Lim Ulrich序列与Lech猜想
Lim Ulrich sequences and Lech's conjecture
摘要：
交换代数中长期存在的Lech猜想指出，对于Noether局部环的平坦局部扩张$(R,\mathfrak{m})\to (S,\mathfrak{n})$，我们有一个关于Hilbert-Samuel重数的不等式：$e(R)\leq e(S)$。一般来说，当$\dim R>3$，甚至在相等的特征时，猜想是完全开放的。本文证明了Lech猜想在所有维数下的成立，条件是$(R,\mathfrak{m})$是局部化在齐次极大理想上的完满域上的标准分次环.我们引入了lim Ulrich序列和弱lim Ulrich序列的概念。粗略地说，这些序列的有限生成模，不一定是科恩-麦考利，但渐近行为像乌尔里希模块。我们证明了这些序列的存在隐含着Lech猜想。虽然Ulrich模的存在性在非常有限的情况下是已知的，我们构造了弱lim Ulrich序列的所有标准分次域上的完满域的正特征。
英文摘要：
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: $e(R)\leq e(S)$. 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.
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。
