Collectanea Mathematica

, Volume 69, Issue 3, pp 451–477

# The structure of the inverse system of level K-algebras

• Shreedevi K. Masuti
• Laura Tozzo
Article

## Abstract

Macaulay’s inverse system is an effective method to construct Artinian K-algebras with the additional properties of being, for example, Gorenstein, level, or having any specific socle type. Recently, Elias and Rossi (Adv Math 314:306–327, 2017) gave the structure of the inverse system of d-dimensional Gorenstein K-algebras for any $$d>0$$. In this paper we extend their result by establishing a one-to-one correspondence between d-dimensional level K-algebras and suitable submodules of the divided power ring. We give several examples to illustrate our result.

## Keywords

Macaulay’s inverse system Level K-algebras Divided power ring Gorenstein K-algebras

## Mathematics Subject Classification

13A02 13H10 13J05 13F20

## Notes

### Acknowledgements

The first author was supported by INdAM COFOUND Fellowships cofounded by Marie Curie actions, Italy. We thank our advisor M. E. Rossi for suggesting the problem and providing many useful ideas throughout the preparation of this manuscript. We thank Juan Elias for providing us the updated version of Inverse-syst.lib and clarifying our doubts in Singular. We would also like to thank Alessandro De Stefani for providing us the proof of Proposition 4(b) in the one-dimensional case, and Aldo Conca and Matteo Varbaro for useful discussions on the examples of level rings.

## References

1. 1.
Bertella, V.: Hilbert function of local Artinian level rings in codimension two. J. Algebra 321, 1429–1442 (2009)
2. 2.
Boij, M.: Artin level algebras. Ph.D. thesis, Royal Institute of Technology, Stockholm (1994)Google Scholar
3. 3.
Boij, M.: Betti numbers of compressed level algebras. J. Pure Appl. Algebra 134(2), 111–131 (1999)
4. 4.
Boij, M.: Artin level modules. J. Algebra 226(1), 361–374 (2000)
5. 5.
Boij, M.: Reducible family of height three level algebras. J. Algebra 321(1), 86–104 (2009)
6. 6.
Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised edn. Cambridge University Press, Cambridge (1998)
7. 7.
Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)
8. 8.
Ciliberto, C.: Algebra lineare. Bollati Boringhieri, Turin (1994)Google Scholar
9. 9.
Conca, A.: Divisor class group and canonical class of determinantal rings defined by ideals of minors of a symmetric matrix. Arch. Math. 63(3), 216–224 (1994)
10. 10.
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-1-0: a computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2016)
11. 11.
De Stefani, A.: Artinian level algebras of low socle degree. Commun. Algebra 42, 729–754 (2014)
12. 12.
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
13. 13.
Elias, J., Iarrobino, A.: The Hilbert function of a Cohen–Macaulay local algebra: extremal Gorenstein algebras. J. Algebra 110(2), 344–356 (1987)
14. 14.
Elias, J.: Inverse-syst.lib: singular library for computing Macaulay’s inverse systems. www.ub.edu/C3A/elias/inverse-syst-v.5.2.lib. arXiv:1501.01786 (2015)
15. 15.
Elias, J., Rossi, M.E.: The structure of the inverse system of Gorenstein $$K$$-algebras. Adv. Math. 314, 306–327 (2017)
16. 16.
Emsalem, J.: Géométrie des points épais. Bull. Soc. Math. France 106(4), 399–416 (1978)
17. 17.
Fouli, L.: A study on the core of ideals. Ph.D. thesis, Purdue University (2006)Google Scholar
18. 18.
Fröberg, R.: Connections between a local ring and its associated graded ring. J. Algebra 111(2), 300–305 (1987)
19. 19.
Gabriel, P.: Objets injectifs dans les catégories abéliennes. Séminaire P. Dubriel 1958-1959 17, 1–32 (1959)Google Scholar
20. 20.
Geramita, A.V., Harima, T., Migliore, J.C., Shin, Y.S.: The Hilbert function of a level algebra. Mem. Am. Math. Soc. 186(872), vi+139 (2007)
21. 21.
Goto, S.: On the Gorensteinness of determinantal loci. J. Math. Kyoto Univ. 19(2), 371–374 (1979)
22. 22.
Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ (2016)
23. 23.
Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
24. 24.
Huneke, C., Trung, N.V.: On the core of ideals. Compos. Math. 141(1), 1–18 (2005)
25. 25.
Iarrobino, A.: Compressed algebras: artin algebras having given socle degrees and maximal length. Trans. Am. Math. Soc. 285(1), 337–378 (1984)
26. 26.
Iarrobino, A.: Associated graded algebra of a Gorenstein Artin algebra. Mem. Am. Math. Soc. 107(514), viii+115 (1994)
27. 27.
Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1994). (Revised reprint of the 1916 original; With an introduction by Paul Roberts)Google Scholar
28. 28.
Mantero, P., Xie, Y.: Generalized stretched ideals and Sally’s conjecture. J. Pure Appl. Algebra 220(3), 1157–1177 (2016)
29. 29.
Molinelli, S., Tamone, G.: On the Hilbert function of certain rings of monomial curves. J. Pure Appl. Algebra 101(2), 191–206 (1995)
30. 30.
Polini, C., Ulrich, B.: A formula for the core of an ideal. Math. Ann. 331(3), 487–503 (2005)
31. 31.
Rossi, M.E., Valla, G.: Cohen–Macaulay local rings of embedding dimension $$e+d-3$$. Proc. Lond. Math. Soc. (3) 80(1), 107–126 (2000)
32. 32.
Rossi, M.E., Valla, G.: Hilbert Functions of Filtered Modules. Lecture Notes of the Unione Matematica Italiana, vol. 9. Springer-Verlag, Berlin (2010)Google Scholar
33. 33.
Stanley, R.: Cohen–Macaulay complexes. In: Higher combinatorics (Proceedings of NATO Advanced Study Institute, Berlin, 1976). NATO Advanced Study Institute Series. Series C: Mathematical and Physical Sciences, vol. 31, pp. 51–62. Reidel, Dordrecht (1977)Google Scholar
34. 34.
Stanley, R.: Combinatorics and Commutative Algebra. Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston Inc, Boston (1996)Google Scholar
35. 35.
Xie, Y.: Formulas for the multiplicity of graded algebras. Trans. Am. Math. Soc. 364(8), 4085–4106 (2012)

## Authors and Affiliations

1. 1.Dipartimento di MatematicaUniversità di GenovaGenoaItaly
2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany