论文标题
Lim Ulrich序列和Lech的猜想
Lim Ulrich sequences and Lech's conjecture
论文作者
论文摘要
The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq e(S)$.通常，当$ \ dim r> 3 $，即使在同等特征上，猜想是开放的。在本文中，我们证明了Lech在各个方面的猜想，但前提是$（R，\ Mathfrak {M}）$是一个标准的分级环，这是一个位于同质最大理想的完美场上。我们介绍了Lim Ulrich和Limlim Lim Ulrich序列的概念。大概说这些是有限生成的模块的序列，这些模块不一定是cohen--macaulay，而渐近地表现得像乌尔里希模块。我们证明这些序列的存在意味着Lech的猜想。尽管乌尔里希模块的存在在非常有限的情况下是已知的，但我们在正式的正状特征上为所有标准级别域构建了弱的ulrich序列。
The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq e(S)$. In general the conjecture is wide open when $\dim R>3$, even in equal characteristic. In this paper, we prove Lech's conjecture in all dimensions, provided $(R,\mathfrak{m})$ is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen--Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech's conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect fields of positive characteristic.
