The Two-Square Lemma and the Connecting Morphism 3

Home , Cokernel, Commutative diagram

arXiv:1103.5251v1 [math.CT] 27 Mar 2011 e od n phrases and words Key emti set fOeao hoy fteSbra Branc Siberian the Sciences. of of Academy Theory” Pr Russian Operator Integration of the Aspects School and Geometric Res Scientific (NSh-6613.2010.1), Basic Leading Federation the for sian for Foundation Program Russian Maintenance the State by supported Partially morphism connecting oncigmrhs n[]ivle raeinvrino pca ca special a of 3]. version Lemma preabelian [6, Fay–Hardie-Hilton a of involves Lemma [8] Two-Square con The in general [8]). morphism in (see counter connecting assumptions guaranteed nonadditive “semi-stability” be extra their cannot without in gories [10]), and categories [13] homological categories dis quasi-abelian in even “modula and “exactness” some analogs of weaker -C Ker (pullbacks) their the pushouts or of under 1 proper (epimorphisms), exactness stability key 13, the and The for 9, “strictness” required 10]). 8, are diagram [4, [3, initial e.g., the e.g., in (see, (see, phisms categories categories nonabelian additive of of classes classes for authors ( pullba kernels (respectively, that pushouts and under co category “survive” of preabelian not (respectively, a do in monomorphism cokernels) re coincide and main not The kernel do of diagram. morphism) abe initial notions T the an the categories, on in that preabelian assumptions holds of additional always se context without It general homological fails more obtain complexes. the to of in possible However, sequences it exact makes short which from Lemma Snake so-called ahmtc ujc Classification. Subject Mathematics vnteeitneo oncigmrhs,vldi bla categor abelian in valid morphism, connecting a of existence the Even s by studied was case nonabelian the in Lemma Snake the of validity The algebra homological in assertions diagram important most the of One H W-QAELMAADTECONNECTING THE AND LEMMA TWO-SQUARE THE Abstract. pt ino w entoso oncigmrhs fteSn the of morphism connecting a of prov definitions also two (in We of and sign 1994. 1989 in to Generalov in up by Hilton categories and preabelian Hardie, for Fay, by categories abelian eoti eeaiaino h w-qaeLmapoe fo proved Lemma Two-Square the of generalization a obtain We raeinctgr,kre,ckre,plbc,pushou pullback, cokernel, kernel, category, preabelian : AOLVKOPYLOV YAROSLAV Introduction MORPHISM 18A20 1 jc Qaiofra nlssand Analysis “Quasiconformal oject n uirSinit fteRus- the of Scientists Junior and s n h a atBac fthe of Branch East Far the and h ac Gat0-1012a,the 09-01-00142-a), (Grant earch h equivalence the e pca case) special a k Lemma. ake cks). eSaeLemma Snake he monomorphisms raeincate- preabelian ] n nsome in and 4]) iy [10]. rity” iso h mor- the of ties enladepi- and kernel tuto fthe of struction ,SaeLemma, Snake t, kr-sequence oker incategory. lian respectively, at Gran- part, sn are asons r eo the of se e (and ies quences everal sthe is 2 YAROSLAV KOPYLOV

Theorem 0.1 (The Two-Square Lemma). Suppose that the following diagram in an abelian category has exact rows:

ψ ϕ A −−−−→ B −−−−→ C    α  γ β (1) A′ −−−−→ B′ −−−−→ C′ . ψ′ ϕ′ Let Q′ −−−−→σ C   ′ γ σ (2) B′ −−−−→ C′ ϕ′ be a pullback and let ψ A −−−−→ B   α τ (3) A′ −−−−→ Q τ ′ be a pushout. Then (i) there exists a unique θ : Q → B′ such that θτ = β, θτ ′ = ψ′; (ii) there exists a unique ρ : B → Q′ such that σρ = ϕ, σ′ρ = β; (iii) there exists a unique η : Q → Q′ such that ητ = ρ, σ′η = θ, σητ ′ =0. The proof in [6] remains valid in any preabelian category. The Two-Square Lemma of [6] also claims that if ψ′ is a monomorphism then so is η and if ϕ is an epimorphism then so is η. In [8], Generalov proved the following assertion: Theorem 0.2. Consider a diagram of the form (1) in a preabelian category. If ψ′ is a semi-stable kernel and ϕ is a semi-stable cokernel then η is an isomorphism. Below we study the question when η is monic, epic, a kernel, a cokernel in a preabelian category. The article is organized as follows. In Section 1, we give basic deﬁnitions and facts about preabelian categories. In Section 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 η. In Section 3, we prove the equivalence of two deﬁnitions of a connecting morphism of the Snake Lemma in a preabelian category.

