A non-Golod ring with a trivial product on its Koszul homology
A non-Golod ring with a trivial product on its Koszul homology
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发表时间：
2015-11
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通讯作者：
Lukas Katthan
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作者：
Lukas Katthan
我们提出了一个单名理想a S，使得S/a不是Golod，即使其Koszul同调的乘积是平凡的。这是Berglund和Jöllenbeck的一个著名结果的反例（这个错误可以追溯到Jöllenbeck早期文章中的错误）。在积极的一面，我们证明了如果R是一个单项环，使得对于所有r ≤ max（2，reg R− 2），稀疏Massey积为零，则R是Golod环。特别地，如果R是维数至多为3的单纯复形的Stanley-Reisner环，则R是Golod当且仅当其Koszul同调的乘积是平凡的。此外，我们还证明了：如果k-可定向流形的Stanley-Reisner环是Golod，那么k-可定向流形的三角剖分是2-邻域的.这推广了Iriye和Kishimoto最近的一个结果。
We present a monomial ideal a ⊂ S such that S/a is not Golod, even though the product in its Koszul homology is trivial. This constitutes a counterexample to a well-known result by Berglund and Jöllenbeck (the error can be traced to a mistake in an earlier article by Jöllenbeck). On the positive side, we show that if R is a monomial ring such that the rary Massey product vanishes for all r ≤ max(2, reg R− 2), then R is Golod. In particular, if R is the Stanley-Reisner ring of a simplicial complex of dimension at most 3, then R is Golod if and only if the product in its Koszul homology is trivial. Moreover, we show that if ∆ is a triangulation of a k-orientable manifold whose Stanley-Reisner ring is Golod, then ∆ is 2-neighborly. This extends a recent result of Iriye and Kishimoto.