ON THE FINITISTIC GLOBAL DIMENSION CONJECTURE FOR ARTIN ALGEBRAS
ON THE FINITISTIC GLOBAL DIMENSION CONJECTURE FOR ARTIN ALGEBRAS
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DOI：
10.1090/fic/045/15
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
Kiyoshi Igusa;G. Todorov
中科院分区：
文献类型：
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作者：
Kiyoshi Igusa;G. Todorov
我们找到了一个简单的条件，它蕴涵了有限代数整体维数的有限性。由此，我们得到了根立方零代数有限整体维数猜想的一个简短证明。同样的条件也适用于表示维数小于或等于3的代数。因此，有限维猜想在这种情况下也成立。设Λ是一个阿提尼代数（可交换阿提尼环上有限长度的代数）。然后，有限全局维数猜想表明，对于有限pd的所有fg（左）Λ-modules的有限射影维数（pd）存在一个统一界findimΛ。这个猜想暗示了中山猜想。一些已知的有限全局维猜想成立的情况是根立方零情况[GZ]和单项式关系情况[GKK]（参见[IZ]， [BFGZ]）。当有限pd的模的范畴在所有fg模的范畴中变有限时，这个猜想也是成立的[AR]。然而，反过来是不正确的[IST]。本文给出了对任意代数上根方根为零的所有模的有限格猜想的一个简短证明。这是根方根为零情况的推广因为在这种情况下所有的合子都有根方根为零。在[Z-H]中可以找到有限全局维猜想状态的全面概述。作为主要定理的另一个结果，我们证明了弱表示维数不超过3的代数的有限维猜想，从而证明了表示维数repdimΛ≤3的代数的有限维猜想。表征维度的概念是由M. Auslander在他的《玛丽女王笔记》（Queen Mary Notes）中提出的[A1]，他和其他许多人都认为这个维度的边界是3。O. Iyama证明了它总是有限的[I]，许多代数类已知具有repdimΛ = 3，最近的一类是具有相同根的有限表示型代数的子代数[EHIS]。主要定理的证明是基于以下众所周知的基本观察。引理1（拟合的引理）a)设M是noether环R上的一个模，设f: M→M是M的自同态。然后对于NSF 90 02512支持的任何研究，NSF 90 09590支持的研究
We find a simple condition which implies finiteness of finitistic global dimension for artin algebras. As a consequence we obtain a short proof of the finitistic global dimension conjecture for radical cubed zero algebras. The same condition also holds for algebras of representation dimension less then or equal to three. Hence the finitistic dimension conjecture holds in that case as well. Let Λ be an Artin algebra (an algebra of finite length over a commutative Artinian ring). Then the finitistic global dimension conjecture states that there exists a uniform bound called findimΛ for the finite projective dimensions (pd) of all f.g. (left) Λ-modules of finite pd. This conjecture would imply the Nakayama conjecture. Some of the known cases in which the finitistic global dimension conjecture holds are the radical cubed zero case [GZ] and the monomial relation case [GKK] (see also [IZ], [BFGZ]). The conjecture is also true in the case when the category of modules of finite pd is contravariantly finite in the category of all f.g. modules [AR]. However, the converse is not true [IST]. In this paper we give a short proof of the finitistic gl dim conjecture for all modules of radical square zero over any Artin algebra. This is a generalization of the radical cubed zero case since all syzygies have radical square zero in that case. A thorough overview of the state of the finitistic global dimension conjecture can be found in [Z-H]. As another consequence of the main theorem we prove the finitistic dimension conjecture for algebras with weak representation dimension at most 3, and consequently for algebras with representation dimension repdimΛ ≤ 3. The notion of representation dimension was introduced by M. Auslander in his Queen Mary Notes [A1], and he and many others expect this dimension to be bounded by 3. O. Iyama showed that it is always finite [I], many classes of algebras are known to have repdimΛ = 3, the most recent class being subalgebras of algebras of finite representation type with the same radical [EHIS]. The proof of the main theorem is based on the following well-known elementary observation. Lemma 1 (Fitting’s Lemma). a) Let M be a module over a Noetherian ring R and let f : M → M be an endomorphism of M . Then for any Research supported by NSF 90 02512 Research supported by NSF 90 09590