Semi-invariants in surgery

Davis, James F.; Ranicki, Andrew

doi:10.1007/bf00533988

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…By the universal property of cokernels, we observe that there is a surjective map v : U d−2 (Λ) → coker(ϕ d ) such that the lower-third square in Diagram ( 13) is commutative. By taking kernels of all the maps between the bottom and the middle exact sequences of Diagram (13), we obtain the exact sequence in the top row. In particular, we see that…”

Section: An Exact Sequencementioning

confidence: 99%

“…Let W i (A σ − free) be Balmer's Witt groups of finite free modules over A (see [5] for this notation). The following result may be compared to [13,Theorem 5.2].…”

Section: Appendix a Witt Groups Of Clifford Algebrasmentioning

confidence: 99%

“…If K 0 (A) → K 0 ( Ā) is an isomorphism, then both the left and right vertical maps are isomorphisms. The second vertical map is injective for i odd by [13,Theorem 5.2]. Since Ā is semisimple, we have W i ( Āσ ) = 0 for i odd by [31](or [5,Theorem 4.2]).…”

Section: Appendix a Witt Groups Of Clifford Algebrasmentioning

confidence: 99%

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Witt groups of smooth projective quadrics

Xie

2019

Advances in Mathematics

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Section: An Exact Sequencementioning

confidence: 99%

“…Let W i (A σ − free) be Balmer's Witt groups of finite free modules over A (see [5] for this notation). The following result may be compared to [13,Theorem 5.2].…”

Section: Appendix a Witt Groups Of Clifford Algebrasmentioning

confidence: 99%

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Witt groups of smooth projective quadrics

Xie

2019

Advances in Mathematics

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“…is an isomorphism, then both the left and the right vertical maps are isomorphisms. The second vertical map is injective for i odd by [8,Theorem 5.2]. Since Ā is semisimple, we have W i ( Āσ ) = 0 for i odd by [22](or [4,Theorem 4.2]).…”

Section: Groups Of Clifford Algebrasmentioning

confidence: 99%

Grothendieck-Witt groups of quadrics

Xie¹

2016

Preprint

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Let k be a commutative ring containing 1 2 . In this paper, we construct certain homotopy fibration sequences for Grothendieck-Witt spectra of smooth quadric hypersurfaces over k. As an application, we compute Witt groups of smooth quadric hypersurfaces over certain kinds of bases. Contents 1. Introduction 1 2. Review: Grothendieck-Witt theory 4 3. Review: Semi-orthogonal decomposition on D b Q 8 4. The canonical involution 11 5. Grothendieck-Witt spectra of odd dimensional quadrics 13 6. Witt groups of odd dimensional quadrics 18 7. An exact sequence 21 8. Grothendieck-Witt spectra of even dimensional quadrics 22 9. Witt groups of even dimensional quadrics 26 10. Witt groups of Quadrics defined by m⟨−1⟩ ⊥ n⟨1⟩ 29 Appendix A. Witt groups of Clifford algebras 32 References 34

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“…In this case, the odd-indexed Witt groups are the L h 2i+1 (R) in the notation of [DR]. The good news is that the lift of information is now possible, by a result of Davis-Ranicki, see [DR,Cor. 5.2 (ii)], which says that in the following diagram, the map on the left is a monomorphism :…”

Section: Introductionmentioning

confidence: 99%