Lectures on Inverse Systems of Local Rings, Part I
Speaker: Juan Elias (University of Barcelona)

Time: 9h00, 19/2/2014
Venue: Room 6, Building A14, 18 Hoang Quoc Viet Cau Giay, Ha noi

Abstract: The aim of these three lectures is to present some recent results on the classi cation of Artin Gorenstein and level algebras by using the inverse system of Macaulay. Let R = k[x1,...,xn] be the ring of the formal series and let S = k[y1,...,yn] be a polynomial ring. Injective envelope of k as R-module is isomorphic to S; the structure of S as R-module if de fined by derivation. Macaulay established an one-to one correspondence between the Gorenstein Artin algebras A = R/I and monogen submodules of the polynomial ring S. This correspondence is a particular case of Matlis duality. Macaulay's correspondence establish a dictionary
between the algebraic-geometric properties of Artin Gorenstein algebras A and the algebraic properties of its inverse system F or the geometric properties of the variety defi ned by F = 0.

In the first lecture we will review the main results on the Macaulay's correspondence and how can be applied to the classi cation of Artin algebras. In the second lecture we will present the main results obtained on the classi cation of short Artinian Gorenstein algebras and level algebras. We will study the link between the classi cation of some Artinian Gorenstein algebras and the classi cation of some projective hypersurfaces. In the third lecture we will consider the computational problems appearing in the results presented in the previous lectures. In particular we consider the problem of computing the Betti numbers of an ideal by considering only its inverse system. Finally, we will present some of the main open problems on inverse systems.


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