Mathematics – Commutative Algebra
Scientific paper
2010-05-07
Mathematics
Commutative Algebra
This version includes a new theorem (Theorem 1.8), in which results due to D'Anna and Shapiro are completed and strengthened.
Scientific paper
A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\gets S$ 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 local ring obtained by factoring out the diagonal image of $V$ in the fiber product $R\times_TS$. 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.
Ananthnarayan H.
Avramov Luchezar L.
Moore Frank W.
No associations
LandOfFree
Connected sums of Gorenstein local rings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Connected sums of Gorenstein local rings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Connected sums of Gorenstein local rings will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-303303