Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Local Cohomology
  • Cited by 18
Publisher:
Cambridge University Press
Online publication date:
December 2012
Print publication year:
2012
Online ISBN:
9781139044059
DOI:
https://doi.org/10.1017/CBO9781139044059
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
Cambridge Studies in Advanced Mathematics (136)
Information

Book description

This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum–Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton–Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.

Reviews

Review of the first edition:‘… Brodmann and Sharp have produced an excellent book: it is clearly, carefully and enthusiastically written; it covers all important aspects and main uses of the subject; and it gives a thorough and well-rounded appreciation of the topic's geometric and algebraic interrelationships … I am sure that this will be a standard text and reference book for years to come.'

Liam O'Carroll Source: Bulletin of the London Mathematical Society

Review of the first edition:‘The book is well organised, very nicely written, and reads very well … a very good overview of local cohomology theory.'

Source: Newsletter of the European Mathematical Society

Review of the first edition:‘… a careful and detailed algebraic introduction to Grothendieck's local cohomology theory.'

Source: L'Enseignement Mathematique

'… the book opens the view towards the beauty of local cohomology, not as an isolated subject but as a tool helpful in commutative algebra and algebraic geometry.'

Source: Zentralblatt MATH

'From the point of view of the reviewer (who learned all his basic knowledge about local cohomology reading the first edition of this book and doing some of its exercises), the changes previously described (the new Chapter 12 concerning canonical modules, the treatment of multigraded local cohomology, and the final new section of Chapter 20 about locally free sheaves) definitely make this second edition an even better graduate textbook than the first. Indeed, it is well written and, overall, almost self-contained, which is very important in a book addressed to graduate students.'

Alberto F. Boix Source: Mathematical Reviews

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
References
[1] Y., Aoyama, ‘On the depth and the projective dimension of the canonical module’, Japanese J. Math. 6 (1980) 61–66.
[2] Y., Aoyama, ‘Some basic results on canonical modules’, J. Math. Kyoto Univ. 23 (1983) 85–94.
[3] H., Bass, ‘On the ubiquity of Gorenstein rings’, Math. Zeit. 82 (1963) 8–28.
[4] M., Brodmann and C., Huneke, ‘A quick proof of the Hartshorne–Lichtenbaum Vanishing Theorem’, Algebraic geometry and its applications (Springer, New York, 1994), pp. 305–308.
[5] M., Brodmann and J., Rung, ‘Local cohomology and the connectedness dimension in algebraic varieties’, Comment. Math. Helvetici 61 (1986) 481–490.
[6] M., Brodmann and R. Y., Sharp, ‘Supporting degrees of multi-graded local cohomology modules’, J. Algebra 321 (2009) 450–482.
[7] W., Bruns and J., Herzog, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Revised Edition (Cambridge University Press, Cambridge, 1998).
[8] F. W., Call and R. Y., Sharp, ‘A short proof of the local Lichtenbaum– Hartshorne Theorem on the vanishing of local cohomology’, Bull. London Math. Soc. 18 (1986) 261–264.
[9] C., D'cruz, V., Kodiyalam and J. K., Verma, ‘Bounds on the a-invariant and reduction numbers of ideals’, J. Algebra 274 (2004) 594–601.
[10] D., Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150 (Springer, New York, 1994).
[11] D., Eisenbud and S., Goto, ‘Linear free resolutions and minimal multiplicity’, J. Algebra 88 (1984) 89–133.
[12] J., Elias, ‘Depth of higher associated graded rings’, J. London Math. Soc. (2) 70 (2004) 41–58.
[13] F., Enriques, Le superficie algebriche (Zanichelli, Bologna, 1949).
[14] G., Faltings, ‘ ‘Über die Annulatoren lokaler Kohomologiegruppen’, Archiv der Math. 30 (1978) 473–476.
[15] G., Faltings, ‘Algebraisation of some formal vector bundles’, Annals of Math. 110 (1979) 501–514.
[16] G., Faltings, ‘Der Endlichkeitssatz in der lokalen Kohomologie’, Math. Annalen 255 (1981) 45–56.
[17] R., Fedder and K., Watanabe, ‘A characterization of F -regularity in terms of F -purity’, Commutative algebra: proceedings of a microprogram held June 15 — July 2, 1987, Mathematical Sciences Research Institute Publications 15 (Springer, New York, 1989), pp. 227–245.
[18] D., Ferrand and M., Raynaud, ‘Fibres formelles d'un anneau local Noethérien’, Ann. Sci. École Norm. Sup. 3 (1970) 295–311.
[19] H.-B., Foxby, ‘Gorenstein modules and related modules’, Math. Scand. 31 (1972) 267–284.
[20] W., Fulton and J., Hansen, ‘A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings’, Annals of Math. 110 (1979) 159–166.
[21] P., Gabriel, ‘Des catégories abéliennes’, Bull. Soc. Math. France 90 (1962) 323–448.
[22] S., Goto and K., Watanabe, ‘On graded rings, II (ℤn-graded rings)’, Tokyo J. Math. 1 (1978) 237–261.
[23] A., Grothendieck, ‘Sur la classification des fibrés holomorphes sur la sphère de Riemann’, American J. Math. 79 (1957) 121–138.
[24] A., Grothendieck, ‘Éléments de géométrie algébrique IV: étude locale des schémas et des morphismes de schémas’, Institut des Hautes Études Scientifiques Publications Mathématiques 24 (1965) 5–231.
[25] A., Grothendieck, Local cohomology, Lecture Notes in Mathematics 41 (Springer, Berlin, 1967).
[26] A., Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Séminaire de Géométrie Algébrique du Bois-Marie 1962 (North-Holland, Amsterdam, 1968).
[27] J., Harris, Algebraic geometry: a first course, Graduate Texts in Mathematics 133 (Springer, New York, 1992).
[28] R., Hartshorne, ‘Complete intersections and connectedness’, American J. Math. 84 (1962) 497–508.
[29] R., Hartshorne, ‘Cohomological dimension of algebraic varieties’, Annals of Math. 88 (1968) 403–450.
[30] R., Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, New York, 1977).
[31] M., Herrmann, E., Hyry and T., Korb, ‘On a-invariant formulas’, J. Algebra 227 (2000) 254–267.
