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Gerstenhaber algebra structure in singular Hochschild cohomology
汪正方博士(巴黎七大)
2018-01-01 12:13  华东师范大学

摘要:
Let A be an associative algebra over a commutative ring k such that A is projective as a k-module. Then the Hochschild cohomology HH^m(A, A) can be viewed as the Hom-space Hom_{D^b(A\otimes_k A^{op})}(A, A[m]) in the bounded derived category D^b(A\otimes_k A^{op}).We replace D^b(A\otimes_k A^{op}) by the singular category D_{sg}(A\otimes_k A^{op}),which is the Verdier quotient of D^b(A\otimes_k A^{op}) by the full subcategory Perf(A\otimes_k A^{op}) consisting ofperfect complexes of A\otimes_k A^{op}-modules and define the singular Hochschild cohomology HH_{sg}^m(A, A) to be the Hom-space Hom_{D_{sg}(A\otimes_k A^{op})}(A, A[m]) for anyinteger m. In this talk, we prove that HH_{sg}^*(A, A) has a Gerstenhaber algebra structure.We also provide a prop interpretation for the graded Lie bracket (a joint work with G. Zhou). Several examples will be presented on how to compute HH_{sg}^*(A, A). If time permits, we will talk about the invariance of Gerstenhaber algebras under singular equivalences.