The Hochschild cocycle corresponding to a long exact sequence
Let \(A\) be an associative algebra over a field \(k\) and \(M,N\) right \(A\)-modules. Then there is a natural isomorphism \(\eta_{MN}:\text{Ext}^ t_ A(M,N)\to H^ t(A,\Hom_ k(M,N))\), where \(H^.\) denotes the Hochschild cohomology of \(A\). Let \[ E=(0\leftarrow M\leftarrow^{g_ 0}Y_ 1\leftarrow^{g_ 1}Y_ 2\leftarrow\cdots\leftarrow Y_ t\leftarrow^{g_ t}N\leftarrow 0) \] be an exact sequence. As a sequence of \(k\)-spaces, it splits, thus there are \(k\)-linear maps \[ M\to^{\gamma_ 0}Y_ 1\to^{\gamma_ 1}Y_ 2\to\cdots\to Y_ t\to^{\gamma_ t}N \] such that \(\gamma_{i-1}\gamma_ i=0\), \(g_{i-1}\gamma_{i-1}+\gamma_ ig_ i=1_{Y_ i}\) for \(1\leq i\leq t\) and \(\gamma_ 0g_ 0=1_ M\), \(g_ t\gamma_ t=1_ N\). With \(\gamma =(\gamma_ 0,...,\gamma_ t)\) we associate a map \(\Omega_{\gamma}: A^{\otimes (t+2)}\to\Hom_ k(M,N)\) defined by \((a_ 0,...,a_{t+1})\Omega_{\gamma}=\bar a_ 0\gamma_ 0\bar a_ 1\gamma_ 1...\bar a_ t\gamma_ t\bar a_{t+1}\) for \(a_ 0,...,a_{t+1}\) in \(A\), where \(\bar a_ i\) denotes the scalar multiplication by \(a_ i\). The authors then show the following theorem: The map \(\Omega_{\gamma}\) is a cocycle and the cohomology classes \([\Omega_{\gamma}]\) and \(\eta([E])\) in \(H^ t(A,Hom_ k(M,N))\) are equal up to sign. They also remark that one reason for their interest in this problem comes from the recursive construction for quasi-hereditary algebras due to Parshall and Scott and that the Hochschild cohomology groups play a prominent role in Tachikawa's discussion of the Nakayama conjecture.
scientific article; zbMATH DE number 4200409