[32] M., Herrmann, S., Ikeda and U., Orbanz, Equimultiplicity and blowing up (Springer, Berlin, 1988).
[33] J., Herzog and E., Kunz, Der kanonische Modul eines Cohen–Macaulay-Rings, Lecture Notes in Mathematics 238 (Springer, Berlin, 1971).
[34] J., Herzog and E., Kunz, Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Sitzungsberichte der Heidelberger Akademie der Wissenschaften Mathematisch-naturwissenschaftliche Klasse, Jahrgang 1971 (Springer, Berlin, 1971).
[35] L. T., Hoa, ‘Reduction numbers and Rees algebras of powers of an ideal’, Proc. American Math. Soc. 119 (1993) 415–422.
[36] L. T., Hoa and C., Miyazaki, ‘Bounds on Castelnuovo–Mumford regularity for generalized Cohen–Macaulay graded rings’, Math. Annalen 301 (1995) 587–598.
[37] M., Hochster, ‘Contracted ideals from integral extensions of regular rings’, Nagoya Math. J. 51 (1973) 25–43.
[38] M., Hochster and C., Huneke, ‘Tight closure, invariant theory and the Briančon–Skoda Theorem’, J. American Math. Soc. 3 (1990) 31–116.
[39] M., Hochster and C., Huneke, ‘Indecomposable canonical modules and connectedness’, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemporary Mathematics 159 (American Mathematical Society, Providence, RI, 1994), pp. 197–208.
[40] G., Horrocks, ‘Vector bundles on the punctured spectrum of a local ring’, Proc. London Math. Soc. (3) 14 (1964) 689–713.
[41] C., Huneke, Tight closure and its applications, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 88 (American Mathematical Society, Providence, RI, 1996).
[42] C., Huneke, ‘Tight closure, parameter ideals and geometry’, Six lectures on commutative algebra (Bellaterra, 1996), Progress in Mathematics 166 (Birkhäauser, Basel, 1998), pp. 187–239.
[43] C. L., Huneke and R. Y., Sharp, ‘Bass numbers of local cohomology modules’, Transactions American Math. Soc. 339 (1993) 765–779.
[44] D., Kirby, ‘Coprimary decomposition of Artinian modules’, J. London Math. Soc. (2) 6 (1973) 571–576.
[45] I. G., Macdonald, ‘Secondary representation of modules over a commutative ring’, Symposia Matematica 11 (Istituto Nazionale di alta Matematica, Roma, 1973) 23–43.
[46] I. G., Macdonald, ‘A note on local cohomology’, J. London Math. Soc. (2) 10 (1975) 263–264.
[47] I. G., Macdonald and R. Y., Sharp, ‘An elementary proof of the non-vanishing of certain local cohomology modules’, Quart. J. Math. Oxford (2) 23 (1972) 197–204.
[48] T., Marley, ‘The reduction number of an ideal and the local cohomology of the associated graded ring’, Proc. American Math. Soc. 117 (1993) 335–341.
[49] E., Matlis, ‘Injective modules over Noetherian rings’, Pacific J. Math. 8 (1958) 511–528.
[50] H., Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8 (Cambridge University Press, Cambridge, 1986).
[51] L., Melkersson, ‘On asymptotic stability for sets of prime ideals connected with the powers of an ideal’, Math. Proc. Cambridge Philos. Soc. 107 (1990) 267–271.
[52] L., Melkersson, ‘Some applications of a criterion for artinianness of a module’, J. Pure and Applied Algebra 101 (1995) 291–303.
[53] E., Miller and B., Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227 (Springer, New York, 2005).
[54] D., Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies 59 (Princeton University Press, Princeton, NJ, 1966).
[55] M. P., Murthy, ‘A note on factorial rings’, Arch. Math. (Basel) 15 (1964) 418–420.
[56] M., Nagata, Local rings (Interscience, New York, 1962).
[57] U., Nagel, ‘On Castelnuovo's regularity and Hilbert functions’, Compositio Math. 76 (1990) 265–275.
[58] U., Nagel and P., Schenzel, ‘Cohomological annihilators and Castelnuovo– Mumford regularity’, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemporary Mathematics 159 (American Mathematical Society, Providence, RI, 1994), pp. 307–328.
[59] D. G., Northcott, Ideal theory, Cambridge Tracts in Mathematics and Mathematical Physics 42 (Cambridge University Press, Cambridge, 1953).
[60] D. G., Northcott, An introduction to homological algebra (Cambridge University Press, Cambridge, 1960).
