论文标题
在Grothendieck戒指及其单元组与Picard组的关系
On the Grothendieck ring and the relation of its group of units with the Picard group
论文作者
论文摘要
本文的第一个主要结果断言，如果$ e $和$ e'$是$ a $的同性恋，那么我们有以下规范的同构同构为$ a $ a $ -modules：$$ ae \ oplus ae \ oplus ae'\ simeq ae/ae/ae/ae/ae/ae（1- e''） a（e+e'-2ee'）。$$此结果在证明Grothendieck环$ k_ {0}（a）$的几个结果中起着重要作用。尤其是，对于任何可交换的环$ a $，我们获得了以下（拆分）亚洲小组的复合体，这些组合在开始和结束时是：$$ \ xymatrix {0 \ ar [r]＆\ pic（a）\ ar [r]＆k_ {0}＆k_ {0}＆k_ {0}（a）（a） $ a^{\ ast} $，我们的意思是$ a $的单位组，以及$ \ mathscr {b}（a）$，我们的意思是$ a $ a $ a $ a $ a $ a $。作为一个应用程序，如果$ a $是一个dedekind域或更一般的noetherian一维环，则我们获得以下Abelian组的拆分精确序列：$ \ xymatrix {0 \ ar [r]＆\ pic（a）\ ar [a）\ ar [r] \ ar [r]＆\ Mathscr {b}（a）\ ar [r]＆0。} $$作为另一个主要结果，对于任何环$ a $，我们获得了Abelian oblian $ \ Mathscr {b}（b}（b}（a）（a）\ simeq \ simeq \ Mathscr {b} b} b} big big big big big（a）的规范同构h_ {0}（a）^{\ ast} $。接下来，我们表明，且仅当诱导的环映射$ k_ {0}（a）（a）\ rightarrow k_ {0}（b）$抬起iDempotents时，我们才表明，$ a \ rightarrow b $ lifts divempotents b $抬起iDempotents。如果此外，$ b $具有有限的最大理想，那么地图$ k_ {0}（a）\ rightArrow k_ {0}（b）$是溢出的。最后，我们表明，当且仅当它的痕迹理想才是环的整个单元理想时，有限生成的投影模块的支持是整个序列。
The first main result of this article asserts that if $e$ and $e'$ are idempotents of a commutative ring $A$, then we have the following canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus A(e+e'-2ee').$$ This result plays an important role in proving several results on the Grothendieck ring $K_{0}(A)$. Especially, for any commutative ring $A$ we obtain the following (split) complex of Abelian groups which is exact at the beginning and end: $$\xymatrix{0\ar[r]&\Pic(A)\ar[r]&K_{0}(A)^{\ast} \ar[r]&\mathscr{B}(A)\ar[r]&0}$$ where by $A^{\ast}$ we mean the group of units of $A$, and by $\mathscr{B}(A)$ we mean the group of idempotents of $A$. As an application, if $A$ is a Dedekind domain or more generally a Noetherian one dimensional ring, then we obtain the following split exact sequence of Abelian groups: $$\xymatrix{0\ar[r]&\Pic(A)\ar[r]&K_{0}(A)^{\ast} \ar[r]&\mathscr{B}(A)\ar[r]&0.}$$ As another main result, for any ring $A$ we obtain the canonical isomorphisms of Abelian groups $\mathscr{B}(A)\simeq\mathscr{B}\big(K_{0}(A)\big)\simeq H_{0}(A)^{\ast}$. Next, we show that a morphism of rings $A\rightarrow B$ lifts idempotents if and only if the induced ring map $K_{0}(A)\rightarrow K_{0}(B)$ lifts idempotents. If moreover, $B$ has finitely many maximal ideals then the map $K_{0}(A)\rightarrow K_{0}(B)$ is surjective. Finally, we show that the support of a finitely generated projective module is the whole prime spectrum if and only if its trace ideal is the whole unit ideal of the ring.
