The multiplicative structure on Hochschild cohomology of a complete intersection. (English) Zbl 1308.13022
Let \(K\) be a commutative (base) ring, let \(A\) be a commutative \(K\)-algebra, and let \(A^{\mathrm{ev}}\) be the enveloping algebra of \(A\) (over \(K\)). Consider the map \(s : \mathrm{Der}_K(A,A) = \mathrm{Ext}^1_{A^{\mathrm{ev}}} (A,A) \to \mathrm{Ext}^{2}_{A^{\mathrm{ev}}}(A,A)\) that sends a derivation \(D\) to the class \([D \circ D]\) of its square under the Yoneda product. In this paper, a more detailed description of the map \(s\) is given: If \(A\) is written \(A = P/I\), where \(I\) is an ideal in a polynomial ring \(P\) with coefficients in \(K\), then it is shown that \(s\) factors through \(\mathrm{Hom}_A(I/I^2,A)\). The first map \(q : \mathrm{Der}_K(A,A) \to \mathrm{Hom}_A(I/I^2,A)\) in this factorization is induced by the Hessian map. If \(A\) is projective as a \(K\)-module, then \(\mathrm{Ext}^{*}_{A^{\mathrm{ev}}}(A,A)\) is isomorphic to the Hochschild cohomology ring \(\mathrm{HH}^*(A/K,A)\), so in this case the result gives information about the squaring operation \(s : \mathrm{HH}^1(A/K,A) \to \mathrm{HH}^2(A/K,A)\).
In the case where \(A\) is a so-called homological complete intersection algebra, it is proved that \(\mathrm{Ext}^{*}_{A^{\mathrm{ev}}}(A,A)\) can be obtained as the cohomology algebra of a certain DG (Clifford) algebra \((\mathrm{Cliff}(q),\partial)\), whose differential can be described quite explicitly. Again, if \(A\) is projective as a \(K\)-module, then this result gives information about (all) the products in Hochschild cohomology ring \(\mathrm{HH}^*(A/K,A)\).
In the case where \(A\) is a so-called homological complete intersection algebra, it is proved that \(\mathrm{Ext}^{*}_{A^{\mathrm{ev}}}(A,A)\) can be obtained as the cohomology algebra of a certain DG (Clifford) algebra \((\mathrm{Cliff}(q),\partial)\), whose differential can be described quite explicitly. Again, if \(A\) is projective as a \(K\)-module, then this result gives information about (all) the products in Hochschild cohomology ring \(\mathrm{HH}^*(A/K,A)\).
Reviewer: Henrik Holm (København)
MSC:
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 13C40 | Linkage, complete intersections and determinantal ideals |
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
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