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Residue currents constructed from resolutions of monomial ideals

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Abstract

Given a free resolution of an ideal J of holomorphic functions, one can construct a vector-valued residue current R, whose annihilator is precisely J. In this paper we compute R in case J is a monomial ideal and the resolution is a cellular resolution in the sense of Bayer and Sturmfels. A description of R is given in terms of the underlying polyhedral cell complex and it is related to irreducible decompositions of J.

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References

  1. Andersson M.: Residue currents and ideals of holomorphic functions. Bull. Sci. Math. 128(6), 481–512 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson M., Wulcan E.: Residue currents with prescribed annihilator ideals. Ann. Sci. École Norm. Sup. 40(6), 985–1007 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Andersson, M., Wulcan, E.: Decomposition of residue currents. Preprint, arXiv:0710.2016

  4. Andersson, M., Samuelsson, H.: Koppelman formulas and existence theorems for the \({\bar \partial}\) -equation on analytic varieties. Preprint, Göteborg, available at arXiv:0801.0710

  5. Bayer D., Sturmfels B.: Cellular resolutions of monomial modules. J. Reine Angew. Math. 502, 123–140 (1998)

    MATH  MathSciNet  Google Scholar 

  6. Bayer D., Peeva I., Sturmfels B.: Monomial resolutions. Math. Res. Lett. 5(1–2), 31–46 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Berenstein, C.A., Gay, R., Vidras, A., Yger, A.: Residue Currents and Bezout Identities. Progress in Mathematics, vol. 114. Birkhäuser Verlag, Basel (1993)

  8. Berenstein C., Yger A.: Effective Bezout identities in Q[z 1,...,z n ]. Acta Math. 166, 69–120 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Berndtsson B., Passare M.: Integral formulas and an explicit version of the fundamental principle. J. Funct. Anal. 84, 358–372 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Björk, J.-E.: Residues and \({\mathcal D}\) -modules. The Legacy of Niels Henrik Abel, pp. 605–651. Springer, Berlin (2004)

  11. Coleff, N.R., Herrera, M.E.: Les Courants Résiduels Associés à une Forme Méromorphe. Lect. Notes in Math, vol. 633. Berlin, Heidelberg, New York (1978)

  12. Dickenstein A., Sessa C.: Canonical representatives in moderate cohomology. Invent. Math. 80, 417–434 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

  14. Eisenbud, D.: The Geometry of Syzygies. A Second Course in Commutative Algebra and Algebraic Geometry, Graduate Texts in Mathematics, vol. 229. Springer, New York (2005)

  15. Khovanskii A.G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11, 289–295 (1978)

    Article  Google Scholar 

  16. Miller, E.: Alexander duality for monomial ideals and their resolutions. Preprint, arXiv:math.AC/9812095

  17. Miller E., Sturmfels B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

    Google Scholar 

  18. Miller E., Sturmfels B., Yanagawa K.: Generic and cogeneric monomial ideals, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). J. Symbolic Comput. 29(4–5), 691–708 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Passare M.: Residues, currents, and their relation to ideals of holomorphic functions. Math. Scand. 62(1), 75–152 (1988)

    MATH  MathSciNet  Google Scholar 

  20. Passare M., Tsikh A., Yger A.: Residue currents of the Bochner-Martinelli type. Publ. Mat. 44, 85–117 (2000)

    MATH  MathSciNet  Google Scholar 

  21. Sturmfels, B.: Solving Systems of Polynomial Equations. In: CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2002)

  22. Swanson, I.: Primary decompositions. Lecture notes. http://www.reed.edu/~iswanson/primdec.pdf

  23. Taylor, D.: Ideals generated by monomials in an R-sequence. Ph.D. thesis, University of Chicago (1966)

  24. Teissier, B.: Monomial Ideals, Binomial Ideals, Polynomial Ideals. Trends in Commutative Algebra, Math. Sci. Res. Inst. Publ., vol. 51, pp. 211–246. Cambridge University Press, Cambridge (2004)

  25. Varchenko A.N.: Newton polyhedra and estimating of oscillating integrals. Funct. Anal. Appl. 10, 175–196 (1976)

    Article  MATH  Google Scholar 

  26. Wulcan E.: Residue currents of monomial ideals. Indiana Univ. Math. J. 56(1), 365–388 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wulcan, E.: Residue currents and their annihilator ideals. Ph.D. thesis, Chalmers University of Technology (2007)

  28. Ziegler G.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

    Google Scholar 

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  1. Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, 412 96, Göteborg, Sweden

    Elizabeth Wulcan

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  1. Elizabeth Wulcan
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Correspondence to Elizabeth Wulcan.

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Wulcan, E. Residue currents constructed from resolutions of monomial ideals. Math. Z. 262, 235–253 (2009). https://doi.org/10.1007/s00209-008-0371-0

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  • DOI: https://doi.org/10.1007/s00209-008-0371-0

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