[61] D. G., Northcott, Lessons on rings, modules and multiplicities (Cambridge University Press, Cambridge, 1968).
[62] D. G., Northcott, ‘Generalized Koszul complexes and Artinian modules’, Quart. J. Math. Oxford (2) 23 (1972) 289–297.
[63] D. G., Northcott and D., Rees, ‘Reductions of ideals in local rings’, Proc. Cambridge Philos. Soc. 50 (1954) 145–158.
[64] L., O'Carroll, ‘On the generalized fractions of Sharp and Zakeri’, J. London Math. Soc. (2) 28 (1983) 417–427.
[65] A., Ooishi, ‘Castelnuovo's regularity of graded rings and modules’, Hiroshima Math. J. 12 (1982) 627–644.
[66] C., Peskine and L., Szpiro, ‘Dimension projective finie et cohomologie locale’, Institut des Hautes Études Scientifiques Publications Mathématiques 42 (1973) 323–395.
[67] L. J., Ratliff Jr., ‘Characterizations of catenary rings’, American J. Math. 93 (1971) 1070–1108.
[68] D., Rees, ‘The grade of an ideal or module’, Proc. Cambridge Philos. Soc. 53 (1957) 28–42.
[69] I., Reiten, ‘The converse to a theorem of Sharp on Gorenstein modules’, Proc. American Math. Soc. 32 (1972) 417–420.
[70] P., Roberts, Homological invariants of modules over commutative rings, Séminaire de Mathématiques Supérieures (Les Presses de l'Université de Montréal, Montréal, 1980).
[71] J. J., Rotman, An introduction to homological algebra (Academic Press, Orlando, FL, 1979).
[72] P., Schenzel, ‘Einige Anwendungen der lokalen Dualitäat und verallgemeinerte Cohen–Macaulay-Moduln’, Math. Nachr. 69 (1975) 227–242.
[73] P., Schenzel, ‘Flatness and ideal-transforms of finite type’, Commutative algebra, Proceedings, Salvador 1988, Lecture Notes in Mathematics 1430 (Springer, Berlin, 1990), pp. 88–97.
[74] P., Schenzel, ‘On the use of local cohomology in algebra and geometry’, Six lectures on commutative algebra (Bellaterra, 1996), Progress in Mathematics 166 (Birkhäauser, Basel, 1998), pp. 241–292.
[75] P., Schenzel, ‘On birational Macaulayfications and Cohen–Macaulay canonical modules’, J. Algebra 275 (2004) 751–770.
[76] P., Schenzel, N. V., Trung and N. T., Cuong, ‘Verallgemeinerte Cohen– Macaulay-Moduln’, Math. Nachr. 85 (1978) 57–73.
[77] J.-P., Serre, ‘Faisceaux algébriques cohérents’, Annals of Math. 61 (1955) 197–278.
[78] F., Severi, Serie, sistemi d'equivalenza e corrispondenze algebriche sulle varietà algebriche (a cura di F. Conforto e di E. Martinelli, Roma, 1942).
[79] R. Y., Sharp, ‘Finitely generated modules of finite injective dimension over certain Cohen–Macaulay rings’, Proc. London Math. Soc. (3) 25 (1972) 303–328.
[80] R. Y., Sharp, ‘On the attached prime ideals of certain Artinian local cohomology modules’, Proc. Edinburgh Math. Soc. (2) 24 (1981) 9–14.
[81] R. Y., Sharp, Steps in commutative algebra: Second edition, London Mathematical Society Student Texts 51 (Cambridge University Press, Cambridge, 2000).
[82] R. Y., Sharp and M., Tousi, ‘A characterization of generalized Hughes complexes’, Math. Proc. Cambridge Philos. Soc. 120 (1996) 71–85.
[83] J. R., Strooker, Homological questions in local algebra, London Mathematical Society Lecture Notes 145 (Cambridge University Press, Cambridge, 1990).
[84] J., Stäckrad and W., Vogel, Buchsbaum rings and applications (Springer, Berlin, 1986).
[85] K., Suominen, ‘Localization of sheaves and Cousin complexes’, Acta Mathematica 131 (1973) 27–41.
[86] N. V., Trung, ‘Reduction exponent and degree bound for the defining equations of graded rings’, Proc. American Math. Soc. 101 (1987) 229–236.
[87] N. V., Trung, ‘The largest non-vanishing degree of graded local cohomology modules’, J. Algebra 215 (1999) 481–499.
[88] O., Zariski, ‘Complete linear systems on normal varieties and a generalization of a lemma of Enriques–Severi’, Annals of Math. 55 (1952) 552–592.
[89] O., Zariski and P., Samuel, Commutative algebra, Vol. II, Graduate Texts in Mathematics 29 (Springer, Berlin, 1975).

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.