Grothendieck Groups and Picard Groups of Abelian Group Rings
Grothendieck Groups and Picard Groups of Abelian Group Rings
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阿贝尔群环的格洛腾迪克群和皮卡德群
DOI：
10.2307/1970360
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发表时间：
1967
影响因子：
4.9
通讯作者：
M. P. Murthy
中科院分区：
文献类型：
--
作者：
H. Bass;M. P. Murthy
对于交换环A，我们研究了有限生成投射A-模的Grothendieck群K0A和一阶投射模的Picard群Pic(A)。对于KOA中的零阶元，有一个满射Det：K0A Pic(A)，它由r阶射影模的第r次外方定义。我们最初的目的是计算Kj(Zw)，其中w是有限生成的阿贝尔群。我们在？8中做到了这一点。论文的大部分是对一些在其他情况下已经熟悉的技术的相当一般性的讨论，我们使用这些技术来进行计算。我们的程序是将w分解为W0×T，其中W0是有限群，T是自由阿贝尔数，然后记为Zw=A[T]，其中A=Zw0。则A是(Krull)维的Noether环。对于这样的A，我们证明(定理7.8)Det：Ko(A[T])Pic(A[T])是同构的。在？9中，我们也证明了对于秩为2的T，这是一个非稳定的模拟。在？8中，我们证明(定理8.1)
For a commutative ring A we study here the Grothendieck group, K0A, of finitely generated projective A-modules, and the Picard group, Pic (A), of projective modules of rank one (under (?A). Writing koA for the elements of rank zero in KOA, there is an epimorphism det: K0A Pic (A) defined by the rth exterior power of a projective module of rank r. Our original objective was to calculate KJ(Zw), where w is a finitely generated abelian group. We do this in ? 8. The bulk of the paper is a rather general discussion of the techniques, some already familiar in other contexts, which we use to make this calculation. Our procedure is to decompose w as w0 x T, with w0 a finite group and T free abelian, and then to write Zw= A[T] where A = Zw0. Then A is a noetherian ring of (Krull) dimension one. For such an A we show (Theorem 7.8) that det: ko(A[T]) Pic (A[T]) is an isomorphism. In ?9 we prove also a non-stable analogue of this for T of rank <2. In ?8 we show (Theorem 8.1) that