1. Preabelian Categories A preabelian category is an additive category with kernels and cokernels. In a preabelian category, every morphism α admits a canonical decomposition α = (im α)¯α(coim α), where im α = kercoker α, coim α = coker ker α. THE TWO-SQUARE LEMMA AND THE CONNECTING MORPHISM 3

A morphism α is called strict ifα ¯ 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. [5, 11, 13, 19] The following hold in a preabelian category. (i) A morphism α is a kernel if and only if α = im α, a morphism α is a cokernel if and only if α = coim α; (ii) A morphism α is strict if and only if α is representable as α = α1α0, where α0 is a cokernel, α1 is a kernel; in this case, α0 = coim α and α1 = im α; (iii) Suppose that the commutative square C −−−−→α D   g  f (4) A −−−−→ B β is a pullback. Then ker f = α ker g. If f = ker h for some h then g = ker(hβ). In particular, if f is monic then g is monic; if f is a kernel then g is a kernel. In the dual manner, assume that (4) is a pushout. Then coker f = βcoker g. If g = coker e for some e then f = coker(αe). In particular, if g is epic then f is epic; if g is a cokernel then f is a cokernel. A kernel g in a preabelian category is called semi-stable [19] 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 [2, 18, 20, 21] and non-semi-stable kernels are shown in [19]. If all kernels and cokernels are semi- stable then the preabelian category is called quasi-abelian [22]. Lemma 1.2. [7, 19] 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) a pushout of a semi-stable kernel is a semi-stable kernel; a pullback of a semi-stable cokernel is a semi-stable cokernel. If the 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 [17]: if (4) is a pushout and g is a kernel then f is monic; if (4) is a pullback and f is a cokernel then g is epic. Until recently it was unclear whether every P-semi-abelian category is quasi-abelian (Raikov’s Conjecture); this was disproved by Bonet and Dierolf [2] and Rump [20, 21]. It turned out that, for instance, the categories of barrelled and bornological spaces are P-semi-abelian but not quasi-abelian (see [21]). In general preabelian categories, kernels (cokernels) may even push out (pull back) to zero morphisms (see [18, 19]). In [15] 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 α, α = (im α)¯α coim α, the central morphismα ¯ is a bimorphism, that is, monic and epic simultaneously. 4 YAROSLAV KOPYLOV

Lemma 1.3. [14, 15] The following hold in a 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 gf is deﬁned then gf is a kernel; if f,g are cokernels and gf is deﬁned then gf is a cokernel; (iii) if gf is strict and g is monic then f is strict; if gf is strict and g ∈ P then f is strict. The following lemma is due to Yakovlev [23]. Lemma 1.4. For every morphism α in a preabelian category, ker α = ker coim α, coker α = coker im α. a b A sequence ... → B → ... in a preabelian category is said to be exact at B if im a = ker b. 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 [15, Theorem 3] and [12, Lemma 6]. Lemma 2.1. Let p1 q1 A −−−−→ B1 −−−−→ C  r

A −−−−→ B2 −−−−→ C p2 q2 be a commutative diagram in a preabelian category. (i) If p1 = ker q1, q2p2 =0, p2 is monic then r is monic. (ii) Suppose that p1 = ker q1, p2 = ker q2, p2 and im q1 are semi-stable kernels, and q1 is strict. Then r is a semi-stable kernel. The dual assertions also hold.

