论文标题
具有特征依赖性betti数字的单一理想的力量
Powers of monomial ideals with characteristic-dependent Betti numbers
论文作者
论文摘要
我们探讨了单一理想的贝蒂数量对田地特征的依赖性。第一个观察结果是，对于固定的prime $ p $，$ i $ th的betti数量是单一理想的所有足够高功率的特征$ 0 $和特征$ p $的不同之处，或者对于所有高功率而言相同。在我们的主要结果中，我们提供了单一理想的结构和明确的例子，所有功能都具有某些特征性依赖性的贝蒂数，或者其渐近规则性取决于该领域。我们证明，将新变量对单一理想添加了一个单一的理想，可以将特征依赖性传播到所有力量。对于任何给定的Prime $ P $，这都会产生一个优势理想，因此其所有功率的Betti数字超过$ \ Mathbb {q} $，而$ \ Mathbb {z} _p $却不同。此外，我们表明，对于每$ r \ geq 0 $和$ i \ geq 3 $，都有一个单一的理想$ i $，以便在kodiyalam polyenmials $ \ mathfrak p_3（i），\ ldots，\ ldots，\ mathfrak p_ p_ p_ p_ p_ p_ p_ p_ p_ p_ p_ p_ p_ rdots $ \ mathfrak p_3（i+r p_ interivation $ \ geq r $ of geq r $中我们还提供了相关结果的摘要，并推测了其他组合定义的理想的行为。
We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime $p$ either the $i$-th Betti number of all high enough powers of a monomial ideal differs in characteristic $0$ and in characteristic $p$ or it is the same for all high enough powers. In our main results we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal, allows to spread the characteristic dependence to all powers. For any given prime $p$, this produces an edge ideal such that the Betti numbers of all its powers over $\mathbb{Q}$ and over $\mathbb{Z}_p$ are different. Moreover, we show that, for every $r \geq 0$ and $i \geq 3$ there is a monomial ideal $I$ such that some coefficient in a degree $\geq r$ of the Kodiyalam polynomials $\mathfrak P_3(I),\ldots,\mathfrak P_{i+r}(I)$ depends on the characteristic. We also provide a summary of related results and speculate about the behaviour of other combinatorially defined ideals.
