CC BY
14
УДК 512.66
The Two-Square Lemma and the Connecting Morphism in a Preabelian Category
Yaroslav A. Kopylov*
Sobolev Institute of Mathematics, pr. Akad. Koptyuga, 4, Novosibirsk, 630090
Russia and
Novosibirsk State University Pirogova, 2, Novosibirsk, 630090
Russia
Received 23.02.2012, received in revised form 16.03.2012, accepted 16.04.2012
We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994■ We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.
Keywords: strict morphism, preabelian category, pullback, pushout, semi-stable (co)kernel, Snake Lemma, connecting morphism.
Introduction
One of the most important diagram assertions in homological algebra is the so-called Snake Lemma which makes it possible to obtain homological sequences from short exact sequences of complexes. It always holds in an abelian category. However, in the more general context of preabelian categories, The Snake Lemma fails without additional assumptions on the initial diagram. The main reasons are that the notions of kernel and monomorphism (respectively, of cokernel and epimorphism) do not coincide in a preabelian category and that kernels (respectively, cokernels) do not "survive" under pushouts (respectively, pullbacks).
The question of the validity of the Snake Lemma in the nonabelian case was studied by several authors for classes of additive categories (see, e.g., [1-5]) and in some classes of nonabelian categories (see, e.g., [6,7]). The key properties of the morphisms in the initial diagram required for the exactness of the Ker-Coker-sequence are "strictness" and stability under pushouts (pullbacks) of some monomorphisms (epimorphisms), or their weaker analogs "exactness" and "modularity" [7].
Even the existence of a connecting morphism in the Ker-Coker-sequence, valid in abelian categories (and even in quasi-abelian categories [4] and in their nonadditive counterpart, Grandis homological categories [7]), cannot be guaranteed in general preabelian categories without extra "semi-stability" assumptions (see [2]). The construction of the connecting morphism in [2] involves a preabelian version of a special case of the Two-Square Lemma of Fay-Hardie-Hilton [8, Lemma 3].
Theorem 0.1 (The Two-Square Lemma). Suppose that the following diagram in an abelian
* yakop @math.nsc.ru © Siberian Federal University. All rights reserved
category has exact rows:
A
A'
Let
be a pullback and let
V'
Q'
a'
B' A
a A'
B
ß B'
C
C
7
C
(1)
(2)
v'
V
C'
B
Q
(3)
be a pushout. Then
(i) there exists a unique morphism 6 : Q ^ B' such that 6t = ß, 6t' = ;
(ii) there exists a unique morphism p : B ^ Q' such that ap = p, a'p = ß;
(iii) there exists a unique morphism n : Q ^ Q' such that nt = p, a'n = 6,
0.
The proof in [8] remains valid in any preabelian category. The Two-Square Lemma of [8] also claims that if is a monomorphism then so is n and if ip is an epimorphism then so is n
In [2], Generalov proved the following assertion:
Theorem 0.2. Consider a diagram of the form (1) in a preabelian category. If is a semi-stable kernel and p is a semi-stable cokernel then n is an isomorphism.
Below we study the question when n is a monomorphism, an epimorphism, a kernel, a cokernel in a preabelian category.
The article is organized as follows. In Sec. 1., we give basic definitions and facts about preabelian categories. In Sec. 2., we prove the main assertion of the article, Theorem 2.1, explaining what conditions on the initial diagram (1) guarantee each of the above-mentioned properties of n. In Sec. 3., we prove the equivalence up to sign of two definitions of a connecting morphism of the Ker-Coker -sequence in a preabelian category.
V*
v
a
V
a
7
T
T
1. Preabelian Categories
A preabelian category is an additive category with kernels and cokernels. In a preabelian category, every morphism a admits a canonical decomposition
a = (im a)a(coim a), where im a = ker coker a, coim a = coker ker a.
A morphism a is called strict if a is an isomorphism. A preabelian category is abelian if and only if every morphism in it is strict. Note that
strict monomorphisms = kernels, strict epimorphisms = cokernels.
Lemma 1.1. [4,9-11] The following hold in a preabelian category.
