The global homological dimensions of trivial extensions of rings
The global homological dimensions of trivial extensions of rings
复制标题
环的平凡扩张的全局同调维数
DOI：
10.1016/0021-8693(76)90078-8
复制
发表时间：
1976
影响因子：
0.9
通讯作者：
C. Löfwall
中科院分区：
文献类型：
--
作者：
C. Löfwall
设R是环，M是R上的双模.平凡扩张R × M是阿贝尔群R@ M，乘法由（I，m）（r '，m'）=（r '，rm'+ mr '）给出. Palmer和鲁什在[ll]中给出了R x M的整体同调维数的几乎一般公式。然而，他们的证明是不完整的。他们想证明谱序列退化，但他们只给出了一阶微分为零的证明。本文证明了高阶微分也为零。此外，我们的方法（与[III]中的方法非常相似）给出了R x M（以及wgldim R x M）的全局同调维数的完全一般公式。鲁什的一个想法是，“multiple-Tot-”应该与计算wgldim R x M相关。他在[lo]中找到了这个定义（“多焦油”在那里是在一个更复杂的情况下定义的，这在本说明中将不作研究）。设U是左R-模，V是右R-模。Tor（V，M，M，...，M，U）如下。选择U的一个射影分解P，选择复形M@ P的一个射影分解（要做到这一点，请注意复形范畴是一个具有足够射影的阿贝尔范畴，或者使用[1]最后一章中给出的方法），取这个分解的相关单复形，应用函子M@ m，等等，最后应用函子V@。求出相应复形的同调。我们可以证明这个定义与投射分解的选择无关。本文的主要结果之一是：wgldim R x M< n+ Tor，（V，M，.，M，U）= 0（M的p个拷贝）对于所有p，q> 0使得p+ q= n+ 1和所有左R-模U和所有右R-模V. Palm&和鲁什在[111]中对M做了以下假设（这通常不成立）：M有分解g，由平坦的左（或右）R-模的R-双模组成。(ThiS例如，如果R是交换的，M是对称的。）在这种情况下，我们有Tor（V，M，...，M，U）= 287
Let R be a ring, and M a bimodule over R. The trivial extension R x M is as abelian group R@ M and the multiplication is given by (I, m)(r’, m’)=(rr’, rm’+ mr’). Palmer and Roos give in [ll] an almost general formula for the global homological dimension of R x M. However, their proof is incomplete. They want to prove that a spectral sequence degenerates, but they only give a proof of the fact that the first differential is zero. In this note, we prove that also the higher differentials are zero. Moreover, our methods (very similar to those in [Ill) give a completely general formula for the global homological dimension of R x M (and also for wgldim R x M). It was an idea of Roos that “multiple-Tot-” should be relevant to compute wgldim R x M. He found the definition in [lo](“multiple-Tar” is defined there in a more complicated situation, which will not be studied in this note). Let U be a left R-module, and V a right R-module. The definition of Tor (V, M, M,..., M, U) is as follows. Choose a projective resolution P, of U, choose a projective resolution of the complex M@ P,(to do this, observe that the category of complexes is an abelian category with enough projectives or use the method given in the last chapter of [l]), take the associated simple complex of this resolution, apply the functor M@ m, and so on. Apply at last the functor V@. and take the homology of the corresponding complex. One can show that the definition is independent of the choice of projective resolutions. One of the main results in this note is: wgldim R x M< n+ Tor,(V, M,..., M, U)= 0 (p copies of M) for allp, q> 0 such that p+ q= n+ 1 and all left R-modules U, and all right R-modules V. Palm& and Roos make the following assumption on M in [l 11 (which is not true in general): M has a resolution g, consisting of R-bimodules that are flat as left (or right) R-modules.(ThiS is true for instance if R is com-mutative and M is symmetric.) In this case, we have Tor (V, M,..., M, U)= 287