Abstract
A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.
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References
J. Alperin, L. Evens, Representations, resolutions, and Quillen’s dimension theorem,J. Pure Appl. Algebra 22 (1981), 1–9.
D. Anick, A counterexample to a conjecture of Serre,Ann. of Math. 115 (1982), 1–33.
M. André, Hopf algebras with divided powers,J. Algebra 18 (1971), 19–50.
E. F. Assmus, Jr., On the homology of local rings,Ill. J. Math. 3 (1959), 187–199.
M. Auslander, M. Bridger,Stable module theory, Mem. Amer. Math. Soc. 94 (1969).
L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions,Amer. J. Math. 103 (1981), 1–31.
L. L. Avramov, Local algebra and rational homotopy, Homotopie algébrique et algèbre locale (J.-M. Lemaire, J.-C. Thomas, eds.),Astérisque, vol. 113–114, Soc. Math. France, Paris, 1984, p. 15–43.
L. L. Avramov, Modules of finite virtual projective dimension,Invent. math. 96 (1989), 71–101.
L. L. Avramov, Homological asymptotics of modules over local rings,Commutative algebra (M. Hochster, C. Huneke, J. Sally, eds.),MSRI Publ., vol. 15, Springer, New York, 1989, p. 33–62.
L. L. Avramov, Local rings over which all modules have rational Poincaré series,J. Pure Appl. Algebra 91 (1994), 29–48.
L. L. Avramov, A. R. Kustin, M. Miller, Poincaré series of modules over local rings of small embedding codepth or small linking number,J. Algebra 118 (1988), 162–204.
L. L. Avramov, L.-C. Sun, Cohomology operators defined by a deformation,J. Algebra, to appear.
D. J. Benson, J. F. Carlson, Projective resolutions and Poincaré duality complexes,Trans. Amer. Math. Soc. 342 (1994), 447–488.
N. Bourbaki,Algèbre. III, Nouvelle édition, Paris, Hermann, 1970.
N. Bourbaki,Algèbre commutative. IX, Paris, Masson, 1983.
R.-O. Buchweitz, G.-M. Greuel, F. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II,Invent. math. 88 (1987), 165–182.
J. A. Eagon, M. Hochster, R-sequences and indeterminates,Quart. J. Math. Oxford Ser. (2)25 (1974), 61–71.
D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations,Trans. Amer. Math. Soc. 260 (1980), 35–64.
D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity,J. Algebra 88 (1984), 89–133.
Y. Félix, S. Halperin, C. Jacobsson, C. Löfwall, J.-C. Thomas, The radical of the homotopy Lie algebra,Amer. J. Math.,110 (1988), 301–322.
V. N. Gasharov, I. V. Peeva, Boundedness versus periodicity over commutative local rings,Trans. Amer. Math. Soc. 320 (1990), 569–580.
A. Grothendieck, Éléments de géométrie algébrique. IV2,Publ. Math. IHES 24 (1965).
T. H. Gulliksen, A change of rings theorem, with applications to Poincaré series and intersection multiplicity,Math. Scand. 34 (1974), 167–183.
T. H. Gulliksen, On the deviations of a local ring,Math. Scand. 47 (1980), 5–20.
J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections,J. Pure Appl. Algebra 71 (1991), 187–202.
A. R. Kustin, S. M. Palmer, The Poincaré series of every finitely generated module over a codimension 4 almost complete intersection is a rational function,J. Pure Appl. Algebra 95 (1994), 271–295.
S. MacLane,Homology, Grundlehren Math. Wiss., vol. 114, Springer, Berlin, 1963.
Yu. I. Manin, Some remarks on Koszul algebras and quantum groups,Ann. Inst. Fourier (Grenoble)37 (1987), 191–205.
H. Matsumura,Commutative ring theory, Stud. Adv. Math., vol. 8, Cambridge, Univ. Press, 1986.
V. B. Mehta,Endomorphisms of complexes and modules over Golod rings, Ph. D. Thesis, Univ. of California, Berkeley, 1976.
J. W. Milnor, J. C. Moore, On the structure of Hopf algebras,Ann. of Math. (2)81 (1965), 211–264.
M. Nagata,Local rings, New York, Wiley, 1962.
J. Shamash, The Poincaré series of a local rings,J. Algebra 12 (1969), 453–470.
G. Sjödin, Hopf algebras and derivations,J. Algebra 64 (1980), 218–229.
L.-C. Sun, Growth of Betti numbers over local rings of small embedding codepth or small linking number,J. Pure Appl. Algebra 96 (1994), 57–71.
J. Tate, Homology of Noetherian rings and of local rings,Ill. J. Math. 1 (1957), 14–27.
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The first author was partly supported by NSF Grant No. DMS-9102951.
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Avramov, L.L., Gasharov, V.N. & Peeva, I.V. Complete intersection dimension. Publications Mathématiques de L’Institut des Hautes Scientifiques 86, 67–114 (1997). https://doi.org/10.1007/BF02698901
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DOI: https://doi.org/10.1007/BF02698901