Mapping cones of monomial ideals over exterior algebras. (English) Zbl 1539.13039
Let \(K\) be a field and \(E:=K\langle e_1,\ldots,e_n\rangle\) be the exterior algebra of a finite dimensional \(K\)-vector space \(V\) with basis \(\{e_1,\ldots,e_n\}.\) The aim of the paper is to find a minimal graded free resolution of a monomial ideal with linear quotients from \(E\), by using the iterated mapping cones procedure. It is proved that if \(I\) is a monomial ideal with linear quotients, the mapping cone gives a minimal graded free resolution of \(E/I\) and moreover the basis of each free module in the resolution is described. A formula for the graded Betti numbers is thus obtained, as well as some formulas for the graded Poincaré series, the complexity and the depth of \(I.\) Finally, the authors introduce the class of t-spread strongly stable ideals in \(E\) and they compute their graded Betti numbers using the above quoted results.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 15A75 | Exterior algebra, Grassmann algebras |
References:
| [1] | Amata, L., Crupi, M. (2018). ExteriorIdeals: a package for computing monomial ideals in an exterior algebra. JSAG8:71-79. DOI: . · Zbl 1409.13001 |
| [2] | Aramova, A., Avramov, L. L., Herzog, J. (2000). Resolutions of monomial ideals and cohomology over exterior algebras. Trans. Amer. Math. Soc.352(2):579-594. DOI: . · Zbl 0930.13011 |
| [3] | Aramova, A., Herzog, J., Hibi, T. (1997). Gotzmann theorems for exterior algebra and combinatorics. J. Algebra191:174-211. DOI: . · Zbl 0897.13030 |
| [4] | Björner, A., Wachs, M. L. (1996). Shellable nonpure complexes and posets, I. Trans. Amer. Math. Soc.348(4):1299-1327. DOI: . · Zbl 0857.05102 |
| [5] | Crupi, M., Ficarra, A. (2023). A note on minimal resolutions of vector-spread Borel ideals. An. St. Univ. Ovidius Constantia Seria Matematica31(2):71-84. DOI: . |
| [6] | Dochtermann, A., Mohammadi, F. (2014). Cellular resolutions from mapping cones. J. Combin. Theory Ser. A128:180-206. DOI: . · Zbl 1301.05379 |
| [7] | Eisenbud, D. (1995). Commutative Algebra: With A View Toward Algebraic Geometry. Graduate Texts in Mathematics, Vol. 150. New York: Springer. DOI: . · Zbl 0819.13001 |
| [8] | Eliahou, S., Kervaire, M. (1990). Minimal resolutions of some monomial ideals. J. Algebra129:1-25. DOI: . · Zbl 0701.13006 |
| [9] | Ene, V., Herzog, J., Qureshi, A. A. (2019). t-spread strongly stable monomial ideals. Commun. Algebra47(12):5303-5316. DOI: . · Zbl 1426.13004 |
| [10] | Evans, G., Charalambous, H. (1995). Resolutions obtained by iterated mapping cones. J. Algebra176:75-754. DOI: . · Zbl 0840.13005 |
| [11] | Ferraro, L., Hardesty, A. (2023). The Eliahou-Kervaire resolution over a skew polynomial ring. Commun. Algebra. DOI: . |
| [12] | Ficarra, A. (2023). Vector-spread monomial ideals and Eliahou-Kervaire type resolution. J. Algebra615:170-204. DOI: . · Zbl 1505.13015 |
| [13] | Grayson, D. R., Stillman, M. E. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2. |
| [14] | Herzog, J., Hibi, T. (1999). Componentwise linear ideals. Nagoya Math. J. 153:141-153. DOI: . · Zbl 0930.13018 |
| [15] | Herzog, J., Hibi, T. (2011). Monomial Ideals. Graduate Texts in Mathematics, Vol. 260. London, UK: Springer-Verlag. DOI: . · Zbl 1206.13001 |
| [16] | Herzog, J., Takayama, Y., (2002). Resolutions by mapping cones. Homol. Homotopy Appl.4(2):277-294. DOI: . · Zbl 1028.13008 |
| [17] | Kämpf, G. (2010). Module theory over exterior algebra with applications to combinatorics. Dissertation zur Erlangung des Doktorgrades, Fachbereich Mathematik/Informatik, Universität Osnabrück. |
| [18] | Mastroeni, M., McCullough, J., Osborne, A., Rice, J., Willis, C. (2022). Depth and singular varieties of exterior edge ideals. arXiv:2208.03366 |
| [19] | Matsumura, H. (1987). Commutative Ring Theory, (Reid, M., trans.) Cambridge Studies in Advanced Mathematics, Vol. 8. Cambridge, UK: Cambridge University Press. DOI: . · Zbl 0634.46007 |
| [20] | Peeva, I. (2011). Graded Syzygies. Algebra and Applications, Vol. 14. London, UK: Springer-Verlag. DOI: . · Zbl 1213.13002 |
| [21] | Thieu, D. P. (2013). On graded ideals over the exterior algebra with applications to hyperplane arrangements. Dissertation zur Erlangung des Doktorgrades, Fachbereich Mathematik/Informatik, Universität Osnabrück. |
| [22] | VandeBogert, K. (2021). Iterated mapping cones for strongly Koszul algebras. arXiv:2104.00037 |
| [23] | Weibel, C. A. (1994). An introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. DOI: . · Zbl 0797.18001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
