自由分解与标记族
Free resolutions and marked families
AI总结：
本文研究拟稳定模U上标记基生成子模的U-分解，证明分量线性理想的U-分解极小性等价条件，建立相关函子同构，并给出深度≥2理想的标记基与第一合冲模标记基的互逆对应关系。
摘要：
设$\u211a$为一个域，$A$为一个Noetherian $\u211a$-代数。在2020年M. Albert、C. Bertone、M. Roggero与W. M. Seiler的一篇论文中证明了：给定$R=\u211a[x_0,\u2026,x_n]$上的拟稳定模$U \u2282 R^m$，任何由$U$上的标记基生成的子模$M\u2286 (R\u2297 A)^m$都具有一种同样用标记基描述的特殊自由分解，称为$M$的{\textit{U-分解}}。本文首先研究了U-分解的极小性及其结构。当$M$为理想且$A=\u211a$时，我们证明：$M$是分量线性的当且仅当它的U-分解在变量的线性变换下是极小的。随后，我们采用函子方法构造U-分解，证明了与该分解自然关联的某些函子是同构的。这些同构源于U-分解中第$i$个合冲模的标记基可以用$M$的标记基的系数来表示这一事实。此外，当$M$是深度至少为2的理想时，这种对应关系可以逆转：在这种情况下，$M$本身的标记基可以用它的第一个合冲模的标记基的系数来表示。
英文摘要：
Let $\mathbb{K}$ be a field and $A$ a Noetherian $\mathbb{K}$-algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module $U \subset R^m$ with $R=\mathbb{K}[x_0,\dots,x_n]$, any submodule $M\subseteq (R\otimes A)^m$ generated by a marked basis over $U$ admits a special free resolution described in terms of marked bases as well, called the {\em $U$-resolution of $M$}. In this paper, we first investigate the minimality of the $U$-resolution and its structure. When $M$ is an ideal and $A=\mathbb{K}$, we show that $M$ is componentwise linear if and only if its $U$-resolution is minimal, up to a linear change of variables. Then, adopting a functorial approach to the construction of the $U$-resolution, we prove that certain functors naturally associated with the resolution are isomorphic. These isomorphisms arise from the fact that the marked basis of the $i$-th syzygy module in the $U$-resolution can be expressed in terms of the coefficients of the marked basis of $M$. Moreover, when $M$ is an ideal of depth at least 2, this correspondence can be reversed: in this case, the marked basis of $M$ itself can be written in terms of the coefficients of the marked basis of its first syzygy module.
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