In the sense of Palamodov, a preabelian category is semi-abelian if for every morphism the natural morphism between the cokernel of its kernel and the kernel of its cokernel is simultaneously a monomorphism and an epimorphism. In this article we present several conditions which are all equivalent to semi-abelianity. First we consider left and right semi-abelian categories in the sense of Rump and establish characterizations of these notions via six equivalent properties. Then we use these properties to deduce the characterization of semi-abelianity. Finally, we investigate two examples arising in functional analysis which illustrate that the notions of right and left semiabelian categories are distinct and in particular that such categories occur in nature.A priori all categories above are not accessible for standard methods of homological algebra since there might be different choices to define "(short) exact sequences" and it is moreover not clear if basic results like the snake, five, horseshoe, or comparison lemmas remain true in the given category (cf. Grandis [9], Kopylov, Kuz minov [16,17] or Kopylov [15]). On the other hand, the reason for considering the categories above is in fact the hope that a certain, for example purely analytic, problem can be formulated in the language of exact sequences and then eventually be solved with the help of purely abstract methods of homological algebra.The key notion for handling this situation on an abstract level -apart from ad hoc solutions -was invented by Quillen [26] (see Bühler [5]) and is that of an exact category. Given any additive category, one chooses a so-called exact structure, i.e. a 2010 Mathematics Subject Classification: Primary 18A20; Secondary 46M18.
We consider some problems concerning the Lp,q-cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L k p (M ) into L k+1 q (M ). In the second part, we prove that the first Lp,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q.
Mathematics Subject Classification (2000). 58A12, 46E30, 22E25.
We study the validity of the Snake Lemma (the existence and exactness of the Ker-Cokersequence) in a P-semi-abelian category. We also obtain a generalization of the Snake Lemma in a quasiabelian category.
Abstract. Vanishing results for reduced Lp,q-cohomology are established in the case of twisted products, which are a generalization of warped products. Only the case q ≤ p is considered. This is an extension of some results by Gol ′ dshtein, Kuz ′ minov and Shvedov about the Lp-cohomology of warped cylinders. One of the main observations is the vanishing of the "middledimensional" cohomology for a large class of manifolds.Mathematics Subject Classification. 58A10, 58A12.
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