Decomposing Gorenstein Rings as Connected Sums
Abstract
In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and if so, what are its components. We answer these questions in the Artinian case and investigate conditions on the ring which force it to be indecomposable as a connected sum. We further give a characterization for Gorenstein Artin local rings to be decomposable as connected sums, and as a consequence, obtain results about its Poincare series and minimal number of generators of its defining ideal. Finally, in the graded case, we show that the indecomposable components appearing in the connected sum decomposition are unique up to isomorphism.
Keywords
Cite
@article{arxiv.1406.7600,
title = {Decomposing Gorenstein Rings as Connected Sums},
author = {H. Ananthnarayan and Ela Celikbas and Jai Laxmi and Zheng Yang},
journal= {arXiv preprint arXiv:1406.7600},
year = {2017}
}
Comments
4 major changes: 1) Uniqueness of decomposition is addressed, more conditions for decomposition are studied (Sections 4 and 5). 2) Associated graded rings related material removed, to appear in a separate paper. 3) Results are proved more generally, not just for algebras over a field. 4) One more co-author, Jai Laxmi, has also contributed to this version