Tilting, cotilting, and serially tilted rings
Tilting, cotilting, and serially tilted rings
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DOI：
10.1080/00927879008823985
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发表时间：
1990
影响因子：
0.7
通讯作者：
R. Colby;K. Fuller
中科院分区：
文献类型：
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作者：
R. Colby;K. Fuller
1586科尔比、FULLER和Ringel [12]通过计算倾斜模和相应的倾斜代数得到了一类有趣的有限型代数。在这里，我们提供了一些已知的结果的倾斜和cotilting理论更一般的环，并利用它们获得的描述的倾斜模noether系列环，其自同态环，我们称之为系列倾斜环，和范畴S-mod和mod-S的系列倾斜环S。Miyashita [16]和Cline，Parshall，and Scott [7]研究了倾斜模的推广，并得到了任意环的著名倾斜定理（例如见[4]）的部分推广。在第1节中，我们提出了一个相当完整的版本的倾斜定理（与简化的证明）的任意环R;我们证明，如果R是左遗传和S=结束（RT），那么T诱导分裂扭转理论的S-Mod完全类似于代数的情况。科尔比[B]研究了Noether环上的余倾斜模的概念，得到了倾斜定理的一个对偶形式.在此基础上，在第二节中我们证明了左Noether遗传环上的倾斜模也是余倾斜模，并且如果（R，T，S）是R左遗传的余倾斜三元组，则倾斜模T在mod-S上导出了一个分裂挠理论，该理论用R中的T-挠和T-自反模来描述mod-S。在第三节中，我们证明了Noether序列环R上的倾斜模是R的Artin因子环上的投射模与倾斜模的直和，并证明了在遗传Noether序列环上，倾斜模和余倾斜模是同一个，第4节包含从前面几节导出的序列倾斜Noether环的特征，Happel和Ringel对倾斜到三角矩阵代数的代数的特征[12]，Fuller和Haack [10]，定理21和沃菲尔德关于素Noether序列环的刻划[22]。这类环是整体维数为5 2的Noether环，其生成的左模和右模完全由tilting-cotilting理论和沃菲尔德对R是Noether序列环的R-mod的描述[22]所描述.此外，与序列情况不同，它们的左和右结构彼此之间几乎没有相似之处。
1586 COLBY AND FULLER and Ringel [12] obtained an interesting class of algebras of finite type by calculating the tilting modules and the corresponding tilted algebras. Here we provide extensions of some of the known results on tilting and cotilting theory to more general rings, and employ them to obtain descriptions of the tilting modules over noetherian serial rings, their endomorphism rings which we call serially tilted rings, and the categories S-mod and mod-S for a serially tilted ring S. Miyashita [16] and Cline, Parshall, and Scott [7] have investigated generalizations of tilting modules and have obtained generalizations of portions of the well known Tilting Theorem (see [4], for example) for arbitrary rings. In Section 1 we present a rather complete version of the Tilting Theorem (with simplified proofs) for an arbitrary ring R; and we prove that if R is left hereditary and S= End (RT) then T induces a splitting torsion theory on S-Mod entirely analogous to the algebra case. Colby [B] has investigated the notion of cotilting modules over noetherian'rings and obtained a dual version of the Tilting Theorem. Building on his results, in Section 2 we show that tilting modules over left noetherian hereditary rings are also cotilting modules, and if (R, T, S) is a cotilting triple with R left hereditary then a tilting module T induces a splitting torsion theory on mod-S that describes mod-S in terms of the T-torsion and T-reflexive modules in R-mod. In Section 3 we show that a tilting module over a noetherian serial ring R is a direct sum of a projective module and a tilting module over an artinian factor ring of R, and we prove that, over hereditary noetherian serial rings, tilting modules and cotilting modules are one-and-the-same, Section 4 contains a characterization of serially tilted noetherian rings derived from the preceding sections, an extension of Happel and Ringel's characterization of algebras that tilt to triangular matrix algebras [12], Fuller and Haack's [10, Theorem 21, and Warfield's characterization of prime noetherian serial rings [22]. These rings are noetherian rings of global dimension 5 2 whose finitely generated left and right modules are completely described by the tilting-cotilting theory and Warfield's description [22] of R-mod for R a noetherian serial ring. Moreover, unlike the serial case, their left and right structures bear little or no resemblance to each other.