Abstract
Local cohomology is a useful tool in several branches of commutative algebra and algebraic geometry. The main aim of this series of lectures is to illustrate a few of these techniques. The material presented in the sequel needs some basic knowledge about commutative resp. homological algebra. The basic chapters of the textbooks [9], [28], and [48] are a recommended reading for the preparation. The author’s intention was to present applications of local cohomology in addition to the examples in these textbooks as well as those of [7].
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References
M. Auslander: Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–645.
M. Auslander, D. A. BuchsbauM: Codimension and multiplicity, Ann. of Math. 68 (1958), 625–657.
L. L. Avramov: Infinite free resolutions,this volume.
M. Brodmann: Asymptotic stability of Ass(M/IThM), Proc. Amer. Math. Soc. 74 (1979), 16–18.
M. Brodmann, J. Rung: Local cohomology and the connectedness dimension in algebraic varieties, Comment. Math. Rely. 61 (1986), 481–490.
M. Brodmann, R. Y. Sharp: ‘Local Cohomology — An algebraic introduction with geometric applications’, Cambr. Univ. Press, to appear.
W. Bruns, J. Herzog: ‘Cohen-Macaulay rings’, Cambr. Univ. Press, 1993.
L. Burch: Codimension and analytic spread, Proc. Cambridge Phil. Soc. 72 (1972), 369–373.
D. Eisenbud: ‘Commutative Algebra (with a view towards algebraic geometry)’, Springer-Verlag, 1995.
D. Eisenbud, S. GôTO: Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.
O. Gabber: Non negativity of Serre’s intersection multiplicities, Preprint, I.H.E.S., 1995.
M. Green: Koszul homology and the geometry of projective varieties, J. Diff. Geometry 19 (1984), 125–171.
A. Grothendieck: Elements de geometrie algebrique, M k, Publ. Math., I.H.E.S. 11 (1961).
A. Grothendieck: ‘Local cohomology’, notes by R. Hartshorne, Lect. Notes in Math., 41, Springer, 1967.
R. Hartshorne: Complete intersections and connectedness, Amer. J. Math. 84 (1962), 497–508.
R. Hartshorne: Cohomological dimension of algebraic varieties, Ann. of Math. 88 (1968), 403–450.
R. Hartshorne, A. Ogus: On the factoriality of local rings of small embedding codimension, Comm in Algebra 1 (1974), 415–437.
R. Heitmann: A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. 29 (1993), 94–97.
M. Hochster: ‘Topics in the homological theory of modules over commutative rings’, Cbms Regional Conference Series 24, Amer. Math. Soc., 1975.
M. Hochster, C. Huneke: Indecomposable canonical modules and connectedness,Proc. Conf. Commutative Algebra (Eds.: W. Heinzer, C. Huneke, J. Sally), Contemporary Math. 159 (1994), 197–208.
C. Huneke: On the associated graded ring of an ideal, Illinois J. Math. 26 (1982), 121–137.
C. Huneke, G. Lyubeznik: On the vanishing of local cohomology modules, Invent. math. 102 (1990), 73–93.
C. Huneke, R. Wiegand: Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), 449–476.
P. Jorgensen: Non-commutative Castelnuovo-Mumford regularity, Preprint no. 8, Kopenhagen Univ., 1996.
T. Kawasaki: On the Macaulayfication of quasi-projective schemes, Preprint, Tokyo Metrop. Univ., 1996.
S. Lichtenbaum: On the vanishing of Tor in regular local rings, Ill. J. Math. 10 (1966), 220–226.
J. Lipman: Equimultiplicity, reduction, and blowing up, Commutative Algebra: Analytical Methods, Lect. Notes Pure Appl. Math. 68 (1982), 111–147.
H. Matsumura: ‘Commutative ring theory’, Cambridge University Press, 1986.
S. Mcadam: Asymptotic prime divisors and analytic spreads, Proc. Amer. Math. Soc. 80 (1980), 555–559.
S. Mcadam, P. Eakin: The asymptotic Ass, J. Algebra 61 (1979), 71–81.
D. Mumford: ‘Lectures on Curves on an Algebraic Surface’, Ann. of Math. Studies No. 59, Princeton University Press, 1966.
M. P. Murthy: A note on factorial rings, Archiv der Math. 15 (1964), 418–420.
M. Nagata: On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51–64.
M. Nagata: A treatise on the 14th problem of Hilbert, Mem. Coll. Sci. Kyoto Univ. 30 (1956), 57–82.
M. Nagata: ‘Local rings’, Interscience, 1962.
D. G. Northcott, D. Rees: Reductions of ideals in local rings, Proc. Cambr. Phil. Soc. 50 (1954), 145–158.
D. Rees: On a problem of Zariski, Illinois J. Math. 2 (1958), 145–149.
P. Roberts: Le théorème d’intersection, C. R. Acad. Sc. Paris, Sér. I, 304 (1987), 177–180.
C. Peskine, L. Szpiro: Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. 42 (1973), 77–119.
L. J. Ratliff, JR.: On prime divisors of I“E, n large, Michigan Math. J. 23 (1976), 337–352.
P. Schenzel: ‘Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe’, Lect. Notes in Math., 907, Springer, 1982.
P. Schenzel: Finiteness of relative Rees rings and asymptotic prime divisors, Math. Nachr. 129 (1986), 123–148.
P. Schenzel: Filtrations and Noetherian symbolic blow-up rings, Proc. Amer. Math. Soc. 102 (1988), 817–822.
P. Schenzel: Applications of Koszul homology to numbers of generators and syzygies, J. Pure Appl. Algebra 114 (1997), 287–303.
J. P. Serre: ‘Algèbre locale. Multiplicités’, Lect. Notes in Math., 11, Springer, 1965.
R. Y. Sharp: Some results on the vanishing of local cohomology modules, Proc. London Math. Soc. (3) 30 (1975), 177–195.
I. Swanson: Linear equivalence of ideal topologies, Preprint, New Mexico State University, 1997.
C. Weibel: ‘An Introduction to Homological Algebra’, Cambr. Univ. Press, 1994.
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Schenzel, P. (1998). On the Use of Local Cohomology in Algebra and Geometry. In: Elias, J., Giral, J.M., Miró-Roig, R.M., Zarzuela, S. (eds) Six Lectures on Commutative Algebra. Progress in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0329-4_4
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