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Associated Graded Rings and Connected Sums

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 نشر من قبل H Ananthnarayan
 تاريخ النشر 2017
  مجال البحث
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In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincare series of Q.



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A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $Rto Tgets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the emph{connected sum} $R#_TS$ is defined to be the l ocal ring obtained by factoring out the diagonal image of $V$ in the fiber product $Rtimes_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R#_TS$. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra $Ext^*_{R#_kS}(k,k)$ is an amalgam of the algebras $Ext^*_{R}(k,k)$ and $Ext^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring $R#_TS$ almost never is complete intersection.
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