Relative Hochschild cohomology, rigid algebras, and the Bockstein. (English) Zbl 0603.16021

In this paper the authors exhibit in all positive characteristics, finite dimensional rigid associative algebras with nontrivial infinitesimal deformations thus confirming for positive characteristics, a 25 year old conjecture.
The details are as follows: If \(\Sigma\) is a triangulation of a space X, then one can associate to \(\Sigma\), and any commutative ring k an incidence matrix ring. The authors show that for this k-algebra \(k\Sigma\), the Hochschild cohomology \(H^.(k\Sigma,k\Sigma)\) equals the simplicial cohomology \(H^.(\Sigma,k)\) which follows from their beautiful result: If S is a separable subalgebra of an algebra A, then the Hochschild cohomology of A is identical with that computed relative to S.
Next, a deformation of an algebra A is a multiplicative 2-cocycle valued in \(1+tA[[ t]]\) and its leading term is an additive 2-cocycle valued in A. The later is called the infinitesimal of the deformation. The authors aim to construct algebras which admit nontrivial infinitesimal changes in their structure but no nontrivial global changes, i.e. to construct a rigid algebra (no nontrivial deformations) having \(H^ 2(A,A)\neq 0\). The natural way to move from additive to multiplicative structure via exponentiation by looking at \(k\Sigma =A\), seems to work nicely except when k has characteristic p, when the exponential becomes undefined at the coefficient of \(t^ p\) showing the primary obstruction to deformation of \(k\Sigma\), which lead the authors to study the topological Bockstein map for the general coefficient ring with many of its properties. When \(k=F_ p=Z/pZ\) this is just the classical Bockstein (the cohomology connecting homomorphism induced by \(0\to Z/pZ\to Z/p^ 2Z\to Z/pZ\to 0)\) which however in general is not k-linear.
Using Bockstein they find conditions which then allowed them to control \(H^.(\Sigma,W)\) and hence the deformation theory of \(k\Sigma\). At this point, finding examples mentioned earlier became a matter of finding appropriate spaces i.e. with \(H^ 2(\Sigma,k)\neq 0\) and the Bockstein \(H^ 2(\Sigma,k)\to H^ 3(\Sigma,k)\) a monomorphism.
Reviewer: S.A.Huq


16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55U20 Universal coefficient theorems, Bockstein operator
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
16S80 Deformations of associative rings
16P10 Finite rings and finite-dimensional associative algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16Exx Homological methods in associative algebras
Full Text: DOI


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