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Arithmetic of blowup algebras. (English) Zbl 0813.13008

London Mathematical Society Lecture Note Series. 195. Cambridge: Cambridge University Press. 329 p. (1994).
Let \(M\) denote a finitely generated module over a commutative Noetherian ring \(A\). In the early times of commutative algebra [see the work of D. Hilbert and F. S. Macaulay] the notion of the first module of syzygies \(S_ 1(M)\) of \(M\) is an important construction. Moreover, the chain of syzygies of \(M\) defined by \(S_{i+1} (M) : = S_ 1 (S_ i(M))\), \(i \geq 0\), with \(S_ 0 = M\), leads to the basics of homological algebra, in particular, to those notions like Cohen-Macaulay module or Gorenstein ring.
During the last three decades the notion of symmetric algebra, first studied by A. Micali [Ann. Inst. Fourier 14, No. 2, 33-88 (1964; Zbl 0152.026)] in a systematic way, became a powerful tool in commutative algebra and algebraic geometry. It is defined as the polynomial ring \(A[T]\) over the generators of \(M\) modulo the linear relations coming from syzygies of \(M\). In the case of \(M = I\) an ideal of \(A\) the symmetric algebra \(S(I)\) is just the first approximation of the Rees ring \(R(I)\) which is defined as the polynomial ring modulo all the relations, i.e., also those of higher degrees in the generators. The Rees algebra \(R(I)\), introduced in the early work of D. Rees, has its significance in the fact that it is the graded ring whose projective spectrum gives the blowing up of \(\text{Spec} A\) along \(V(I)\). It has the effect that the ideal \(I\) is locally principal in \(\text{Proj} R(I)\). Two other algebras related to the symmetric algebra are the symbolic blowup of a prime ideal and the factorial closure \(B(M)\) of \(S(M)\). Both of these algebras are not always Noetherian.
The main theme of the book is to describe local, homological, global properties of these algebras, their ideal theory and the relations among them. More precisely, the author starts with the description of the Krull dimension of \(S(M)\) which depends on the Fitting ideals of \(M\). The notion of a \(d\)-sequence introduced by C. Huneke [Adv. Math. 46, 249-278 (1982; Zbl 0505.13004)] is an important generalization of the notion of a regular sequence. For an ideal generated by a \(d\)-sequence it follows that \(S(I)\) and \(R(I)\) coincide. The construction of the approximation complexes, a family of chain complexes derived from Koszul complexes, is made by lack of methods to find the chain of syzygies of the symmetric powers of the module \(M\). The acyclicity of these approximation complexes is related to properties of \(d\)-sequences and further generalizations of regular sequences. – In a separate chapter the arithmetic of Rees algebras, i.e., the normality, Serre’s condition \((S_ k)\), and the Cohen-Macaulayness (resp. Gorensteinness) of \(R(I)\) is considered. Several recent results about Hilbert functions of primary ideals are included and new classes of Cohen-Macaulay rings are constructed. Another part of the book is devoted to the relation of Koszul homology and linkage. It describes invariants of the linkage class of complete intersections. In greater detail there is an investigation of the Koszul homology of ideals associated to graphs. Moreover the author provides a smooth entry to the more advanced work about linkage [see C. Huneke and B. Ulrich, Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)]. – Furthermore the canonical module, resp. the canonical class in the divisor class group is described. The study of the divisor class group leads to the consideration of the factoriality of \(S(M)\) and the factoriality conjecture which says: If for a regular local ring \(A\) the symmetric algebra \(S(M)\) is a factorial domain, then the projective dimension of \(M\) is not larger than 1, i.e., \(S(M)\) is a complete intersection. This is proved whenever the enveloping algebra of \(S(M)\) is a domain. The affirmative answer to the conjecture suggests that factoriality is very seldom for \(S(M)\) while \(B(M)\), the factorial closure, is always factorial, but in general not always of finite type. A consideration of particular cases when it is finite is included. This is related to the study of symbolic blowups via ideal transforms. It includes the explicit description of symbolic blowups as well as the outline of a counterexample of P. Roberts [J. Algebra 132, No. 2, 461-473 (1990; Zbl 0716.13013)] to Hilbert’s 14th problem.
The material of the book is based on: (1) results in the literature, presented in an original way, (2) the author’s own results published in several research papers over the last years, and (3) new results which did not appear elsewhere. It is not only the contents but also the fresh style of presentation which gives the book the attitude of a gourmet’s line to insights. The book is intended as an introduction to present day research for advanced students in commutative algebra and algebraic geometry as well as a reference for questions around Rees algebras and related topics. This intention is underlined by the rather extensive bibliography of more than 300 items. – Two of the highlights are the instructive examples and the constructive point of view in the whole presentation of the material. To this end the author includes also a chapter on computer algebra in commutative algebra beginning with an introduction of Gröbner bases. It covers algorithms and recipes to deal with basic constructions in commutative algebra, as primary decomposition, ideal transforms, integral closure etc. So the book is not only an introduction to present day research on blowup rings, it serves also as a basis how to handle computational questions in different fields of commutative algebra and algebraic geometry.

MathOverflow Questions:

Blowups of Cohen-Macaulay varieties

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C20 Class groups
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13D02 Syzygies, resolutions, complexes and commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13C40 Linkage, complete intersections and determinantal ideals
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