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• W3021361210 abstract "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 as long as $dim R>3$, 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 sequence. 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 field of positive characteristic." @default.
• W3021361210 created "2020-05-13" @default.
• W3021361210 creator A5079675868 @default.
• W3021361210 date "2020-05-05" @default.
• W3021361210 modified "2023-09-27" @default.
• W3021361210 title "Lim Ulrich sequence: proof of Lech's conjecture for graded base rings" @default.
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