Proof. (i) Assume that rx = 0 and prove that then x = 0. We have q1x = q2rx = 0. Since p1 = ker q1, this gives x = p1y for some y. Then p2y = rp1y = rx = 0. Since p2 is monic, y=0 and thus x = p1y = 0. ′ ′′ ′ ′′ ′′ → (ii) Decompose q1 as q1 = q1q1 , q2 = q2q2 , where qj = coim qj : Kj C, j = ′ 1, 2. By assumption, q1 = im q1. Since coim q1 = coker p1 and coim q2 = coker p2, there is a unique morphism w : K1 → K2 such that w coim q1 = (coim q2)r. For this ′ ′ ′ w, q1 = q2w. Since q1 is a semi-stable kernel by hypothesis, so is w (Lemma 1.2). Consider the pushout p1 A −−−−→ B1   p2 u2 (5) u1 B2 −−−−→ F. Since u1rp1 = u1p2 = u2p1, we have (u1r − u2)p1 = 0. Therefore, there exists a unique morphism s : K1 → F with the property u1r − u2 = s coim q1. Consider the pushout w K1 −−−−→ K2   s ′ (6) s ′ F −−−−→w S. THE TWO-SQUARE LEMMA AND THE CONNECTING MORPHISM 5

′ ′ Put µ = w u1 − s coim q2. We infer

′ ′ ′ ′ ′ µr = (w u1 − s coim q2)r = w u2 + w s coim q1 − s (coim q2)r ′ ′ ′ ′ = w u2 + w s coim q1 − w s coim q1 = w u2. ′ ′ Thus, µr = w u2. Since p2 and w are semi-stable kernels, so are u2 and w . Now, ′ by Lemma 1.2(ii), µr = 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.

We will also need the following preabelian version of Lemma 1 in [6]. This also generalizes Lemma 1.1(iii). Lemma 2.2. If in the commutative diagram ϕ B −−−−→ C    γ β (7) A′ −−−−→ B′ −−−−→ C′ ψ′ ϕ′ the square ϕ′β = γϕ is a pullback and the bottom row of (7) is exact then there is a unique morphism ψ : A′ → B such that βψ = ψ′, ϕψ =0. If, in addition, ψ¯′ is epic then the sequence ′ ψ ϕ A −→ B −→ C (8) is exact. The dual assertion about pushouts also holds. Proof. The existence and uniqueness follows from the equalities ϕ′ψ′ = γ0. Now, suppose that ψ¯′ is epic. Then, by Lemma 1.1(iii), β ker ϕ = ker ϕ′ = im ψ′. Put ψ = (ker ϕ)ψ¯′coim ψ′. Then coker ψ = coker ker ϕ = coim ϕ, which is the exactness of sequence (8).

