Tangent algebras

Simis, Aron; Ulrich, Bernd; Vasconcelos, Wolmer

doi:10.1090/S0002-9947-2011-05161-5

Tangent algebras
HTML articles powered by AMS MathViewer

by Aron Simis, Bernd Ulrich and Wolmer V. Vasconcelos

Trans. Amer. Math. Soc. 364 (2012), 571-594

DOI: https://doi.org/10.1090/S0002-9947-2011-05161-5

Published electronically: September 14, 2011

PDF | Request permission

Abstract:

One studies the Zariski tangent cone $T_X\stackrel {\pi }{\longrightarrow } X$ to an affine variety $X$ and the closure $\overline {T}_X$ of $\pi ^{-1}(\textrm {Reg}(X))$ in $T_X$. One focuses on the comparison between $T_X$ and $\overline {T}_X$, giving sufficient conditions on $X$ in order that $T_X=\overline {T}_X$. One considers, in particular, the question of when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Another problem considered here is to understand the impact of the Cohen–Macaulayness or normality of $\overline {T}_X$ on the local structure of $X$.

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
Robert Berger, Differentialmoduln eindimensionaler lokaler Ringe, Math. Z. 81 (1963), 326–354 (German). MR 152546, DOI 10.1007/BF01111579
J. P. Brennan, M. V. Pinto, and W. V. Vasconcelos, The Jacobian module of a Lie algebra, Trans. Amer. Math. Soc. 321 (1990), no. 1, 183–196. MR 958883, DOI 10.1090/S0002-9947-1990-0958883-0
Lindsay Burch, On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941–948. MR 229634, DOI 10.1017/s0305004100043620
Alberto Corso, Claudia Polini, and Bernd Ulrich, Core of projective dimension one modules, Manuscripta Math. 111 (2003), no. 4, 427–433. MR 2002819, DOI 10.1007/s00229-002-0329-1
Shiro Goto and Futoshi Hayasaka, Finite homological dimension and primes associated to integrally closed ideals, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3159–3164. MR 1912992, DOI 10.1090/S0002-9939-02-06436-5
Jürgen Herzog, Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul, Math. Z. 163 (1978), no. 2, 149–162 (German). MR 512469, DOI 10.1007/BF01214062
Jürgen Herzog, Deformationen von Cohen-Macaulay Algebren, J. Reine Angew. Math. 318 (1980), 83–105 (German). MR 579384, DOI 10.1515/crll.1980.318.83
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
J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661–683. MR 737891, DOI 10.1090/S0002-9947-1984-0737891-6
Jooyoun Hong, Sunsook Noh, and Wolmer V. Vasconcelos, Integrally closed modules and their divisors, Comm. Algebra 33 (2005), no. 12, 4719–4733. MR 2188337, DOI 10.1080/00927870500334723
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, Aron Simis, and Wolmer Vasconcelos, Reduced normal cones are domains, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 95–101. MR 999985, DOI 10.1090/conm/088/999985
Mark R. Johnson, Depth of symmetric algebras of certain ideals, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1581–1585. MR 1814083, DOI 10.1090/S0002-9939-00-05742-7
Joseph Lipman, On the Jacobian ideal of the module of differentials, Proc. Amer. Math. Soc. 21 (1969), 422–426. MR 237511, DOI 10.1090/S0002-9939-1969-0237511-0
Paul Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J. Algebra 132 (1990), no. 2, 461–473. MR 1061491, DOI 10.1016/0021-8693(90)90141-A
Günter Scheja and Uwe Storch, Über differentielle Abhängigkeit bei Idealen analytischer Algebren, Math. Z. 114 (1970), 101–112 (German). MR 263808, DOI 10.1007/BF01110319
Aron Simis, Remarkable graded algebras in algebraic geometry, Monografías del Instituto de Matemática y Ciencias Afines [Monographs of the Institute of Mathematics and Related Sciences], vol. 7, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. A paper from the 12th Escuela Latinoamericana de Matemáticas (XII-ELAM) held in Lima, June 28–July 3, 1999. MR 2007333
Aron Simis, Two differential themes in characteristic zero, Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD, 2001) Contemp. Math., vol. 324, Amer. Math. Soc., Providence, RI, 2003, pp. 195–204. MR 1986124, DOI 10.1090/conm/324/05741
Aron Simis, Karen E. Smith, and Bernd Ulrich, An algebraic proof of Zak’s inequality for the dimension of the Gauss image, Math. Z. 241 (2002), no. 4, 871–881. MR 1942243, DOI 10.1007/s00209-002-0444-4
Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Tangent star cones, J. Reine Angew. Math. 483 (1997), 23–59. MR 1431841, DOI 10.1515/crll.1997.483.23
Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Rees algebras of modules, Proc. London Math. Soc. (3) 87 (2003), no. 3, 610–646. MR 2005877, DOI 10.1112/S0024611502014144
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
Aron Simis and Wolmer V. Vasconcelos, Krull dimension and integrality of symmetric algebras, Manuscripta Math. 61 (1988), no. 1, 63–78. MR 939141, DOI 10.1007/BF01153583
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
Wolmer V. Vasconcelos, Ideals generated by $R$-sequences, J. Algebra 6 (1967), 309–316. MR 213345, DOI 10.1016/0021-8693(67)90086-5
Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840, DOI 10.1017/CBO9780511574726
Wolmer Vasconcelos, Integral closure, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Rees algebras, multiplicities, algorithms. MR 2153889
W. V. Vasconcelos and R. Villarreal, On Gorenstein ideals of codimension four, Proc. Amer. Math. Soc. 98 (1986), no. 2, 205–210. MR 854019, DOI 10.1090/S0002-9939-1986-0854019-X
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