# zbMATH — the first resource for mathematics

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

##### 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
Full Text:
##### References:
 [1] Bloch, S., Algebraic K-theory and crystalline cohomology, (), 187-268 · Zbl 0388.14010 [2] Browder, W., Torsion in H-spaces, Ann. of math., 74, 24-51, (1961) · Zbl 0112.14501 [3] Coffee, J., Filtered and associated graded rings, Bull. amer. math. soc., 78, 1948, (1972) [4] Demazure, M., Lectures on p-divisible groups, () · Zbl 0247.14010 [5] Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of math., 78, 2, 267-288, (1963) · Zbl 0131.27302 [6] Gerstenhaber, M., On the deformation of rings and algebras, Ann. of math., 79, 2, 59-103, (1964) · Zbl 0123.03101 [7] Gerstenhaber, M., On the deformation of rings and algebras IV, Ann. of math., 99, 2, 257-276, (1974) · Zbl 0281.16016 [8] Gerstenhaber, M.; Schack, S.D., Simplicial cohomology is Hochschild cohomology, J. pure and appl. algebra, 30, 143-156, (1983) · Zbl 0527.16018 [9] MacLane, S., Homology, (1967), Springer Berlin · Zbl 0133.26502 [10] Orzech, M.; Small, C., The Brauer group of commutative rings, () · Zbl 0302.13001 [11] Richardson, R.W., On the rigidity of semi-direct products of Lie algebras, Pacific J. math., 22, 339-344, (1967) · Zbl 0166.30301 [12] Serre, J.-P., Algèbre locale: multiplicités, () · Zbl 0091.03701 [13] Spanier, E.H., Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303 [14] Steenrod, N.E., Products of cocycles and extensions of mappings, Ann. of math., 48, 290-320, (1947) · Zbl 0030.41602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.