Theorem 2.1. Consider a commutative diagram with exact rows of the kind (1) in a preabelian category. Preserve all notations of Theorem 0.1. Then the following hold: (i) If ψ′ is a semi-stable kernel and, in the canonical decomposition of ϕ, ϕ = (im ϕ)¯ϕ coim ϕ, ϕ¯ is a monomorphism then η is a monomorphism. If ϕ is a semi-stable cokernel and, in the canonical decomposition of ψ′, ψ′ = (im ψ′)ψ¯′coim ψ′, ψ¯′ is an epimorphism then η is an epimorphism. (ii) If ψ′ and im ϕ are semi-stable kernels and ϕ is strict then η is a semi-stable kernel. If ϕ and coim ψ′ are semi-stable cokernels and ψ′ is strict then η is a semi-stable cokernel. Proof. (i) Since ψ′ = σ′τ ′ is a semi-stable kernel, so is τ ′ (Lemma 1.2). The commutative diagram ψ ϕ A −−−−→ Q′ −−−−→ C   α τ A′ −−−−→ Q −−−−→ C τ ′ ση 6 YAROSLAV KOPYLOV has a pushout on the left and an exact top row, and is such thatϕ ¯ is monic. By Lemma 1.1(iii), we infer that τ ′ = ker(ση). Assume now that ηz = 0 for some z : Z → Q. We have σηz = 0, and hence z = τ ′z′ for some z′. We infer ′ ′ ′ ′ ′ ψ z = θτ z = θz = σ ηz =0. Since ψ′ is a monomorphism, z = 0. Thus, η is a monomorphism. The second assertion of (i) is dual to the first. (ii) We have already noticed that τ ′ = ker(ση). Note also that ητ ′ = ker σ. Indeed, we have the commutative diagram ′ ητ A′ −−−−→ Q′ −−−−→σ C   ′ γ σ A′ −−−−→ B′ −−−−→ C′ ψ′ ϕ in which ψ′ = ker ϕ′ and the square on the right is a pullback. Hence, ητ ′ = ker σ. Since τψ = τ ′α is a pushout, we have (coker τ ′)τ = coker ϕ = coim ψ. Hence, ′ (im ϕ)(coker τ )τ = (im ϕ)coim ϕ = ϕ = σητ. Moreover, (im ϕ)(coker τ ′)τ ′ = 0 and σητ ′ = 0. We infer that ′ ′ ′ (ση − (im ϕ)coker τ )τ =0, (ση − (im ϕ)coker τ )τ =0. Since the zero morphism 0 : Q → C is the only morphism y for which yτ = 0 and yτ ′ = 0, we infer that (im ϕ)coker τ ′ − ση = 0. Therefore, the morphism ση = (im ϕ)coker τ ′ is strict. Now, we arrive at the commutative diagram ′ ση A′ −−−−→τ Q −−−−→ C  η A′ −−−−→ Q′ −−−−→ C, ητ ′ σ where τ ′ = ker(ση), ητ ′ = ker σ, ητ ′ a semi-stable kernel (because ψ′ = σ′ητ ′ is a semi-stable kernel), ση is strict, and im (ση) = im ϕ is a semi-stable kernel. By Lemma 2.1, we see that η is a semi-stable kernel. The first assertion of (ii) is proved, and the second is dual to the first. Observe that the only thing we really need from the semi-stability of ψ′ (or ϕ) in the proof of Theorem 2.1(i) is the implication ′ ′ ψ is a kernel =⇒ τ is a kernel (ϕ is a cokernel =⇒ σ is a cokernel). By Lemma 1.3(i), this assertion holds for arbitrary kernels (cokernels) in a P-semi- abelian category. Thus, we have Corollary 2.1. Consider a commutative diagram with exact rows of the kind (1) in a P-semi-abelian category. Then the following hold. (i) If ψ′ is a kernel then η is a monomorphism. If ϕ is a cokernel then η is an epimorphism. (ii) If ψ′ is a semi-stable kernel and ϕ is a cokernel (or if ψ′ is a kernel and ϕ is a semi-stable cokernel) then η is an isomorphism. THE TWO-SQUARE LEMMA AND THE CONNECTING MORPHISM 7

3. Two Definitions of a Connected Morphism Consider the commutative diagram ψ ϕ A −−−−→ B −−−−→ C −−−−→ 0    α  γ β (9) 0 −−−−→ A′ −−−−→ B′ −−−−→ C′ ψ′ ϕ′ with ψ′ = ker ϕ′ and ϕ = coker ψ in a preabelian category. As in the abelian case, (9) gives rise to two parts of a Ker -Coker -sequence (the composition of two consecutive arrows is zero):

ε ζ Ker α → Ker β → Ker γ and τ θ Coker α → Coker β → Coker γ. In contrast to the case of an abelian category (or even a Grandis-homological [10] or a quasi-abelian [13]) category, for preabelian categories, it is in general impossible to construct a natural connecting morphism δ : Ker γ → Coker α. We will duscuss two constructions of δ, one going back to Andr´e–MacLane, and the other based on the Two-Square Lemma, which was proposed by Fay–Hardie–Hilton in [6] for abelian categories and adapted to the preabelian case by Generalov in [8].

