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Algebraic linkage and homological algebra

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Abstract

We prove that some of the main results of linkage theory can be extended to a more general context in homological algebra. Our main result states that, under suitable circumstances, if one has a morphism among two objects in an abelian category, both equipped with a good resolution, then there is a canonical procedure to build up a good resolution for the cokernel of the dual morphism.

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References

  1. Di Gennaro, V., Franco, D.: Factoriality and Néron–Severi groups. Commun. Contemp. Math. 10(5), 745–764 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Di Gennaro, V., Franco, D.: Monodromy of a family of hypersurfaces. Ann. Sci. Éc. Norm. Supér., \(4^{e}\) série 42(3), 517–529 (2009)

  3. Di Gennaro, V., Franco, D.: Noether–Lefschetz theory and Néron–Severi group. Int. J. Math. 23(1), 12 (2012). https://doi.org/10.1142/S0129167X11007483

    MATH  Google Scholar 

  4. Di Gennaro, V., Franco, D.: Noether–Lefschetz theory with base locus. Rend. Circ. Mat. Palermo (2) 63(2), 257–276 (2014). https://doi.org/10.1007/s12215-014-0156-8

    MathSciNet  MATH  Google Scholar 

  5. Di Gennaro, V., Franco, D.: Néron–Severi group of a general hypersurface. Commun. Contemp. Math. 19(01), 1650004 (2017). https://doi.org/10.1142/S0219199716500048

    MathSciNet  MATH  Google Scholar 

  6. Di Gennaro, V., Franco, D.: A speciality theorem for curves in \(\mathbb{P}^{5}\) contained in Noether–Lefschetz general fourfolds. Ric. Mat. 66(2), 509–520 (2017). https://doi.org/10.1007/s11587-016-0316-6

    MathSciNet  MATH  Google Scholar 

  7. Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. In: Axler, S., Ribet, K. (eds.) Graduate Texts in Mathematics, vol. 150. Springer-Verlag, Berlin (1995)

    Google Scholar 

  8. Ellia, Ph., Franco, D.: On codimension two subvarieties of in \(\mathbb{P}^{5}\) and \(\mathbb{P}^{6}\). J. Algebr. Geom. 11(3), 513–533 (2002)

  9. Ellia, Ph., Franco, D., Gruson, L.: On subcanonical surfaces of \(\mathbb{P}^{4}\). Math. Z. 251(2), 257–265 (2005). https://doi.org/10.1007/s00209-005-0796-7

  10. Ellia, Ph., Franco, D., Gruson, L., Smooth divisors of projective hypersurfaces: Comment. Math. Helv. 83, 371–385 (2008). https://doi.org/10.4171/CMH/128

  11. Franco, D., Kleiman, S.L., Lascu, A.T.: Gherardelli linkage and complete intersections. Mich. Math. J. 48, 271–279 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Grothendieck, A.: Sur quelques points d’algèbre homologique I. Tohoku Math. J. 9(2), 119–221 (1957)

    MathSciNet  MATH  Google Scholar 

  13. Gruson, L., Peskine, Ch.: Genre des courbes de l’espace projectif. In: Olson, L.D. (ed.) Algebraic Geometry, vol. 687, pp. 31–59. Springer-Verlag, New York (1978)

    Google Scholar 

  14. Hartshorne, R.: Algebraic geometry. In: Axler, S., Ribet, K. (eds.) Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York (1977)

    Google Scholar 

  15. Hilton, P.J., Stammbach, U.: A course in homological algebra. In: Axler, S., Ribet, K. (eds.) Graduate Texts in Mathematics, vol. 4. Springer-Verlag, New York (1997)

    Google Scholar 

  16. Iversen, B.: Cohomology of Sheaves. Springer. xii, 464 p. ISBN 3-540-16389-1; 978-3-642-82783-9/ebook (1986)

  17. Martin-Deschamps, M., Perrin, D.: Sur la classification des courbes gauches, Astérisque No 184-85 (1990)

  18. Peskine, Ch., Szpiro, L.: Liaison des variétés algébriques. Invent. Math. 26(1), 271–302 (1974)

    MathSciNet  MATH  Google Scholar 

  19. Popescu, S., Ranestad, K.: Surfaces of degree 10 in the projective fourspace via linear systems and linkage. J. Algebr. Geom. 5(1), 13–76 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Rao, P.: Liaison among curves in \({\mathbb{P}}^3\). Invent. Math 50(3), 205–217 (1978/79)

  21. Rao. P.: Liaison equivalence classes. Math. Ann. 258(2), 169–173 (1981/82)

  22. Weibel, C.A.: An introduction to homological algebra. In: Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge, UK (1994)

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126, Naples, Italy

    Davide Franco & Luciano Amito Lomonaco

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  1. Davide Franco
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  2. Luciano Amito Lomonaco
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Correspondence to Davide Franco.

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Franco, D., Lomonaco, L.A. Algebraic linkage and homological algebra. Ricerche mat 69, 283–292 (2020). https://doi.org/10.1007/s11587-019-00462-y

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