Modules of generalized fractions and balanced big Cohen-Macaulay modules

Sharp, Rodney Y.; Zakeri, H.

doi:10.1017/cbo9781107325517.007

Cited by 11 publications

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“…Y is torsionfree and commutes with base change for all i ≤ k. [17], (3.3) and [4], §1, and the claim follows from (1.3).…”

Section: Corollary Suppose X/y Is Arithmetically S K+2 Ie S and mentioning

confidence: 87%

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On the cohomology of regular differential forms and dualizing sheaves

Hübl¹,

Sun

1998

Proc. Amer. Math. Soc.

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Abstract. If Y is a local Dedekind scheme and X/Y is a projective CohenMacaulay variety of relative dimension 1, then R 1 f * ω 1 X/Y is torsionfree if and only if X/Y is arithmetically Cohen-Macaulay for a suitable embedding in P n k . If X is regular then R 1 f * ω 1 X/Y is torsionfree whenever the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field.Regular differential forms play an important role in the local and global duality theory of varieties and morphisms, and have been studied extensively over the last few years (cf.[12], [5]). Given a complex projective manifold X and a morphism f : X → Y to a reduced variety Y , J. Kollar has shown that R i f * ω X is torsionfree for any i ∈ N (cf.[10]). Little is known in the case of mixed characteristics, though in particular in the case of arithmetic varieties the torsionfreeness of the cohomology of canonical sheaves is of great interest. It appears in Bloch's work on the de Rham discriminant [2], and the second author has shown that it allows one to control the ramification of the base change of arithmetic surfaces, hence giving explicit formulas and estimates for the self-intersection number of canonical sheaves (cf. [14]).In this note we develop some general criteria which reduce the problem of torsionfreeness of the cohomology of canonical sheaves to a question on homogeneous coordinate rings. Using the special situation in the case of arithmetic surfaces, we are able to settle the question in the case that the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field.The work was done while the second author was a visiting fellow at the ICTP. He would like to express his thanks to the mathematics section of the ICTP for its hospitality. Both authors are grateful to Professor M. S. Narasimhan for helpful discussions. §1. Regular differential forms and residual complexes Let Y be an integral noetherian scheme and let f : X → Y be a projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension d and with geometrically connected fibres. We assume that X is reduced and f is generically smooth, i.e. that the extension K(X)/K(Y ) of meromorphic functions is a direct

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“…Y is torsionfree and commutes with base change for all i ≤ k. [17], (3.3) and [4], §1, and the claim follows from (1.3).…”

Section: Corollary Suppose X/y Is Arithmetically S K+2 Ie S and mentioning

confidence: 87%

“…Thus all sequences in A S are poor S ⊗ D E-sequences by [17], (3.3). Any homogeneous S ⊗ D E-regular sequence {F 1 , F 2 } in S + is also an S/πS = Hom S (S/πS, S ⊗ E)-regular sequence by the graded version of [7], (E.9).…”

Section: Corollary Suppose That X/y Is Globally a Complete Intersectmentioning

confidence: 99%

On the cohomology of regular differential forms and dualizing sheaves

Hübl¹,

Sun

1998

Proc. Amer. Math. Soc.

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“…(ii) D(H n x R (D(U (ȳ ) −n−1 n+1 M))) ∼ = Hom R (H n x R (R), D(D(U (ȳ ) −n−1 n+1 M))).Proof. In view of Lemma 3.16 of[14],U(ȳ) −n−1 n+1 M ∼ = U(ȳ) −n−1 n+1 R ⊗ R M. Hence…”

mentioning

confidence: 85%

“…. , x n is a regular sequence on M, by using the exactness theorem for generalized fractions (see Theorem 3.1 of [14] or Theorem 3.3 of [8]), we have the exact sequence…”

Section: Matlis Dual Of Local Cohomology Modulesmentioning

confidence: 99%

On the Matlis duals of local cohomology modules and modules of generalized fractions

Khashyarmanesh

2010

Proc Math Sci

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Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM = M. Let D(−) := Hom R (−, E) be the Matlis dual functor, where E := E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x 1 , . . . , x n is a regular sequence on M contained in a, then H n (x 1 ,...,xn)R D(H n a (M))) is a homomorphic image of D(M), where H i b (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H n (x 1 ,...,xn)R (D(H n a (M)))) ∼ = D(D(M)).

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“…Now the present first author and H. Zakeri introduced a concept of module of generalized fractions in commutative algebra in [23]. Since that paper appeared, there have been several further papers, such as [3], [4], [5], [6], [8], [9], [14], [15], [18], [22], [24], [25], [26], [27] and [28], which have shown that this concept has many interactions with topics of recent and current interest in commutative algebra, especially in commutative Noetherian ring theory. In particular, there are strong links between Cousin complexes and these modules of generalized fractions: it was shown in [18, (3.4)] that, for an i4-module M such that Ass(M) has only finitely many minimal members and a filtration & of Spec(i4) which admits M, the Cousin complex C(SF, M) mentioned above is actually isomorphic to a complex of modules of generalized fractions in the sense of [23].…”

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confidence: 99%

Generalized fractions and Hughes' gradetheoretic analogue of the Cousin complex

Sharp

Yassi

1990

Glasgow Math. J.

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Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [19, Section 2]: it is a complex of A-modules and A-homomorphismswith the property that, for each n ∈ N0 (we use N0 to denote the set of non-negative integers),Cohen–Macaulay rings can be characterized in terms of the Cousin complex: A is a Cohen–Macaulay ring if and only if C(A) is exact [19, (4.7)]. Also, the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring (see [19,(5.4)]).

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Modules of generalized fractions and balanced big Cohen-Macaulay modules

Cited by 11 publications

References 0 publications

On the cohomology of regular differential forms and dualizing sheaves

On the cohomology of regular differential forms and dualizing sheaves

On the Matlis duals of local cohomology modules and modules of generalized fractions

Generalized fractions and Hughes' gradetheoretic analogue of the Cousin complex

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