3.1. The André–MacLane construction. According to [1], the following construction, described in [16, p. 203] for abelian categories, is due to André–MacLane. It was used in [13, 14] for quasi-abelian and P -semi-abelian categories. Let s X −−−−→ Ker γ   u ker γ (10) B −−−−→ C ϕ be a pullback and let ′ ψ A′ −−−−→ B′    v coker α (11) Coker α −−−−→ Y t be a pushout. Instead of semi-stability properties of universal nature, impose on our situation appropriate ad hoc “modularity” conditions a la Grandis [10]: Assumptions A. I n (10) s is epic and in (11) t is a kernel. Assumptions A are fulfilled in a preabelian category if ψ′ is a semi-stable kernel and ϕ is a semi-stable cokernel. In a P-semi-abelian category, the semi-stability of ψ′ is already enough. Since (11) is a pushout, (coker t)v = coker ψ′ = coim ϕ′. If we put (im ϕ′)¯ϕ′ = χ then ϕ′ = χ(coker t)v. We have ′ vβψ = vψ α = t(coker α)α =0. 8 YAROSLAV KOPYLOV

Therefore, vβ = nϕ for some unique n. In the dual manner, ψ′βu = 0, and hence βu = ψ′m for a unique morphism m. We infer ′ ′ (coker t)n(ker γ)s = (coker t)nϕu = (coker t)vβu = (coker ψ )ψ m =0. Since s is epic, this implies that (coker t)n ker γ = 0. Since t = kercoker t, we conclude that n ker γ = tδI for some unique δI . This morphism δI is characterized uniquely by the property

tδI s = vβu. (12) By duality, consider Assumptions A∗. I n (10) s is a cokernel and in (11) t is monic. In this case, we also obtain a morphism δI characterized by (12). Therefore, the two morphisms coincide provided that s is a cokernel and t is a kernel simultaneously.

3.2. The Fay–Hardie–Hilton–Generalov construction. Consider diagram (9) and suppose the fulﬁllment of one of the following conditions (i) and (ii). (i) The ambient category is preabelian, ψ′ is a semi-stable kernel, and ϕ is a semi-stable cokernel; (ii) The ambient category is P-semi-abelian and ψ′ is a semi-stable kernel or ϕ is a semi-stable cokernel. Below we use all notations of the previous subsection and Section 2. Generalov’s Theorem (Theorem 0.2) or Theorem 2.1 for (i) and Corollary 2.1 for (ii) imply that in these cases the morphism η : Q → Q′ of [6] is an isomorphism, ′ and so we may assume that Q = Q , η = idQ. Since (3) is a pushout, coker τ = (coker α)τ ′; by duality, since (2) is a pullback, ker σ′ = σ(ker γ). Put ′ δII = (coker τ)ker σ .

Theorem 3.1. The equality δII = −δI holds.

Proof. Prove that the morphism −δII meets (12), that is, that tδII s = −vβu. We have ′ ′ ′ ′ ′ ′ ′ (vσ − nσ)τ = vσ τ − nστ = vψ = tcoker α = tδ1τ , ′ (vσ − nσ)τ = vβ − nϕ = vβ − vβ =0.

′ ′ ′ ′ ′ Thus, (tδ1 − (vσ − nσ))τ =0, (tδ1 − (vσ − nσ))τ = 0, Hence, since γσ = ϕ σ ′ ′ is a pullback, this implies that tδ1 = vσ − nσ. By duality, δ2s = τu − τ m. Consequently,

′ ′ tδII s = tδ1δ2s = (vσ − nσ)(τu − τ m) ′ ′ ′ ′ ′ = vσ τu − vσ τ m − nστu + nστ m = vβu − vψ m − vβu = −vβu. The theorem is proved.

Even having a connecting morphism δ : Ker γ → Coker α, we in general cannot assert the exactness of the corresponding Ker-Coker-sequence. This exactness usually requires additional conditions like strictness or semi-stability (see [8, 9, 10, 13, 14]. THE TWO-SQUARE LEMMA AND THE CONNECTING MORPHISM 9

