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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bicohomology theory

Author: Donovan H. Van Osdol
Journal: Trans. Amer. Math. Soc. 183 (1973), 449-476
MSC: Primary 18H15
MathSciNet review: 0323873
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Abstract: Given a triple T and a cotriple G on a category $ \mathcal{D}$ such that T preserves group objects in $ \mathcal{D}$, let P and M be in $ \mathcal{D}$ with M an abelian group object. Applying the ``hom functor'' $ \mathcal{D}( - , - )$ to the (co)simplicial resolutions $ {G^ \ast }P$ and $ {T^ \ast }M$ yields a double complex $ \mathcal{D}({G^ \ast }P,{T^ \ast }M)$. The nth homology group of this double complex is denoted $ {H^n}(P,M)$, and this paper studies $ {H^0}$ and $ {H^1}$. When $ \mathcal{D}$ is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of $ {H^0}$ and $ {H^1}$ is given. The applications include a solution of the singular extension problem for sheaves of algebras.

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Keywords: Triple, principal homogeneous object, cotriplable, distributive law, double complex, homology, algebras, sheaves
Article copyright: © Copyright 1973 American Mathematical Society