Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals


  • Lars Winther Christensen
  • Oana Veliche
  • Jerzy Weyman



Recent work on generic free resolutions of length $3$ attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade $3$ perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.


Avramov, L. L., Kustin, A. R., and Miller, M., Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra 118 (1988), no. 1, 162–204.

Boij, M. and Laksov, D., Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1083–1092.

Buchsbaum, D. A. and Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485.

Christensen, L. W., Veliche, O., and Weyman, J., Trimming a Gorenstein ideal, J. Commut. Algebra 11 (2019), no. 3, 1–15.

Eagon, J. A. and Northcott, D. G., Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. Ser. A 269 (1962), 188–204.

Fröberg, R. and Laksov, D., Compressed algebras, in “Complete intersections (Acireale, 1983)”, Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 121–151.

Golod, E. S., A note on perfect ideals, in “Algebra” (Kostrikin, A. I., ed.), Moscow State University Press, 1980, pp. 37–39.

Huneke, C. and Ulrich, B., The structure of linkage, Ann. of Math. (2) 126 (1987), no. 2, 277–334.

Józefiak, T., Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 (1978), no. 4, 595–607.

Kaplansky, I., Commutative rings, revised ed., The University of Chicago Press, Chicago, Ill.-London, 1974.

Kunz, E., Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111–115.

Miller, M. and Ulrich, B., Linkage and compressed algebras, in “Proceedings of the conference on algebraic geometry (Berlin, 1985)”, Teubner-Texte Math., vol. 92, Teubner, Leipzig, 1986, pp. 267–275.

Watanabe, J., A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232.

Weyman, J., Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003.

Weyman, J., Generic free resolutions and root systems, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 3, 1241–1296.



How to Cite

Christensen, L. W., Veliche, O., & Weyman, J. (2019). Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals. MATHEMATICA SCANDINAVICA, 125(2), 163–178.