Relative Hochschild cohomology, rigid algebras, and the Bockstein.

*(English)*Zbl 0603.16021In 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.

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

##### MSC:

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 |

##### Keywords:

finite dimensional rigid associative algebras; infinitesimal deformations; triangulation; incidence matrix ring; Hochschild cohomology; simplicial cohomology; separable subalgebra; infinitesimal changes; Bockstein map
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\textit{M. Gerstenhaber} and \textit{S. D. Schack}, J. Pure Appl. Algebra 43, 53--74 (1986; Zbl 0603.16021)

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