References

[1] R. Beyl, The connecting morphism in the kernel-cokernel sequence, Arch. Math. 32 (1979), 305–308. [2] J. Bonet and S. Dierolf, The pullback for bornological and ultrabornological spaces Note Mat. 25 (2006), no. 1, P. 63–67. [3] T. Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. [4] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories. Mathematics and its Applications, 566. Kluwer Academic Publishers, Dordrecht, 2004. [5] I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, Pure and Applied Mathematics, XIX, Interscience Publication John Wiley & Sons, Ltd., London-New York-Sydney, 1968. [6] T. H. Fay, K. A. Hardie, P. J. Hilton, The two-square lemma, Publ. Mat. 33 (1989), no. 1, 133-137. [7] A. I. Generalov, Relative homological algebra in pre-abelian categories. I. Derived categories, Algebra i Analiz 4 (1992), no. 1, 98–119; English translation: St. Petersburg Math. Jour. 4 (1993), no. 1, 93–113. [8] A. I. Generalov, The Ker-Coker sequence for pre-abelian categories (Russian), Abelian groups and modules, no. 11, 12. Tomsk. Gos. Univ., Tomsk, 1994, 78–89. [9] N. V. Glotko and V. I. Kuz′minov, On the cohomology sequence in a semiabelian category (Russian), Sibirsk. Mat. Zh. 43 (2002), no. 1, 41–50; English translation in: Siberian Math. J. 43 (2002), no. 1, 28–35. [10] M. Grandis, On homological algebra, I. Semiexact and homological categories, Dip. Mat. Univ. Genova, Preprint 186 (1991). [11] G. M. Kelly, Monomorphisms, epimorphisms and pullbacks, J. Austral. Math. Soc. 9 (1969), 124–142. [12] Ya. A. Kopylov, The five- and nine-lemmas in P-semi-abelian categories, Sibirsk. Mat. Zh. 50, no. 5, 1097–1104 (2009); English translation: Sib. Math. J. 50, no. 5, 867–873 (2009). [13] Ya. A. Kopylov and V. I. Kuz′minov, On the Ker-Coker sequence in a semiabelian category, Sibirsk. Mat. Zh. 41 (2000), no. 3, 615–624; English translation in: Siberian Math. J. 41 (2000), no. 3, 509–517. [14] Ya. A. Kopylov and V. I. Kuz′minov, The Ker-Coker-sequence and its generalization in some classes of additive categories, Sibirsk. Mat. Zh. 50 (2009) no. 1, 107–117; English translation: Siberian Math. J. 50 (2009), no. 1, 86–95. [15] V. I. Kuz′minov and A. Yu. Cherevikin, Semiabelian categories, Sibirsk. Mat. Zh. 13 (1972), no. 6, 1284–1294; English translation: Siberian Math. J. 13 (1972), no. 6, 895–902. [16] S. Maclane, Categories for the Working Mathematician. Graduate Texts in Math. 5, New York: Heidelberg–Berlin, 1972. [17] V. P. Palamodov, Homological methods in the theory of locally convex spaces, Usp. Mat. Nauk 26 (1971), no. 1 (157), 3–65; English translation: Russ. Math. Surv. 26 (1971), no. 1, 1–64. [18] F. Prosmans, Derived categories for functional analysis, Publ. Res. Inst. Math. Sci. 36 (2000), No. 1, 19–83 [19] F. Richman and E. A. Walker, Ext in pre-Abelian categories, Pacific J. Math. 71 (1977), No. 2, 521–535. [20] W. Rump, A counterexample to Raikov’s conjecture, Bull. London Math. Soc. 40 (2008), no. 6, 985–994. [21] W. Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. of Pure Appl. Algebra 215 (2011), no. 1, 44–52. [22] J.-P. Schneiders, Quasi-Abelian Categories and Sheaves, Mém. Soc. Math. Fr. (N.S.) (1999), no. 76. [23] A. V. Yakovlev, Homological algebra in pre-Abelian categories, Zap. Nauchn. Sem. LOMI 94 (1979), 131–141; English translation: J. Soviet Mathematics, 19 (1982), no. 1, 1060–1067.

Yaroslav Kopylov Sobolev Institute of Mathematics, Pr. Akademika Koptyuga 4, 630090, Novosibirsk, Russia 10 YAROSLAV KOPYLOV

Novosibirsk State University E-mail address: [email protected]