Subintegrality, invertible modules and polynomial extensions

Sadhu, Vivek; Singh, Balwant

doi:10.1016/j.jalgebra.2013.06.029

Cited by 5 publications

(6 citation statements)

References 6 publications

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“…The following result is due to Sadhu and Singh (Theorem 1.5 of [5]) which we use frequently throughout this paper. Proof.…”

Section: The Map I(a B) → I(a[x X −1 ] B[x X −1 ])mentioning

confidence: 99%

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Subintegrality, invertible modules and Laurent polynomial extensions

Sadhu

2015

Proc Math Sci

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Let A ⊆ B be a commutative ring extension. Let I(A, B) be the multiplicative group of invertible A-submodules of B. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension A ⊆ B of integral domains with dim A ≤ 1, so that the natural map I(A, B) → I(A[X, X −1 ], B[X, X −1 ]) is an isomorphism. In the same situation, we show that if dim A ≥ 2, then the condition is necessary but not sufficient. We also discuss some properties of the cokernel of the natural map I(A, B) → I(A[X, X −1 ], B[X, X −1 ]) in the general case.

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“…The following result is due to Sadhu and Singh (Theorem 1.5 of [5]) which we use frequently throughout this paper. Proof.…”

Section: The Map I(a B) → I(a[x X −1 ] B[x X −1 ])mentioning

confidence: 99%

“…Proof. By Lemma 2.2 of [5], it is enough to show that ϕ (A, D, B) is surjective for every subring D of C containing A such that D is finitely generated as an A-algebra. Let such a ring D be given.…”

Section: Proofmentioning

confidence: 99%

Subintegrality, invertible modules and Laurent polynomial extensions

Sadhu

2015

Proc Math Sci

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“…) is the direct sum of I(A, B), n terms of the form LI(A, B) and 2 i n i terms of the form N i I(A, B), 1 ≤ i ≤ n. Since we know from [8] that NI(A, B) = 0 is equivalent to A being seminormal in B (Definition 6.5) we can further conclude: Corollary 1.3. For A ⊂ B, the following are equivalent:…”

Section: Corollary 12 For Every Ring Extension a ⊂ B I(a[t ] B[t ]mentioning

confidence: 85%

“…Since we know from [8] that NI(A, B) = 0 is equivalent to A being seminormal in B (Definition 6.5) we can further conclude: It is immediate from our Main Theorem that LI(A, B) is a torsionfree group (we give a simple proof in Corollary 3.6); it is free abelian of finite rank when A is pseudo-geometric and finite dimensional (Proposition 3.7).…”

Section: Corollary 12 For Every Ring Extension a ⊂ B I(a[t ] B[t mentioning

confidence: 88%

Relative Cartier divisors and Laurent polynomial extensions

Sadhu¹,

Weibel

2016

Math. Z.

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“…Note that NK 0 (f ) ∼ = NPic (f). By Theorem 1.5 of [15], NPic (f) = 0 if and only if f is seminormal. Therefore, NK 0 (f ) = 0 and hence f is not K 0 -regular.…”

Section: On Regularitymentioning

confidence: 98%

On the vanishing of relative negative K-theory

Sadhu

2019

J. Algebra Appl.

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In this article, we study the relative negative K-groups K −n (f ) of a map f : X → S of schemes. We prove a relative version of the Weibel conjecture i.e. if f : X → S is a smooth affine map of noetherian schemes with dim S = d then K −n (f ) = 0 for n > d + 1 and the natural map K −n (f ) → K −n (f × A r ) is an isomorphism for all r > 0 and n > d. We also prove a vanishing result for relative negative K-groups of a subintegral map.

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Subintegrality, invertible modules and polynomial extensions

Cited by 5 publications

References 6 publications

Subintegrality, invertible modules and Laurent polynomial extensions

Subintegrality, invertible modules and Laurent polynomial extensions

Relative Cartier divisors and Laurent polynomial extensions

On the vanishing of relative negative K-theory

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