The Grothendieck ring of a finite group
The Grothendieck ring of a finite group
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有限群的格洛腾迪克环
DOI：
10.1016/0040-9383(63)90025-9
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发表时间：
1963
期刊：
影响因子：
--
通讯作者：
R. G. Swan
中科院分区：
文献类型：
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作者：
R. G. Swan
如果E是加法范畴，则Grothendieck群Kc%？)定义为当%‘[6]，[23]中存在正合列0+A’+A-+A“-+0时，对V的每个对象A有一个生成元[A]，且关系[A]=[A‘]+[A”]的群。如果R是交换环，Rr是有限群，我将用G(RRR)表示有限生成的范畴的Grothendieck群：RN-模[23]。本文的主要目的是得到当R是Dedekind环时G(Rn)的结构信息.如果R是具有商域K的整环，则与K张量给出一个同态I‘：G(RRR)-+G(Kn)[23，$21.在1美元中，我将证明这始终是对的。现在G(Kn)同构于K上7c的表示的特征标环[23，引理(4.L)]。因此，我们可以假设G(Kn)是已知的。然后，主要问题是确定G(RRR)+G(Kn)的核。为了做到这一点，我们需要更多的定义。设P，(R7c)是有限生成投射RRR-模范畴P的Grothendieck群，具有K@Rp是I&自由的性质。存在整数到P(RJT)的自然同态E，E(1)=[RRR]。还有一个自然同态Q：P，(Rn)+2 by Q(P)=n，如果K@Rp在n个生成元上是Kn自由的。由于VE=1，我们有一个自然分裂P，(RRL)=Z0C，，(Rz)其中C，(RX)=ker Q称为RTI的约化射影类群[23，$91，[20]。由于有限生成投射RRR-模范畴是有限生成RRR-模范畴的一个子范畴，因此存在一个自然映射P，(Rn)+G(Rn)by[P]-+[PI.将其限制为C，(Rn)得到一个映射0：C，(Rn)-+G(Rn)。现在可以陈述主要定理，它肯定地回答了DS Rim的一个猜想。
IF E is any additive category, the Grothendieck group Kc%?) is defined to be the group with one generator [A] for each object A of V and relations [A]=[A’]+[A”] whenever there is an exact sequence 0+ A’+ A-+ A”-+ 0 in%’[6],[23]. If R is a commutative ring and rr is a finite group, I will denote by G (Rrr) the Grothendieck group of the category of finitely generated: Rn-modules[23]. The main purpose of this paper is to obtain information about the structure of G (Rn) when R is a Dedekind ring.If R isan integral domain with quotient field K, tensoring with K gives a homomorphism I’: G (Rrr)-+ G (Kn)[23, $21. In $1, I will show that this is always onto. Now G (Kn) is isomorphic to the ring of characters of representations of 7c over K [23, Lemma (4. l)]. Thus we can assume G (Kn) to be known. The main problem is then to determine the kernel of G (Rrr)+ G (Kn). In order to do this, we need a few more definitions. Let P,(R7c) be the Grothendieck group of the category of finitely generated projective Rrr-modules P with the property that K@ RP is I&-free. There is a natural homomorphism E of the integers into P,(RJT) by E (1)=[Rrr]. There is also a natural homomorphism q: P,(Rn)+ 2 by q (P)= n if K@ RP is Kn free on n generators. Since VE= 1, we have a natural splitting P,(Rrl)= Z 0 C,,(Rz) where C,(RX)= ker q is called the reduced projective class group of RTI [23, $91,[20]. Since the category of finitely generated projective Rrr-modules is a subcategory of the category of finitely generated Rrr-modules, there is a natural map P,(Rn)+ G (Rn) by [P]-+[PI. Restricting this to C,(Rn) gives a map 0: C,(Rn)-+ G (Rn). It is now possible to state the main theorem, which answers affirmatively a conjecture of DS Rim.