The global homological dimension of some algebras of differential operators

Björk, Jan-Erik

doi:10.1007/bf01390024

Cited by 31 publications

(17 citation statements)

References 8 publications

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“…An important result of Bjork [2] asserts that, under these conditions, (3.3). Our aim now is to prove that, under the above assumptions, dh 0S>(/)) = n, where ,@:=.…”

Section: 2)mentioning

confidence: 98%

“…This result can be understood as a generalization of Proposition (1.4) to the non-commutative case. Let us observe that the results of [2] cannot be directly applied to this situation because the residue fields at the maximal ideals of A(t) are not ^-algebraic in general (cf. Example (1.9)), and the &-algebraicity is an essential assumption in [2].…”

Section: 2)mentioning

confidence: 99%

“…Let us observe that the results of [2] cannot be directly applied to this situation because the residue fields at the maximal ideals of A(t) are not ^-algebraic in general (cf. Example (1.9)), and the &-algebraicity is an essential assumption in [2]. But Theorems (2.3) and (2.5) will allow us to use the results in [2] to prove the equality dh {2{t)) = n.…”

Section: 2)mentioning

confidence: 99%

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A Note on the Behaviour Under a Ground Field Extension of Quasicoefficient Fields

Macarro

1991

Journal of the London Mathematical Society

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The aim of this paper is to study the behaviour by the ground field extension k->k(t) of the quasicoefficient fields of the localization at the maximal ideals of a certain type of commutative fc-algebras (k a field of characteristic zero), not necessarily finitely generated.The chief motivation for the present work is to be found in the proof of the existence of the Bernstein-Sato polynomial [1,6,3] given by Mebkhout and Narvaez in [9,10]. This proof depends on the following results.(a) Let A be a noetherian, regular, equicodimensional /c-algebra, such that the residue fields at the maximal ideals are algebraic over k. Then A ® k k(t) is equicodimensional of the same dimension as A. (b) Let A be a ^-algebra as above, of dimension n. Let us assume, in addition, that there are x 1 ,...,x n eA,d 1 ,...,S n e Der fc (^4) such that S^x,) = <5 y , and let S) Alk be the ring of A>linear differential operators of A. Then the homological dimension of k(t) ® k @ A/k is equal to the homological dimension of @ A/k . The second result is a consequence of our main theorems (2.3) and (2.5) (see Theorem (3.4)). In [9,10] one can find another proof of (b) independent of Theorems (2.3), (2.5) (see Remark (3.5)). We have thought it useful to publish our results as an issue independent from the source problem, mainly due to their purely algebraic character, and their close relationship with another development in [5,13, 2].In §1 we include some results of Mebkhout and Narvaez [9, 10] about the preservation of equicodimensionality by the ground field extension k^-k(t). These results are used in §2, so we repeat them here for the sake of completeness.In §2 we give the main result of this paper.(2.5). Let A be a noetherian, regular, equicodimensional k-algebra of dimension n, whose residue fields at the maximal ideals are k-algebraic. Let us assume, in addition, that there are x x , ...,x n eA, S lt ...,S n eDer k (A) such that S^x,) = 5 i} . Then, for every maximal ideal m c A{t): = k(t) ® k A, one has that K m = {aeA(t) m \5<(a) = OforallSeDer^A)} (where S e is extended derivation) is a quasicoefficient field of A(t) m .Note that this result is closely related to [8, Theorem 30.1] (see Remark (2.6)). In §3 the preceding results are used to prove (b) above.

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“…An important result of Bjork [2] asserts that, under these conditions, (3.3). Our aim now is to prove that, under the above assumptions, dh 0S>(/)) = n, where ,@:=.…”

Section: 2)mentioning

confidence: 98%

Section: 2)mentioning

confidence: 99%

Section: 2)mentioning

confidence: 99%

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A Note on the Behaviour Under a Ground Field Extension of Quasicoefficient Fields

Macarro

1991

Journal of the London Mathematical Society

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“…Here, we review some facts about D-modules that will be used, and refer the reader to [3,6] for details on these objects of study.…”

Section: D-modulesmentioning

confidence: 99%

Lyubeznik numbers in mixed characteristic

Núñez-Betancourt

Witt

2013

Mathematical Research Letters

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Abstract. We define a family of invariants associated to any local ring whose residue field has prime characteristic; in particular, this includes those of mixed characteristic. These Lyubeznik numbers in mixed characteristic are defined via a surjection from an unramified regular local ring of mixed characteristic. As is true for the Lyubeznik numbers, the "highest-index" Lyubeznik number in mixed characteristic is a well-defined notion. These new invariants and the original Lyubeznik numbers coincide for certain local rings of equal characteristic p > 0; e.g., for Cohen-Macaulay rings and rings of dimension at most two. However, we provide an example where they differ.

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“…Let A be a finitely generated commutative algebra over the field k. In the terminology of [9] . When D is an abelian Lie algebra and a finitely generated free A -module the ring A(D) has been studied by a number of authors [1], [3][4][5].…”

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confidence: 99%

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1984

Proc. Amer. Math. Soc.

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The global homological dimension of some algebras of differential operators

Cited by 31 publications

References 8 publications

A Note on the Behaviour Under a Ground Field Extension of Quasicoefficient Fields

A Note on the Behaviour Under a Ground Field Extension of Quasicoefficient Fields

Lyubeznik numbers in mixed characteristic

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