On the arithmetic and homology of algebras of linear type

Herzog, J.; Simis, A.; Vasconcelos, W. V.

doi:10.1090/S0002-9947-1984-0737891-6

On the arithmetic and homology of algebras of linear type
HTML articles powered by AMS MathViewer

by J. Herzog, A. Simis and W. V. Vasconcelos PDF

Trans. Amer. Math. Soc. 283 (1984), 661-683 Request permission

Abstract:

Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.

References

Luchezar L. Avramov, Complete intersections and symmetric algebras, J. Algebra 73 (1981), no. 1, 248–263. MR 641643, DOI 10.1016/0021-8693(81)90357-4
Lâcezar Avramov and Jürgen Herzog, The Koszul algebra of a codimension $2$ embedding, Math. Z. 175 (1980), no. 3, 249–260. MR 602637, DOI 10.1007/BF01163026
M. Brodmann, Asymptotic stability of $\textrm {Ass}(M/I^{n}M)$, Proc. Amer. Math. Soc. 74 (1979), no. 1, 16–18. MR 521865, DOI 10.1090/S0002-9939-1979-0521865-8
Winfried Bruns, Zur Konstruktion basischer Elemente, Math. Z. 172 (1980), no. 1, 63–75 (German). MR 576297, DOI 10.1007/BF01182780
Edward D. Davis, Ideals of the principal class, $R$-sequences and a certain monoidal transformation, Pacific J. Math. 20 (1967), 197–205. MR 206035
J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
John Fogarty, Truncated Hilbert functors, J. Reine Angew. Math. 234 (1969), 65–88. MR 244268, DOI 10.1515/crll.1969.234.65
Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254
Robin Hartshorne and Arthur Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. MR 347821, DOI 10.1080/00927877408548627
Jürgen Herzog and Ernst Kunz (eds.), Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, Vol. 238, Springer-Verlag, Berlin-New York, 1971. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971. MR 0412177
J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings, J. Algebra 74 (1982), no. 2, 466–493. MR 647249, DOI 10.1016/0021-8693(82)90034-5
J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings. II, J. Algebra 82 (1983), no. 1, 53–83. MR 701036, DOI 10.1016/0021-8693(83)90173-4
J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
Craig Huneke, On the symmetric algebra of a module, J. Algebra 69 (1981), no. 1, 113–119. MR 613861, DOI 10.1016/0021-8693(81)90131-9
Craig Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043–1062. MR 675309, DOI 10.2307/2374083
Craig Huneke, On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268–275. MR 563225, DOI 10.1016/0021-8693(80)90179-9

Commutative rings

Michael Kühl, On the symmetric algebra of an ideal, Manuscripta Math. 37 (1982), no. 1, 49–60. MR 649563, DOI 10.1007/BF01239944
Karsten Lebelt, Freie Auflösungen äusserer Potenzen, Manuscripta Math. 21 (1977), no. 4, 341–355 (German, with English summary). MR 450253, DOI 10.1007/BF01167853
G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315–323. MR 230715
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
Erich Platte, Zur endlichen homologischen Dimension von Differentialmoduln, Manuscripta Math. 32 (1980), no. 3-4, 295–302 (German, with English summary). MR 595423, DOI 10.1007/BF01299606
D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145–149. MR 95843
Maria Evelina Rossi, A note on the symmetric algebras which are Gorenstein rings, Comm. Algebra 11 (1983), no. 22, 2575–2591. MR 733343, DOI 10.1080/00927878308822981
Pierre Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237–249 (French). MR 186702
A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), no. 2, 203–224. MR 610474, DOI 10.2307/2374214
Aron Simis and Wolmer V. Vasconcelos, On the dimension and integrality of symmetric algebras, Math. Z. 177 (1981), no. 3, 341–358. MR 618200, DOI 10.1007/BF01162067
John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 419359, DOI 10.1007/BF01389745
Giuseppe Valla, On the symmetric and Rees algebras of an ideal, Manuscripta Math. 30 (1980), no. 3, 239–255. MR 557107, DOI 10.1007/BF01303330
Udo Vetter, Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln, Arch. Math. (Basel) 24 (1973), 158–161 (German). MR 344518, DOI 10.1007/BF01228192
Jerzy Weyman, Resolutions of the exterior and symmetric powers of a module, J. Algebra 58 (1979), no. 2, 333–341. MR 540642, DOI 10.1016/0021-8693(79)90164-9