Abstract
We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property.
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Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren Math. Wiss., vol. 267. Springer, Berlin (1985)
Avramov, L.L.: Local rings over which all modules have rational Poincaré series. J. Pure Appl. Algebra 91, 29–48 (1994)
Avramov, L.L.: Infinite free resolutions. In: Six lectures on commutative algebra (Bellaterra, 1996), 1–118, Progr. Math., vol. 166. Birkhäuser (1998)
Avramov, L.L., Iyengar, S.B., Şega, L.M.: Free resolutions over short local rings. J. London Math. Soc. 78, 459–476 (2008)
Avramov, L.L., Eisenbud, D.: Regularity of modules over a Koszul algebra. J. Algebra 153, 85–90 (1992)
Avramov, L.L., Peeva, I.: Finite regularity and Koszul algebras. Am. J. Math. 123, 275–281 (2001)
Avramov, L.L., Foxby, H.-B., Herzog, B.: Structure of local homomorphisms. J. Algebra 164, 124–145 (1994)
Backelin, J., Fröberg, R.: Koszul algebras, Veronese subrings, and rings with linear resolutions. Rev. Roumaine Math. Pures Appl. 30, 85–97 (1985)
Bruns, W., Conca, A.: Gröbner bases and determinantal ideals. Commutative algebra, singularities and computer algebra (Sinaia, 2002), 9–66, NATO Sci. Ser. II Math. Phys, vol. 115. Kluwer Acad. Publ., Dordrecht (2003)
Bruns, W., Herzog, J.: Cohen-Macaulay rings. Revised edition, Cambridge Studies Adv. Math, vol. 39. University Press, Cambridge (1998)
Conca, A.: Gröbner bases for spaces of quadrics of low codimension. Adv. App. Math. 24, 111–124 (2000)
Conca, A.: Universally Koszul algebras. Math. Ann. 317, 329–346 (2000)
Conca, A.: Universally Koszul algebras defined by monomials. Rend. Sem. Mat. Univ. Padova 107, 95–99 (2002)
Conca, A., De Negri, E., Rossi, M.E.: Koszul algebra and regularity. In: Peeva, I. (ed.) Commutative Algebra: Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday, pp 285–315. Springer, New York (2013)
Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compos. Math. 129(1), 95–121 (2001)
Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective space. Math. Scand. 89, 201–216 (2001)
Dress, A., Krämer, H.: Bettireihen von Faserprodukten lokaler Ringe. Math. Ann. 215, 79–82 (1975)
Eisenbud, D.: Green’s conjecture: an orientation for algebraists. In: Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, UT, 1990), vol. 2, pp. 51–78, Res. Notes Math. Jones and Bartlett, Boston (1992)
Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf cohomology and resolutions over the exterior algebra. Trans. Am. Math. Soc. 355, 4397–4426 (2003)
Fröberg, R.: On Stanley-Reisner rings. In: Topics in Algebra, vol. 26 Part 2, pp. 57–70. Banach Center Publications (1990)
Henriques, I.B., Şega, L.M.: Free resolutions over short Gorenstein local rings. Math. Z. 267, 645–663 (2011)
Herzog, J.: Algebra retracts and Poincaré series. Manuscr. Math. 21, 307–314 (1977)
Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya. Math. J. 153, 141–153 (1999)
Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161–178 (2000)
Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)
Herzog, J., Iyengar, S.: Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)
Kempf, G.: Syzygies for points in projective space. J. Algebra 145, 219–223 (1992)
Kustin, A., Palmer Slattery, S.: The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function. J. Pure Appl. Algebra 95, 271–295 (1994)
Iyengar, S.B., Römer, T.: Linearity defects of modules over commutative rings. J. Algebra 322, 3212–3237 (2009)
Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)
Roos, J.-E.: Good and bad Koszul algebras and their Hochschild homology. J. Pure Appl. Algebra 201(1–3), 295–327 (2005)
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Dedicated to Ngo Viet Trung on the occasion of his sixtieth birthday
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Conca, A., Iyengar, S.B., Nguyen, H.D. et al. Absolutely Koszul Algebras and the Backelin-Roos Property. Acta Math Vietnam 40, 353–374 (2015). https://doi.org/10.1007/s40306-015-0125-0
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DOI: https://doi.org/10.1007/s40306-015-0125-0