Abstract
Matlis duality and the particular case of Macaulay correspondence provide a dictionary between the Artin algebras and their inverse systems. Inspired in a result of Emsalem we translate the problem of classification of Artin algebras to a problem of linear system of equations on the inverse systems.
The main purpose of these notes is to use this result to classify Artin Gorenstein algebras with Hilbert function {1, 3, 3, 1}, level algebras and compressed algebras. The main results presented in these notes were obtained in collaboration with M.E. Rossi.
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Acknowledgements
I am grateful to Le Tuan Hoa and to Ngo Viet Trung for giving me the opportunity to speak about one of my favorite subjects. I am also grateful to the participants for their kind hospitality and mathematical discussions that made a very interesting and productive month in the city of Hanoi. We thank to Marcela Silva and Roser Homs for their useful comments and remarks. Last but not least, I am greatly indebted to M. E. Rossi for a long time collaboration on Macaulay’s inverse systems and other topics. Some of the main results of these notes are made in collaboration with M. E. Rossi.
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Elias, J. (2018). Inverse Systems of Local Rings. In: Tu CUONG, N., Tuan HOA, L., Viet TRUNG, N. (eds) Commutative Algebra and its Interactions to Algebraic Geometry. Lecture Notes in Mathematics, vol 2210. Springer, Cham. https://doi.org/10.1007/978-3-319-75565-6_2
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