Hochschild cohomology of relation extension algebras

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• 作者：Ibrahim Assem^a ; ^{Ibrahim.Assem@usherbrooke.ca" class="auth_mail" title="E-mail the corresponding author} ; Maria Andrea Gatica^b ; ^{mariaandrea.gatica@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ralf Schiffler^c ; ^{schiffler@math.uconn.edu" class="auth_mail" title="E-mail the corresponding author} ; Rachel Taillefer^d ; ^{Rachel.Taillefer@math.univ-bpclermont.fr" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：16E40 ; 16S70
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：220
• 期：7
• 页码：2471-2499
• 全文大小：584 K

文摘

Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E  . We define a morphism of associative graded algebras φ^⁎:HH^⁎(B)→HH^⁎(C)${\phi }^{⁎}:{\mathrm{HH}}^{⁎}(B)\to {\mathrm{HH}}^{⁎}(C)$ from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler.

In the case of a trivial extension B=C⋉E$B=C⋉E$, we give necessary and sufficient conditions for each e506af14f07d1da69bfed88cf" title="Click to view the MathML source">φⁿ${\phi }^n$ to be surjective. We prove the surjectivity of φ¹${\phi }^1$ for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ¹${\phi }^1$ for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.