(i) A morphism a is a kernel if and only if a = im a, a morphism a is a cokernel if and only if a = coim a.
(ii) A morphism a is strict if and only if a is representable as a = ai a0, where a0 is a cokernel, a1 is a kernel; in this case, a0 = coim a and a1 = im a.
(iii) Suppose that the commutative square
C —-^ D
(4)
A -► B
P
is a pullback. Then ker f = a ker g. If f = ker h for some h then g = ker(h^). If f is a monomorphism then g is a monomorphism; if f is a kernel then g is a kernel.
In the dual manner, assume that (4) is a pushout. Then coker g = (coker f )fi. If g = coker e for some e then f = coker (ae). If g is an epimorphism then f is an epimorphism; if g is a cokernel then f is a cokernel.
A kernel g in a preabelian category is called semi-stable [10] if, for every pushout of the form (4), f is a kernel too. A semi-stable cokernel is defined in the dual way. Examples of non-semi-stable cokernels may be found, for example, in [12-15] and non-semi-stable kernels are shown in [10]. If all kernels and cokernels are semi-stable then the preabelian category is called quasi-abelian [16].
Lemma 1.2. [10,17] The following hold in a preabelian category:
(i) if gf is a semi-stable kernel then so is f; if gf is a semi-stable cokernel then so is g;
(ii) if f and g are semi-stable kernels (cokernels) and the composition gf is defined then gf is a semi-stable kernel (cokernel);
(iii) if (4) is a pushout and f is a semi-stable kernel then so is g; if (4) is a pullback and g is a semi-stable cokernel then so is f.
If a preabelian category satisfies the following two weaker axioms dual to one another then it is called P-semi-abelian or semi-abelian in the sense of Palamodov [18]: if (4) is a pushout and g is a kernel then f is a monomorphism; if (4) is a pullback and f is a cokernel then g is an epimorphism. Until recently it was unclear whether every P-semi-abelian category is quasi-abelian (Raikov's Conjecture); this was disproved by Bonet and Dierolf [12] and Rump [14,15]. It turned out that, for example, the categories of barrelled and bornological locally convex spaces are P-semi-abelian but not quasi-abelian (see [15]). In general preabelian categories, kernels (cokernels) may push out (pull back) even to zero morphisms (see [10,13]).
In [19] Kuz'minov and Cherevikin proved that a preabelian category is P-semi-abelian in the above sense if and only if, in the canonical decomposition of every morphism a, a = (im a)acoim a, the central morphism a is a bimorphism, that is, a monomorphism and an epimorphism simultaneously.
Lemma 1.3. [4,19] The following hold in any P-semi-abelian category:
(i) if gf is a kernel then f is a kernel; if gf is a cokernel then g is a cokernel;
(ii) if f,g are kernels and the composition gf is defined then gf is a kernel; if f,g are cokernels and the composition gf is defined then gf is a cokernel;
(iii) if gf is strict and g is a monomorphism then f is strict; if gf is strict and f is an epimorphism then g is strict.
We observe that, in fact, in a preabelian category, items (i) and (ii) of Lemma 1.3 are equivalent to P-semi-abelianity (see [20] for details).
The following lemma is due to Yakovlev [21].
Lemma 1.4. For every morphism a in a preabelian category, ker a coker im a.
kercoim a, coker a
A B A ... in a preabelian category is said to be exact at B if im a = ker b.
A sequence ...
As follows from Lemma 1.4, this is equivalent to the fact that coker a = coim b.
2. The Two-Square Lemma
We begin with a lemma which, being itself of an independent interest, will be used below. It is a generalization of [19, Theorem 3] and [22, Lemma 6].
Lemma 2.1. Let
A
A
p i
Bi
B2
91
C
C
p 2
92
be a commutative diagram in a preabelian category.
(i) If pi = ker qi, q2p2 = 0, p2 is a monomorphism then r is a monomorphism.
(ii) Suppose that pi = ker qi, p2 = ker q2, p2 and im qi are semi-stable kernels, and qi is strict. Then r is a semi-stable kernel.
The dual assertions also hold.
