论文标题
单一理想的线性化
Linearization of monomial ideals
论文作者
论文摘要
我们介绍了一种称为线性化的结构，该结构将与较大的多项式环中的任何单一理想$ i $相关联$ i $。该结构的主要功能是新的理想$ \ mathrm {lin}（i）$具有线性商。特别是，由于$ \ mathrm {lin}（i）$是单度生成的，因此可以得出$ \ mathrm {lin}（i）$具有线性分辨率。我们研究了这种结构的一些特性，例如它与理想，其贝蒂数字，功能性和组合解释的经典操作相互作用。此外，我们引入了一种称为等式的辅助结构，该辅助结构与任何单一的理想相关联是在一个多项式环中以一个更可变的多项式环中产生的新单一理想。我们研究等式的某些同源性和组合特性，可以看作是众所周知的均质化结构的单一类似物。
We introduce a construction, called linearization, that associates to any monomial ideal $I$ an ideal $\mathrm{Lin}(I)$ in a larger polynomial ring. The main feature of this construction is that the new ideal $\mathrm{Lin}(I)$ has linear quotients. In particular, since $\mathrm{Lin}(I)$ is generated in a single degree, it follows that $\mathrm{Lin}(I)$ has a linear resolution. We investigate some properties of this construction, such as its interplay with classical operations on ideals, its Betti numbers, functoriality and combinatorial interpretations. We moreover introduce an auxiliary construction, called equification, that associates to any monomial ideal a new monomial ideal generated in a single degree, in a polynomial ring with one more variable. We study some of the homological and combinatorial properties of the equification, which can be seen as a monomial analogue of the well-known homogenization construction.
