The trace of the Coxeter matrix and Hochschild cohomology (Q1359184)

Let \(\Lambda\) be a finite-dimensional basic \(k\)-algebra over an algebraically closed field \(k\). Assume that \(\Lambda\) is of finite global dimension. Let \(P(1),\dots,P(n)\) be a complete set of non-isomorphic indecomposable projective (right) \(\Lambda\)-modules. The entries of the Cartan matrix \(C:=C_\Lambda\) of \(\Lambda\) are defined as \(c_{ij}:=\dim_k\Hom_\Lambda(P(i),P(j))\), \(1\leq i,j\leq n\). The Coxeter matrix \(\Phi_\Lambda\) of \(\Lambda\) is defined as \(-C^{-t}_\Lambda C_\Lambda\), where \(C^{-t}_\Lambda\) is the inverse transpose of \(C_\Lambda\). Let \(\Lambda^e:=\Lambda\otimes_k\Lambda^{\text{op}}\) be the enveloping algebra of \(\Lambda\); the latter is, in a natural way, a right module over the former. The \(i\)th Hochschild cohomology group \(H^i(\Lambda)\) can be defined as \(\text{Ext}^i_{\Lambda^e}(\Lambda,\Lambda)\). The main result of the paper relates the trace of the Coxeter matrix of \(\Lambda\) and the Euler-Poincaré characteristic of \(\Lambda\) as a \(\Lambda^e\)-module. More precisely, the following theorem is proved. Theorem: Under the above assumptions on \(\Lambda\), \[ -\text{trace }\Phi_\Lambda=\sum_{i\geq 0}(-1)^i\dim_kH^i(\Lambda). \] The last section of the paper contains some examples and applications of this theorem.