Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture.(English)Zbl 1261.16010
Let \(k\) be an algebraically closed field, and \(A\), \(B\) be finite dimensional \(k\)-algebras. The algebras \(A\) and \(B\) are said to be stably equivalent if there is an equivalence of their associated stable module categories. The Auslander-Reiten conjecture asserts that if \(A\) and \(B\) are stably equivalent, then \(A\) and \(B\) should have the same number of isomorphism classes of non-projective simple modules. The focus of this paper is on the special (but oft occurring) case of a stable equivalence ofMorita type.
Let \(A^e:=A\otimes_kA^{op}\) denote the enveloping algebra of \(A\). The center \(Z(A)\) of \(A\) can be identified with the \(A^e\)-endomorphisms of \(A\), and the projective center of \(A\) is the subset of \(Z(A)\) consisting of those endomorphisms that factor though a projective \(A^e\)-module. From an alternate perspective, the projective center of \(A\) can be identified with the Higman ideal of \(A\). The quotient of the center by the projective center is called the stable center of \(A\). The authors introduce an analogous notion of the 0-degree stable Hochschild homology group \(HH_0^{\text{st}}(A)\) as a subset of the ordinary Hochschild homology \(HH_0(A)\). It is shown that if \(A\) and \(B\) are stably equivalent of Morita type, then \(HH_0^{\text{st}}(A)\cong HH_0^{\text{st}}(B)\). Further, it is shown that the dimension of \(HH_0(A)\) is the dimension of \(HH_0^{\text{st}}(A)\) plus the \(p\)-rank of the Cartan matrix of \(A\) where \(p\) is the characteristic of \(k\).
The main result is that if \(A\) and \(B\) are stably equivalent of Morita type, then \(A\) and \(B\) having the same number of isomorphism classes of simple modules is equivalent to \(HH_0(A)\) and \(HH_0(B)\) having the same dimension. If \(A\) and \(B\) have no semisimple direct summands, then this is further equivalent to \(A\) and \(B\) having the same number of non-projective simple modules. If one supposes that \(A\) or \(B\) is symmetric, then the condition that \(A\) and \(B\) have the same number of isomorphism classes of simple modules is further equivalent to the centers (or projective centers) of \(A\) and \(B\) having the same dimension.
The above results hold over fields of arbitrary characteristic. For \(p>0\), the authors use the existence of a power-\(p\) map on \(HH_0(A)\) to obtain an alternate proof of the results. They also provide an application to stable Külshammer and Reynolds ideals.
Let \(A^e:=A\otimes_kA^{op}\) denote the enveloping algebra of \(A\). The center \(Z(A)\) of \(A\) can be identified with the \(A^e\)-endomorphisms of \(A\), and the projective center of \(A\) is the subset of \(Z(A)\) consisting of those endomorphisms that factor though a projective \(A^e\)-module. From an alternate perspective, the projective center of \(A\) can be identified with the Higman ideal of \(A\). The quotient of the center by the projective center is called the stable center of \(A\). The authors introduce an analogous notion of the 0-degree stable Hochschild homology group \(HH_0^{\text{st}}(A)\) as a subset of the ordinary Hochschild homology \(HH_0(A)\). It is shown that if \(A\) and \(B\) are stably equivalent of Morita type, then \(HH_0^{\text{st}}(A)\cong HH_0^{\text{st}}(B)\). Further, it is shown that the dimension of \(HH_0(A)\) is the dimension of \(HH_0^{\text{st}}(A)\) plus the \(p\)-rank of the Cartan matrix of \(A\) where \(p\) is the characteristic of \(k\).
The main result is that if \(A\) and \(B\) are stably equivalent of Morita type, then \(A\) and \(B\) having the same number of isomorphism classes of simple modules is equivalent to \(HH_0(A)\) and \(HH_0(B)\) having the same dimension. If \(A\) and \(B\) have no semisimple direct summands, then this is further equivalent to \(A\) and \(B\) having the same number of non-projective simple modules. If one supposes that \(A\) or \(B\) is symmetric, then the condition that \(A\) and \(B\) have the same number of isomorphism classes of simple modules is further equivalent to the centers (or projective centers) of \(A\) and \(B\) having the same dimension.
The above results hold over fields of arbitrary characteristic. For \(p>0\), the authors use the existence of a power-\(p\) map on \(HH_0(A)\) to obtain an alternate proof of the results. They also provide an application to stable Külshammer and Reynolds ideals.
Reviewer: Christopher P. Bendel (Menomonie)
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16G10 | Representations of associative Artinian rings |
| 16D90 | Module categories in associative algebras |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
Auslander-Reiten conjecture;Higman ideals;projective center;stable equivalences of Morita type;stable Hochschild homology;transfer maps;simple modules;finite dimensional algebras;stable module categoriesCite
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