The trace of the Coxeter matrix and Hochschild cohomology. (English) Zbl 0894.16006
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 \(c_{ij}:=\dim_k\operatorname{Hom}_\Lambda(P(i),P(j))\), \(1\leq i,j\leq n\). 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.
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.
Reviewer: A.Martsinkovsky (Boston)
MSC:
16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
16G10 | Representations of associative Artinian rings |
16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
Keywords:
finite-dimensional basic algebras; global dimension; indecomposable projective right modules; Cartan matrices; Coxeter matrices; enveloping algebras; Hochschild cohomology groups; Euler-Poincaré characteristicReferences:
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[2] | Happel, D., Hochschild cohomology of Auslander algebras, (Banach Center Publ., 26 (1990), Polish Acad. Sci: Polish Acad. Sci Warszawa), 303-310 · Zbl 0737.16004 |
[3] | Lukas, F., Elementare Moduln über wilden erblichen Algebren, (Dissertation (1992), Univ. Düsseldorf: Univ. Düsseldorf Düsseldorf) · Zbl 0833.16011 |
[4] | Ringel, C. M., Tame Algebras and Integral Gradratic Forms, (Lecture Notes in Math., 1099 (1984), Springer-Verlag: Springer-Verlag Heidelberg) · Zbl 0546.16013 |
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