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Linearity of free resolutions of monomial ideals

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Abstract

We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the “almost linear” case). We also give sharp bounds on Castelnuovo–Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes \(\Delta \) that cannot be extended to the complete \(\mathrm{dim}\Delta \)-skeleton of the simplex on the same variables by shelling.

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References

  1. Bigdeli, M., Herzog, J., Zaare-Nahandi, R.: On the index of powers of edge ideals. Commun. Algebra 46(3), 1080–1095 (2018)

    Article  MathSciNet  Google Scholar 

  2. Boocher, A.: Free resolutions and sparse determinantal ideals. Math. Res. Lett. 19, 805–821 (2012)

    Article  MathSciNet  Google Scholar 

  3. Boocher, A., Grifo, E.: Lower bounds on Betti numbers, preprint, arxiv:2108.05871, (2021)

  4. Constantinescu, A., Kahle, T., Varbaro, M.: Linear syzygies, flag complexes, and regularity. Collect. Math. 67(3), 357–362 (2016)

    Article  MathSciNet  Google Scholar 

  5. Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38(1), 37–55 (2013)

    Article  MathSciNet  Google Scholar 

  6. Dao, H., Takagi, S.: On the relationship between depth and cohomological dimension. Compos. Math. 152(4), 876–888 (2016)

    Article  MathSciNet  Google Scholar 

  7. Dao, H., Vu, T.: Regularity of monomial ideals with linear syzygies, preprint

  8. Dao, H., Doolittle, J., Lyle, J.: Minimal Cohen-Macaulay simplicial complexes. SIAM J. Discrete Math. 34(3), 1602–1608 (2020)

    Article  MathSciNet  Google Scholar 

  9. Francisco, C., Mastroeni, M., Mermin, J., Schweig, J.: Computing Frobenius powers of monomial ideals, preprint

  10. Gasharov, V., Peeva, I., Welker, V.: The lcm lattice in monomial resolutions. Math. Res. Lett. 6(5–6), 521–532 (1999)

    Article  MathSciNet  Google Scholar 

  11. Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Restricting linear syzygies: algebra and geometry. Compositio Math. 141, 1460–1478 (2005)

    Article  MathSciNet  Google Scholar 

  12. Eisenbud, D., Huneke, C., Ulrich, B.: Regularity of Tor and graded Betti numbers. Am. J. Math. 128(3), 573–605 (2006)

    Article  MathSciNet  Google Scholar 

  13. Eisenbud, D., Schreyer, F.-O.: Betti numbers of graded modules and cohomology of vector bundles. J. Am. Math. Soc. 22(3), 859–888 (2009)

    Article  MathSciNet  Google Scholar 

  14. Farrokhi, M., Sadegh, Y., Yazdan Pour, A.: Green-Lazarsfeld index of square-free monomial ideals and their powers, preprint, arxiv:2110.12174, (2021)

  15. Fröberg, R.: On Stanley-Reisner rings, Topics in Algebra, Part 2 (Warsaw, 1988), Banach Center Pub. 26 PWN Warsaw (1990)

  16. Herzog, J., Kühl, M.: On the Betti numbers of finite pure and linear resolutions. Comm. Algebra 12, 1627–1646 (1984)

    Article  MathSciNet  Google Scholar 

  17. Migliore, J., Nagel, U.: A Tour of the Weak and Strong Lefschetz Properties. J. Comm. Alg. 5, 329–358 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Murai, S., Terai, T.: \(h\)-vectors of simplicial complexes with Serre’s \(S_c\) conditions. Math. Res. Lett. 16, 1015–1028 (2009)

    Article  MathSciNet  Google Scholar 

  19. Peeva, I., Velasco, M.: Frames and degenerations of monomial resolutions. Trans. Am. Math. Soc. 363(4), 2029–2046 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author is grateful to the National Science Foundation for partial support. The second author acknowledges support from a Simons Collaborator Grant and from University of Kansas. The authors would like to thank Kangjin Han, Siamak Yassemi and the referee for helpful comments and corrections that improve the readability of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045-7523, USA

    Hailong Dao

  2. Department of Mathematics, University of California at Berkeley and the Mathematical Sciences Research Institute, Berkeley, CA, 94720, USA

    David Eisenbud

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  1. Hailong Dao
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  2. David Eisenbud
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Correspondence to Hailong Dao.

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Dedicated to Juergen Herzog, inspiring mathematician and master of monomials, on the occasion of his 80th birthday!.

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Dao, H., Eisenbud, D. Linearity of free resolutions of monomial ideals. Res Math Sci 9, 35 (2022). https://doi.org/10.1007/s40687-022-00330-6

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