The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set

Gerstenhaber, Murray; Schack, Samuel D.

doi:10.1090/S0002-9947-1988-0965749-X

The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set
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by Murray Gerstenhaber and Samuel D. Schack PDF

Trans. Amer. Math. Soc. 310 (1988), 135-165 Request permission

Abstract:

To each presheaf (over a poset) of associative algebras $\mathbb {A}$ we associate an algebra $\mathbb {A}!$. We define a full exact embedding of the category of (presheaf) $\mathbb {A}$-bimodules in that of $\mathbb {A}!$-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.

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The cohomology of presheaves of algebras

the barycentric subdivision of a small category

The cohomology of presheaves of algebras

Embedding theorems

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