论文标题
Koszul综合体上的迭代映射锥及其用于完成交叉环的应用
Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings
论文作者
论文摘要
令$（R，\ Mathfrak M，\ Mathsf K）$为完整的交叉点本地环，$ k $是$ \ Mathfrak m $的最小发电机，而$ a = h（k）$是其同源代数。我们建立了涉及$ a $组件的直接总和的精确序列，并将这些序列的地图的图像表示为构建$ k $的迭代映射锥的同源物。作为这种迭代映射锥构建的应用，我们恢复了$ r $的残留场$ \ mathsf k $的最低自由分辨率，与泰特（Tate）通过相邻变量和杀死周期构建的知名分辨率无关。通过我们的构造，差分图可以明确表示为以某些组合图案排列的矩阵块。
Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.
