steinitz Exchange Lemma prominent among them. This correspondence outlines the foundational overlap between homological algebra matroid theory. As with the splitting lemma, the proof is left as an exercise to the reader.

Lemma 6.4.6 (Exchange Lemma). Let {f, g} and {fop, gop} be product and coproduct structures, respectively.

1. An unordered pair {f, h} is a product iffg]h is invertible.

2. An unordered pair {fop, hop} is a coproduct iff hop(gop)[ is invertible.

To gain some familiarity with kernel maps, as well as the splitting and exchange lemmas, let us compute some examples.

Example 1: Kernels in R2

Let λbe the canoncial indexed coproduct structure onR2, and let

be any endomorphism. Ifaij =λ]_iT λj, then T(λ, λ) = λ1 λ2 λ]₁ a11 a12 λ]₂ a21 a22 .

Suppose thata116= 0, and let kbe the map R→R2 such that

1(λ, k) = k λ]₁ −a−₁₁1a12 λ]₂ 1 . Since T(λ, λ)1(λ, k) = k λ]₁ 0 λ]₂ a22−a21a−111a12

one has, in particular, that

λ]₁T k= 0.

As the kernel ofλ]₁T has dimension 1, it follows that

I(k) = K(λ]₁T).

Example 2: Preadditive kernels

The construction described in Example 1 is in fact very general. LetW and Wopbe any two objects with coproduct structures λ={λ1, λ2} andλop= (λop₁ , λop₂ ), respectively, and

fix

T :W →Wop.

For convenience, letaij =λopi T λj.

Proposition 6.4.7. If a11 is invertible, then in any preadditive category

k=λ2−λ1a−111a12

is kernel to λop₁ T.

Proof. The matrix representation ofT with respect to λop and λis

T(λop, λ) =

λ1 λ2

λop₁ a11 a12

λop₂ a21 a22

by definition, and it is simple to check that

1(λ, k) =

k λ]₁ −a−₁₁1a12

λ]₂ 1 .

As in Example 1, we have T(λop, λ)1(λ, k) = k λ]₁ 0 λ]₂ a22−a21a−111a12 .

In particular, λop_{T k = 0. In the category finite-dimensional k-linear spaces and maps}

between them, one can confirm that kis a kernel toλopT by checking

I(k) = K(λop₁ T),

via dimension counting. To establish the result for arbitrary preadditive categories, however, we will check the definition directly.

Recall that ef =f[f] is the idempotent projection operator generated by a morphism

f in a (co)product structure ω, and thatP

f∈ωef = 1. Ifh is any morphism such that

0 =λop₁ T h, then 0 =λop₁ T(eλ1+eλ2)h=a11λ ] 1h+a12λ]2h, whence λ]₁h=−a−₁₁1a12λ]2h, and therefore h= (λ2−λ1a11−1a12)λ]2h.

To wit, the diagram • k & & λ]₂h o o h x x • •

commutes. Therefore hfactors through kwhenλop₁ T h= 0. Sinceλ]₂k= 1, we have thatk

is a monomorphism. Thus the factorization is unique. The desired conclusion follows.

Example 3: Cokernels in R2

As in Example 1, letλbe the canonical unindexed coproduct structure on R2, and fix

T :R2 →R2 If aij =λ]iT λj, then T(λ, λ) = λ1 λ2 λ]₁ a11 a12 λ]₂ a21 a22 .

Suppose thata116= 0, and let kop be the map R→R2 such that

1(kop, λ) =

λ1 λ2

Since

1(kop, λ)T(λ, λ) =

λ1 λ2

kop 0 a22−a21a−₁₁1a12 .

one has, in particular, that kopT λ1 = 0. As the image of T λ1 has dimension 1, it follows

that

I(T λ1) = K(kop).

Since, in addition, kis an epimorphism, we have that kop is a cokernel toT λ1.

Example 4: Preadditive cokernels

Like the kernel in Example 1, the cokernel constructed in Example 3 has a natural analog for arbitrary two-element corpoducts. Let W andWop be any two objects with coproduct structures λ={λ1, λ2} and λop= (λop1 , λ

op

2 ), respectively, and fix

T :W →Wop.

As before, setaij =λopi T λj.

The proof of Proposition 6.4.8 entirely dual to that of Proposition 6.4.7. The details are left as an exercise to the reader.

Proposition 6.4.8. If a11 is invertible, then in any preadditive category

is cokernel to T λ1.