arXiv math/06076392006-07-25math.AC

有限长投射维数2模的泛环的分解

The resolution of the universal ring for finite length modules of projective dimension two

Andrew R. Kustin

备注：: See also http://www.math.sc.edu/~kustin

摘要：

Hochster建立了交换诺特环$\Cal R$和形式为$0\to \Cal R^{e}\to \Cal R^{f}\to \Cal R^{g}\to 0$的通用分解$\Bbb U$的存在性，使得对于任何交换诺特环$S$和任何等于$0\to S^{e}\to S^{f}\to S^{g}\to 0$的分解$\Bbb V$，存在唯一的环同态$\Cal R\to S$与$\Bbb V=\Bbb U\otimes_{\Cal R} S$。本文假设$f=e+g$是整数环上的多项式环，并利用自由$\Cal P$-模找到了$\Cal R$的分解$\Bbb F$，其中$\Cal P$是整数环上的多项式环.分辨率$\Bbb F$不是最小的;但它是直接的，坐标自由的，并且与特征无关。此外，可以使用$\Bbb F$来计算$\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$。如果$e$和$g$都至少为5，则$\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$不是自由阿贝尔群;因此，自由$\pmb K\otimes_{\Bbb Z} \Cal P$-模对$\pmb K\otimes_{\Bbb Z} \Cal R$的最小分解中的分次贝蒂数依赖于域$\pmb K$的特征。我们记录的模块中的最小$\pmb K\otimes_{\Bbb Z} \Cal P$决议的$\pmb K\otimes_{\Bbb Z} \Cal R$的模块出现时，解决了约数定义的行列式环的$e\times g$矩阵的$2\times 2$未成年人。

英文摘要：

Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0\to \Cal R^{e}\to \Cal R^{f}\to \Cal R^{g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\Cal R\to S$ with $\Bbb V=\Bbb U\otimes_{\Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ by free $\pmb K\otimes_{\Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{\Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2\times 2$ minors of an $e\times g$ matrix.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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