Proof. (i) Suppose that rx = 0 and show that then x = 0. We have qix = q2rx = 0. Since pi = ker qi, we infer that x = piy for some y. Then p2y = rpiy = rx = 0. Since p2 is a monomorphism, y=0 and, thus, x = piy = 0.
(ii) Represent qi as qi = qiqi', q2 = q'2q2, where qj' = coim qj : Bj ^ Kj, j = 1, 2. By assumption, qi = imqi. Since coimqi = cokerpi and coimq2 = cokerp2, there exists a unique morphism w : Ki ^ K2 with w coim qi = (coim q2)r. For this w, we have qi = q2w. Since, by hypothesis, qi is a semi-stable kernel, w is also a semi-stable kernel (Lemma 1.2).
Consider the pushout
A Bi
B2
F.
Since uirpi = uip2 = u2pi, we have (uir — u2)pi = 0. Therefore, there exists a unique morphism s : Ki ^ F such that uir — u2 = s coim qi.
Consider the pushout
Ki
F
K2
S.
Put p = w'u\ — s'coimq2. We infer
pr = (w'ui — s'coim q2)r = w'u2 + w's coim qi — s'(coim q2)r =
= w'U2 + w's coim qi — w's coim qi = w'u2.
Thus, pr = w' u2. Since p2 and w are semi-stable kernels, so are u2 and w'. Now, by Lemma 1.2(ii), pr = w'u2 is a semi-stable kernel as a composition of semi-stable kernels. Thus, by Lemma 1.2(i), r is a semi-stable kernel. The lemma is proved. □
r
ui
w
s
s
w
We will also need the following preabelian version of Lemma 1 of [8]. It also generalizes Lemma 1.1(iii).
Lemma 2.2. The following hold. (i) If in the commutative diagram
A'
B
ß
B'
C
Y
->■ C'
(5)
the square p>'3 = YV is a pullback and the lower row in (5) is exact then there exists a unique morphism 1 : A' ^ B such that 31 = 1', = 0. If, in addition, 1' is an epimorphism then the sequence
^ r, v
A'
B
C
(6)
is exact.
(ii) If in the commutative diagram
A
A'
B
C
ß
B '
(7)
the square 3t = t'a is a pushout and the upper row in (7) is exact then there exists a unique morphism p' : B' ^ C such that p'3 = p, p't' = 0. If, in addition, p is a monomorphism then the sequence
r>' v'
A'
B '
C
is exact.
Proof. Prove (i) (then (ii) is obtained by duality). The existence and uniqueness follow from the equalities p' t' = 7O. Suppose now that t}' is an epimorphism. Then, by Lemma 1.1(iii), 3ker p = ker p' = im t'. Put t = (ker p)i}'coim t'. Then coker t = coker ker p = coim p, which is the exactness of (6). □
Theorem 2.1. Consider a commutative diagram with exact rows of the form (1) in a preabelian category. Keep the notations of Theorem 0.1. The following hold.
(i) If t' is a semi-stable kernel and, in the canonical decomposition p = (im p)pcoim p of p, the morphism p is a monomorphism then so is rj-
If p is a semi-stable cokernel and, in the canonical decomposition t' = (im t' )t}'coim t' of t', the morphism rp' is an epimorphism then so is n-
(ii) If t' and im p are semi-stable kernels and p is strict then n is a semi-stable kernel.
If p and coim t' are semi-stable cokernels and t' is strict then n is a semi-stable cokernel.
Proof (i) Since t' = ' is a semi-stable kernel, t' is a semi-stable kernel too (Lemma 1.2(i)). In the commutative diagram
A
A
B
Q
C
C
V
V
V
a
V
a
T
T
the left-hand square is a pushout, the upper row is exact, and p is a monomorphism. By Lemma 2.2(ii), we conclude that the sequence
A —A Q —A C
is exact. Consequently, recalling that t' is a kernel, we infer that t' = ker(an). Assume now that nz = 0 for some z : Z a Q. We have anz = 0, and, hence, z = t' z' for some z '. Therefore,
{ 'z ' = 0t ' z ' = 0z = a 'nz = 0.
