Abstract
Linkage of ideals is a very well-studied topic in algebra. It has lead to the development of module linkage which looks to extend the ideas and results of the former. Although linkage has been used extensively to find many interesting and impactful results, it has only been extended to schemes and modules. This paper builds a framework in which to perform linkage from a categorical perspective. This allows a generalization of many theories of linkage including complete intersection ideal linkage, Gorenstein ideal linkage, linkage of schemes and module linkage. Moreover, this construction brings together many different robust fields of homological algebra including linkage, homological dimensions, and duality. After defining linkage and showing results concerning linkage directly, we explore the connection between linkage, homological dimensions, and duality. Applications of this new framework are sprinkled throughout the paper investigating topics including module linkage, horizontal linkage, module theoretic invariants, and Auslander and Bass classes.
Similar content being viewed by others
References
Auslander, M., Bridger, M.: Stable Module Theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, RI, (1969)
Avramov, L.L., Foxby, H.-B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc. 75(2), 241–270 (1997). https://doi.org/10.1112/S0024611597000348
Borceux, F., Bourn, D.: Mal’cev, Protomodular, Homological and Semi-Abelian Categories, Vol. 566 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2004). https://doi.org/10.1007/978-1-4020-1962-3
Brennan, J. P., York, A.: Linkage and Intermediate C-Gorenstein Dimensions, unpublished
Brennan, J.P., York, A.: An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain. J. Pure Appl. Algebra 223(2), 626–633 (2019). https://doi.org/10.1016/j.jpaa.2018.04.011
Christensen, L. W.: Gorenstein dimensions, Vol. 1747 of Lecture Notes in Mathematics, Springer, Berlin, (2000) https://doi.org/10.1007/BFb0103980
Christensen, L.W., Iyengar, S.: Gorenstein dimension of modules over homomorphisms. J. Pure Appl. Algebra 208(1), 177–188 (2007). https://doi.org/10.1016/j.jpaa.2005.12.005
Cuong, N.T., Nam, T.T.: The \(I\)-adic completion and local homology for Artinian modules. Math. Proc. Camb. Philos. Soc. 131(1), 61–72 (2001). https://doi.org/10.1017/S0305004100004771
Dibaei, M.T., Sadeghi, A.: Linkage of modules and the Serre conditions. J. Pure Appl. Algebra 219(10), 4458–4478 (2015). https://doi.org/10.1016/j.jpaa.2015.02.027
Dibaei, M.T., Sadeghi, A.: Linkage of modules with respect to a semidualizing module. Pacific J. Math. 294(2), 307–328 (2018). https://doi.org/10.2140/pjm.2018.294.307
Enochs, E.E., Jenda, O.M.G.: Gorenstein injective and flat dimensions. Math. Japon. 44(2), 261–268 (1996)
Foxby, H.-B.: Gorenstein modules and related modules. Math. Scand. 31(1972), 267–284 (1973). https://doi.org/10.7146/math.scand.a-11434
García-Rozas, J.R., López, I., Oyonarte, L.: Relative attached primes and coregular sequences. Taiwan. J. Math. 17(3), 1095–1114 (2013). https://doi.org/10.11650/tjm.17.2013.3055
Golod, E. S.: \(G\)-dimension and generalized perfect ideals. Trudy Mat. Inst. Steklov. 165, 62–66, algebraic geometry and its applications. (1984)
Hartshorne, R.: Liaison with Cohen–Macaulay modules. Rend. Semin. Mat. Univ. Politec. Torino 64(4), 419–432 (2006)
Holm, H., Jørgensen, P.: Cotorsion pairs induced by duality pairs. J. Commun. Algebra 1(4), 621–633 (2009). https://doi.org/10.1216/JCA-2009-1-4-621
Holm, H., White, D.: Foxby equivalence over associative rings. J. Math. Kyoto Univ. 47(4), 781–808 (2007). https://doi.org/10.1215/kjm/1250692289
Huneke, C., Ulrich, B.: The structure of linkage. Ann. Math. 126(2), 277–334 (1987). https://doi.org/10.2307/1971402
Huneke, C., Ulrich, B.: Algebraic linkage. Duke Math. J. 56(3), 415–429 (1988). https://doi.org/10.1215/S0012-7094-88-05618-9
Huneke, C., Ulrich, B.: Local properties of licci ideals. Math. Z. 211(1), 129–154 (1992). https://doi.org/10.1007/BF02571423
Kustin, A.R., Miller, M., Ulrich, B.: Linkage theory for algebras with pure resolutions. J. Algebra 102(1), 199–228 (1986). https://doi.org/10.1016/0021-8693(86)90137-7
Martin, H.M.: Linkage and the generic homology of modules. Commun. Algebra 28(9), 4285–4301 (2000). https://doi.org/10.1080/00927870008827090
Martsinkovsky, A., Strooker, J.R.: Linkage of modules. J. Algebra 271(2), 587–626 (2004). https://doi.org/10.1016/j.jalgebra.2003.07.020
Migliore, J.C.: Introduction to Liaison Theory and Deficiency Modules. Progress in Mathematics, vol. 165. Birkhäuser Boston Inc, Boston, MA (1998). https://doi.org/10.1007/978-1-4612-1794-7
Nagel, U.: Liaison classes of modules. J. Algebra 284(1), 236–272 (2005). https://doi.org/10.1016/j.jalgebra.2004.09.022
Nagel, U., Sturgeon, S.: Combinatorial interpretations of some Boij–Söderberg decompositions. J. Algebra 381, 54–72 (2013). https://doi.org/10.1016/j.jalgebra.2013.01.027
Ooishi, A.: Matlis duality and the width of a module, Hiroshima Math. J. 6(3), 573–587 (1976) http://projecteuclid.org/euclid.hmj/1206136213
Peskine, C., Szpiro, L.: Liaison des variétés algébriques. I. Invent. Math. 26, 271–302 (1974). https://doi.org/10.1007/BF01425554
Polini, C., Ulrich, B.: Linkage and reduction numbers. Math. Ann. 310(4), 631–651 (1998). https://doi.org/10.1007/s002080050163
Schenzel, P.: Notes on liaison and duality. J. Math. Kyoto Univ. 22(3), 485–498 (1982/83) https://doi.org/10.1215/kjm/1250521732
Tang, X., Huang, Z.: Homological aspects of the dual Auslander transpose. Forum Math. 27(6), 3717–3743 (2015). https://doi.org/10.1515/forum-2013-0196
Tang, X., Huang, Z.: Homological aspects of the dual Auslander transpose, II. Kyoto J. Math. 57(1), 17–53 (2017). https://doi.org/10.1215/21562261-3759504
Ulrich, B.: On licci ideals, in: Invariant theory (Denton, TX, 1986), Vol. 88 of Contemp. Math. Am. Math. Soc. Providence, RI, 1989, pp. 85–94 https://doi.org/10.1090/conm/088/999984
Yoshino, Y., Isogawa, S.: Linkage of Cohen–Macaulay modules over a Gorenstein ring. J. Pure Appl. Algebra 149(3), 305–318 (2000). https://doi.org/10.1016/S0022-4049(98)00167-4
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Henning Krause.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
York, A. A Categorical Approach to Linkage. Appl Categor Struct 29, 447–483 (2021). https://doi.org/10.1007/s10485-020-09623-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-020-09623-9