D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....referee for the detailed reading and numerous suggestions that lead to a more transparent manuscript. 2 Complete modules This section has a preliminary character in which we extend some properties of complete ideals to modules. Most of the facts we are going to review quickly can be traceable to [12]. For basic facts and terminology we shall use [3] and [5] As for notation: let (R, m) be a Noetherian local ring and let M be a finitely generated R module; we set #(M)for the minimal number of generators of M , #(M) for its length if M has a finite composition series. If M is a torsionfree ....

...., and discuss its role. Definition 2.2 Let E be a finitely generated R module of rank r.Theorder determinant of the embedding E is the ideal I defined by the image of the mapping image (# #) I (R ) When the embedding is clear we will write I =det 0 (E) Proposition 2. 3 ([12]) Let F E be torsionfree R modules of rank r and E## R an embedding. Denote by det 0 (F ) and det 0 (E) the corresponding order determinants. F is a reduction of E if and only if det 0 (F ) is a reduction of det 0 (E) Proof. Let S = R(R ) R[T 1 , T r ] and the Rees algebras ....

D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.

....d r ; 4 : In this formula, det 0 (E) denotes the Fitting ideal of E =E. Even for ideals this formula seems to be new. 2 Reductions of a module There are several measures of size attached to a Rees algebra R(E) all derived from ordinary Rees algebras. We briefly recall some of them (see [10], 11] for more details) For basic facts and terminology we shall use [1] and [5] As for notation, let (R# m) be a Noetherian local ring and let M be a finitely generated R module. For a m primary ideal I, the Hilbert Samuel multiplicityofM with respect to I will be denoted by e(I# M ) ....

D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.

....dimA p 1. The proof is obtained through a sequence of reduction steps and eventually makes use of the well known fact that a normal domain is the intersection of its localizations at height one primes. This sort of argument has of course been used before in varied context (cf. e.g. 8] 18] [23]) We then apply this criterion to the case of a homogenous extension A ae B to get a souped up version of the criterion in terms of the local integrality along primes p of R = A 0 = B 0 whose extensions pA p have height at most one. We are also able to derive a theorem by Kleiman and Thorup ( 14, ....

....be m primary R ideals; then J is a reduction of I if and only if e(J) e(I) The next reduction criterion has been proved by McAdam for ideals ( 18, 4.1] and later by Rees for the case of modules ( 23, 2.5] We obtain it as a direct consequence of Corollary 4.2. Proposition 5. 6 ( 8] 13] [23]) Let R be an equidimensional universally catenary Noetherian local ring, let F ae E ae R e be R modules with height ann R e =F 0, and write = F ) Then F is a reduction of E if and only if F p is a reduction of E p for every prime p of R with ann E=F ae p and dimR p = F p ) Gamma e ....

D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.

....local domain and let M and N be R modules with M 6= 0 and N torsion free. If M Omega R N is reflexive, then N is integral over N . Proof. This follows from the general fact that if N is a submodule of M and IM IN for some non zero ideal I of R, then M is integral over N . See, for example, [R]. ....

D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.

....2.1 Let R be a Noetherian ring and E a finitely generated R module having a rank. The Rees algebra R E of E is S E modulo its R torsion submodule. If E is a submodule of a free R module G, some authors define the Rees algebra of E to be the 4 image of the natural map S E S G (see [40] for a similar approach) The two definitions coincide if the assumptions overlap, i.e. for a finitely generated torsionfree module having a rank (i.e. for a finitely generated module E such that K R E is K free and the R map E K R E is injective) as in this situation the kernel of S ....

D. Rees, Reductions of modules, Proc. Camb. Phil. Soc. 101 (1987), 431--449.

....2.1 Let R be a Noetherian ring and E a finitely generated R module having a rank. The Rees algebra R (E) of E is S(E) modulo its R torsion submodule. If E is a submodule of a free R module G, some authors define the Rees algebra of E to be the 4 image of the natural map S(E) S(G) see [40] for a similar approach) The two definitions coincide if the assumptions overlap, i.e. for a finitely generated torsionfree module having a rank (i.e. for a finitely generated module E such that K Omega R E is K free and the R map E K Omega R E is injective) as in this situation the ....

D. Rees, Reductions of modules, Proc. Camb. Phil. Soc. 101 (1987), 431--449.

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D. Rees, Reductions of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431--449.

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