Since {' is a monomorphism, z' = 0, and hence z = 0. Thus, n is a monomorphism. The second assertion in (i) is dual to the first.
(ii) As we have observed, t' = ker(an). Note also that nT' = ker a. Indeed, we have the commutative diagram
A' -——A Q' -—-A C
Y
A' -► B' -► C',
in which {' = ker p', {' is an epimorphism, and the square on the right is a pullback. By Lemma 2.2(i), we infer that the sequence
A' —A Q' —A C
is exact. Since {' = a'nT' is a semi-stable kernel, by Lemma 1.2(i) nT' is also a semi-stable kernel. Consequently, nT' = ker a.
Since the square t{ = t'a is a pushout, we have (coker t')t = coker { = coim p. Therefore,
(im p)(coker t ' )t = (im p)coim p = p = a'qT.
Moreover, (im p)(coker t')t' = 0 and a'qT' = 0. Hence,
(an — (im p)coker t ' )t = 0, (an — (im p)coker t ' )t ' = 0.
Since the zero morphism 0 : Q a C is the only morphism y with yT = 0 and yT' = 0, we see that (im p)coker t' — an = 0. Therefore, the morphism an = (im p)coker t' is strict. We come to the commutative diagram
A' -—U Q ——a C n
A ————A Q ————A C,
nT ' <7
where t' = ker(an), nT' = ker a, nT is a semi-stable kernel, the morphism an is strict, and im(an) = im p is a semi-stable kernel. Applying Lemma 2.1, we see that n is a semi-stable kernel.
The first assertion in (ii) is proved, and the second follows by duality.
The theorem is proved. □
Note that the only thing we really need from the semi-stability of {' (or p) in the proof of Theorem 2.1(i) is the implication
{' is a kernel t' is a kernel (resp., p is a cokernel a is a cokernel).
By Lemma 1.3(i), this assertion holds also for arbitrary kernels (respectively, cokernels) in a P-semi-abelian category. Thus, we have:
er
Corollary 2.1. Consider a commutative diagram with exact rows of the form (1) in a P-semi-abelian category. The following hold.
(i) If ф' is a kernel then n is a monomorphism. If p is a cokernel then n is an epimorphism.
(ii) If ф' is a semi-stable kernel and p is a cokernel (or if ф' is a kernel and p is a semi-stable cokernel) then n is an isomorphism.
3. Two Definitions of a Connecting Morphism
Consider the commutative diagram
A -—- B -—- C -► 0
(8)
0 -> A' -> B' -> C'
where ф' = ker p' and p = coker ф, in a preabelian category.
As in the abelian case, (8) gives rise to two parts of a Ker-Coker-sequence (the composition of two consecutive arrows is zero):
Ker а Ker ß Ker 7
and
Coker а Coker ß — Coker 7.
In contrast to the case of an abelian category (or even a Grandis-homological [7] or a quasi-abelian [4] category), for preabelian categories, it is in general impossible to construct a natural connecting morphism 5 : Ker 7 — Coker a. We will duscuss two constructions of 5, one going back to Andre-MacLane, and the other based on the Two-Square Lemma, which was proposed by Fay-Hardie-Hilton in [8] for abelian categories and adapted to the preabelian case by Generalov in [2].
в
Y
a
3.1 The André—MacLane Construction
According to [23], the following construction, described in [24, p. 203] for abelian categories, is due to Andre-MacLane. It was used in [4,5] for quasi-abelian and P-semi-abelian categories. Lets
X -► Ker y
be a pullback and let
ker y
B
C
V
A
coker a
Coker a
B
Y
(9)
(10)
be a pushout.
Instead of semi-stability conditions of universal nature, impose on our situation ad hoc "modularity" conditions a la Grandis [7]:
u
-Ф
v
t
Assumptions A. In (9), s is an epimorphism and in (10), t is a kernel. Assumptions A are fulfilled in a semi-abelian category if ф' is a semi-stable kernel and < is a semi-stable cokernel. In a P-semi-abelian category, the semi-stability of ф' is already enough.
