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

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)].

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 |

### Citations:

Zbl 1118.14016
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\textit{O. Villamayor U.}, Rev. Mat. Iberoam. 24, No. 1, 213--242 (2008; Zbl 1151.14013)

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