Rees algebras on smooth schemes: Integral closure and higher differential operator. (English) Zbl 1151.14013

Villamayor U., Orlando

Rev. Mat. Iberoam. 24, No. 1, 213-242 (2008).

A Rees algebra over an algebraic variety \(W\) (over a field \(k\), of any characteristic) is defined by a sequence of coherent sheaves of ideals \(I_j \subset {\mathcal O}_{W}\) such that there is an affine cover \( \{ U_i \} \) of \(W\) such that, for all \(i\), \(\bigoplus I_n(U_i) \, W^i\) ( a subring of the polynomial ring \({\mathcal O}_{Z}(U_i) [W]\)) is a finitely generated \({\mathcal O}_Z(U_i)\)-algebra. We shall assume \(I_{j+1}\subset I_j\) for all \(j\), which is harmless if we work modulo integral closure. If \(W\) is smooth over \(k\) and the Rees algebra \(\mathcal G\) as above also satisfies the condition: “for all \(i\), if \(h \in {\mathcal O}_{Z}(U_i)\) and \(D\) is any differential operator over \(k\), defined on \(U_i\), of order \(\leq r \leq n\) then \(D(h) \in I_{n-r}\)”, we say that \(\mathcal G\) is a differential algebra, relative to \(k\). (The reference to \(k\) will be omitted in the sequel, if it is clear). It is hoped that differential algebras will play a relevant role in the study of resolution of singularities in arbitrary characteristic, namely that they will provide a useful substitute in certain constructions carried out in characteristic zero with the aid of derivatives.
The present paper discusses important foundational results on the theory of Rees and differential algebras. Some of the topics that are covered include: (1) the basic, purely algebraic theory, (2) the basic concepts in the context of Algebraic Geometry, (3) the notion of the differential algebra \(G(\mathcal G)\) generated by a Rees algebra \(\mathcal G\) over a smooth \(k\)-variety, (4) the restriction of a differential algebra \(\mathcal G\) to a smooth subvariety, (5), the singular set of a Rees (or differential) algebra, and related results, (6) integral closures of these algebras. Another concept (probably very important in inductive steps in resolution of singularities) is that of coefficients. If \(Z\) is a smooth subvariety of the \(k\)-smooth variety \(V\), \(\mathcal G\) is a Rees algebra over \(V\), \(x\) is a closed point of \(Z\), in the presence of local retraction \(\pi:V \to Z\) (defined near \(x\)) one may define a Rees algebra over \(Z\), the algebra of coefficients Coeff(\(\mathcal G\)). At the level of completions of local rings this algebra can be described in terms of coefficients in certain power series expansions. The assumption on the existence of the local retraction \(\pi\) is not restrictive if one works with the etale topology. A number of properties of this concept are shown, specially the fact that, if \(\mathcal G\) is a differential algebra, then \(G({\text{Coeff}}(\mathcal G))\) is independent of the local retraction used.
The author has used these results in other works, e.g., in O. Villamayor [Adv. in Math. 213, 687–733 (2007; Zbl 1118.14016)].

Reviewer: Augusto Nobile (Baton Rouge)

MSC:

14E15	Global theory and resolution of singularities (algebro-geometric aspects)
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
32S45	Modifications; resolution of singularities (complex-analytic aspects)
13N10	Commutative rings of differential operators and their modules

Keywords:

Rees algebra; differential algebra; integral closure; differential operator; coefficients

Citations:

Zbl 1118.14016

References:

[1]	Encinas, S. and Villamayor, O.: Rees algebras and resolution of singularities. In Actas del “XVI Coloquio Latinoamericano de Álgebra” (Colonia del Sacramento, Uruguay, 2005) , 1-24. Biblioteca de Rev. Mat. Iberoamericana. Rev. Mat. Iberoamericana, Madrid, 2007.
[2]	Fernández-Lebrón, M. and Narváez-Macarro, L.: Hesse-Schmidt derivations and coefficient fields in positive characteristics. J. Algebra 265 (2003), no. 1, 200-210. · Zbl 1099.13518
[3]	Giraud, J.: Sur la théorie du contact maximal. Math. Z. 137 (1974), 285-310. · Zbl 0275.32003
[4]	Giraud, J.: Contact maximal en caractéristique positive. Ann. Scien. École Norm. Sup. (4) 8 (1975), 201-234. · Zbl 0306.14004
[5]	Grothendieck, A. and Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV . Inst. Hautes Études Sci. Publ. Math. 32 . Press Univ. de France, Paris, 1967.
[6]	Hironaka, H.: Idealistic exponents of a singularity. In Algebraic geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, 1976) , 52-125. John Hopkins University Press, Baltimore, Md., 1977. · Zbl 0496.14011
[7]	Hironaka, H.: Theory of infinitely near singular points. J. Korean Math. Soc. 40 (2003), no. 5, 901-920. · Zbl 1055.14013
[8]	Hironaka, H.: Three key theorems on infinitely near singularities. In Singularités Franco-Japonaises , 87-126. Sémin. Congr. 10 . Soc. Math. France, Paris, 2005. · Zbl 1093.14021
[9]	Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 819-909. · Zbl 1170.14012
[10]	Kollár, J.: Lectures on resolution of singularities . Annals of Mathematics Studies 166 . Princeton University Press. Princeton, NJ, 2007. · Zbl 1113.14013
[11]	Lejeune Jalabert, M. and Teissier, B.: Cloture intégrale des idéaux et équisingularité. Michigan Math. J. 28 (1981), 97-116.
[12]	Lipman, J. and Teissier, B.: Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals. Michigan Math. J. 28 (1981), 97-116. · Zbl 0464.13005
[13]	Matsumura, H.: Commutative Algebra . Second edition. Mathematics Lecture Notes Series. Benjamin, Cumming Publishing Company, 1980. · Zbl 0441.13001
[14]	Mount, K. R. and Villamayor Sr., O.: Taylor Series and higher derivations . Impresiones del Departamento de Matematicas 18 . Universidad de Buenos Aires, 1968 (reimpresión 1979).
[15]	Narváez-Macarro, L: A note on the behaviour under a ground field extension of quasicoefficient fields. J. London Math. Soc. (2) 43 (1991), 12-22. · Zbl 0687.14015
[16]	Traves, W.: Localization of the Hasse-Schmidt algebra. Canad. Math. Bull. 46 (2003), no. 2, 304-309. · Zbl 1096.13528
[17]	Vasconcelos, W. V.: Arithmetic of blowup algebras . London Mathematical Society. Lecture Note Series 195 . Cambridge University Press, Cambridge, 1994. · Zbl 0813.13008
[18]	Villamayor, O.: Differential operators on smooth schemes and embbeded singularities. Rev. Un. Mat. Argentina 46 (2005), no. 2, 1-18. · Zbl 1112.14015
[19]	Villamayor, O.: Hypersurface singularities in positive characteristic. Adv. in Math. 213 (2007), no. 2, 687-733. · Zbl 1118.14016
[20]	Włodarczyk, J.: Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc. 18 (2005), 779-822. · Zbl 1084.14018
[21]	Yakutieli, A.: An explicit construction of the Grothendieck residual complex . Astérisque 208 . Soc. Math. France, 1992.
[22]	Youssin, B.: Newton Polyhedra without coordinates. Mem. Amer. Math. Soc. 87 (1990), no. 433, i-vi, 1-74. · Zbl 0709.14028
[23]	Zariski, O. and Samuel, P.: Commutative Algebra. Vol II . The University Series in Higher Mathematics. D. Van Nostrand Co., Princeton, N. J.-Toronto-London-New York, 1960. · Zbl 0121.27801