Since square (10) is a pushout, (cokert)v = cokerф' = coim<'. Putting (im<')<' = x, we have <' = x(coker t)v. We infer
vßiß = vф'a = t(coker a) a = 0.
Therefore, vß = n< for some unique n. In the dual manner, <'ßu = 0, and, hence, ßu = ф'm for a unique morphism m. We have
(coker t)n(ker y)s = (coker t)n<u = (coker t)vßu = (coker ф' )ф'm = 0.
Since s is an epimorphism, this implies that (coker t)n ker 7 = 0. Since t = ker coker t, we conclude that n ker 7 = tSj for some unique Sj. The morphism Sj is uniquely characterized by the property
tSjs = vßu. (11)
By duality, consider
Assumptions A*. In (9), s is a cokernel and, in (10), t is a monomorphism. In this case, we also obtain a morphism Sj defined by (11). Therefore, the two morphisms coincide if s is a cokernel and t is a kernel.
3.2 The Fay—Hardie—Hilton—Generalov Construction
Consider the diagram (8) and assume the fulfillment of one of the following conditions (i) and (ii).
(i) The ambient category is preabelian, ф is a semi-stable kernel, and < is a semi-stable cokernel.
(ii) The ambient category is P-semi-abelian and ф' is a semi-stable kernel or < is a semi-stable cokernel.
Below we use all notations of the previous subsection and Section 2..
From Generalov's Theorem (Theorem 0.2) or Theorem 2.1 for (i) or Corollary 2.1 for (ii) it follows that, in these cases, the morphism n : Q ^ Q' is an isomorphism and, therefore, we may assume that Q = Q', n = idQ. Since (3) is a pushout, coker a = (coker т)т'; since (2) is a pullback, ker 7 = a (ker a' ). Put
Sjj = (coker т) ker a'. Theorem 3.1. The equality Sjj = — Sj holds.
Proof. Prove that — Sjj satisfies (11), i.e., that tSJJs = —vßu.
Following [2], put for brevity Si = coker т, S2 = ker a'. Then, by definition, Sjj = S1S2. We have the following "multiplication table":
ат = ат ' = 0; а 'т = ß; а'т ' = ф '.
Hence,
(va' — na^ ' = va ' т' — 'аат ' = гоф' = tcoker a = tSi т',
(va ' — na^ = vß — n< = vß — vß = 0.
Thus, (tSi — (va' — na))T' = 0, (t5i — (va' — na))T = 0. Therefore, since the square T't = t'a is a pushout, this implies that tS1 = va' — na. By duality, 52s = tu — t'm. Consequently,
tSH s = tSi§2S = (va ' — na)(TU — t ' m)
= va' tu — va' t'm — naTU + naT' m = vf3u — vt'm — vf3u = —v3u.
The theorem is proved. □
Even having a connecting morphism 5 : Ker 7 ^ Coker a, we in general cannot assert that the corresponding Ker-Coker-sequence is exact. For its exactness, one usually has to impose extra conditions like strictness or semi-stability (see [2-5,7]).
The author is indebted to the referee for valuable remarks.
Acknowlegdments. The author was partially supported by the Russian Foundation for Basic Research (Grants 09-01-00142-a, 12-01-00873-a), the State Maintenance Program for the Leading Scientific Schools and Junior Scientists of the Russian Federation (Grants NSh-6613.2010.1, NSh-921.2012.1), and the Integration Project "Quasiconformal Analysis and Geometric Aspects of Operator Theory" of the Siberian and Far Eastern Branches of the Russian Academy of Sciences.
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Лемма о двух квадратах и связывающий морфизм в предабелевой категории
Ярослав А. Копылов
В 'работе получено обобщение леммы о двух квадратах, доказанной для абелевых категорий Фэем, Харди и Хилтоном в 1989 г. и (в специальном случае) для предабелевых категорий Генераловым в 1994 г. Также доказана эквивалентность с точностью до знака двух определений связывающего морфизма в лемме о змее (Ker-Coker-последовательности).
Ключевые слова: строгий морфизм, предабелева категория, (ко)универсальный квадрат, полустабильное (ко)ядро, лемма о змее, связывающий морфизм.
