A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

Bachmann, Tom; Wickelgren, Kirsten

doi:10.48550/arxiv.2002.01848

Cited by 6 publications

(20 citation statements)

References 26 publications

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“…. This is a result of [3] (see Theorem 2.18, Proposition 2.32 and Theorem 7.6), namely, that for a rank m vector bundle V on smooth k-scheme Y of dimension m and a section s of V with an isolated zero at some closed point p ∈ Y , the local Euler class…”

Section: Quadratic Conductor Formulasmentioning

confidence: 96%

“…Since k is perfect, the extensions k(x i )/k are all separable. In this case, the identification of f * with the trace map on GW(−) is proven in [3,Corollary 8.5].…”

Section: The Quadratic Euler Characteristic Of Vanishing Cyclesmentioning

confidence: 97%

“…This has been treated by Eisenbud-Levine [11] and Khimshiasvili [17], where they use the signature of a quadratic form on the Jacobian ring defined by Scheja-Storch [28]. The Scheja-Storch form has been recognized as giving the local quadratic Euler chararcteristic of the section of Ω X /k defined by df over an arbitrary field (of characteristic = 2) by Kass-Wickelgren [14] and Bachmann-Wickelgren [3].…”

Section: Introductionmentioning

confidence: 99%

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Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Levine¹,

Lehalleur²,

Srinivas³

2021

Preprint

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Let k be a perfect field and let GW(k) be the Grothendieck-Witt ring of (virtual) non-degenerate symmetric bilinear forms over k. We develop methods for computing the quadratic Euler characteristic χ(X/k) ∈ GW(k) for X a smooth hypersurface in a projective space and in a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface X in P n+1 to the cone over a smooth hyperplane section of X; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture that generalizes these computations to similar types of degenerations. Finally, we give an interpretation of the quadratic conductor formulas in terms of Ayoub's nearby cycles functor.M. LEVINE, S. PEPIN LEHALLEUR, AND V. SRINIVASAlthough this gives a quite concrete expression for χ(X/k), explicit computations are still not easy; it is the purpose of this note to give other descriptions more amenable to computation.As is well known, the primitive Hodge cohomology of a smooth hypersurface X ⊂ P n+1 is computable as suitable graded pieces of the Jacobian ring J(F ) := k[X 0 , . . . , X n+1 ]/(. . . , ∂F/∂X i , . . .) of the defining equation F , at least if k has characteristic zero. This goes back to results of Griffiths [5] and was pursued further in Carlson-Griffiths [6], where they showed a certain compatibility of this identification with respect to product structures. We will review these results here. This material has been treated elswhere, for instance, Steenbrink [33] handled the case of hypersurfaces in a weighted projective space over a characteristic zero field (without an explicit discussion of products), and the additive theory in the generality discussed here may be found in the article of Dolgachev [9].Combined with the results of [20] computing the quadratic Euler characteristic in terms of Hodge cohomology, relating the multiplicative structure on the Jacobian ring with that of the Hodge cohomology gives a quite explict description of the quadratic Euler characteristic of a smooth hypersurface in a projective space, as well as for smooth hypersurfaces in a weighted projective space P(a 0 , . . . , a n+1 ) assuming that the weighted degree e and all the a i are prime to the characteristic and that e is divisible by the lcm of the a i .A motivating problem underlying these computations is the search for a quadratic replacement of the conductor formulas of Milnor, Deligne, Bloch, Saito, Kato-Saito, and others. One considers a flat proper morphism f : X → Spec O with O a dvr having parameter t, with X a regular scheme and with generic fiber X K smooth over the quotient field K of O. A "conductor formula" is an expression for the difference χ top c (X K )−χ top c (X 0 ) in terms of algebraic invariants of f (χ top c (−) is the Euler characteristic of compactly supported cohomology). For a morphism of relative dimension zero, this is local ramification theory, where in positive and mixed characteristic, the Swa...

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Section: Quadratic Conductor Formulasmentioning

confidence: 96%

“…Since k is perfect, the extensions k(x i )/k are all separable. In this case, the identification of f * with the trace map on GW(−) is proven in [3,Corollary 8.5].…”

Section: The Quadratic Euler Characteristic Of Vanishing Cyclesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Levine¹,

Lehalleur²,

Srinivas³

2021

Preprint

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“…The Euler classes that we work with were introduced in [KW21] and further studied in [BW20]. See [KW21, Section 1.1] and the introduction of [BW20] for a discussion of related notions of Euler classes in arithmetic geometry.…”

Section: Local Degrees and Euler Classesmentioning

confidence: 99%

“…General approach and outline. Our goal is to prove an equality in GW(k), the Grothendieck-Witt group of isomorphism classes of non-degenerate symmetric bilinear forms over k. One side of this equation will be given by an Euler class [KW21,BW20], which is valued in GW(k) in the context of motivic homotopy theory. The other side of this equation will consist of a sum of local contributions, which are analogs of the local Brouwer degree [Mor12, KW19,KW21].…”

Section: Introductionmentioning

confidence: 99%

Conics meeting eight lines over perfect fields

Cameron¹,

Galimova²,

Gu³

et al. 2021

Preprint

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Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.

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η$\eta$‐Periodic motivic stable homotopy theory over Dedekind domains

Bachmann¹

2022

Journal of Topology

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We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer-Witt 𝐾-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence, we lift the fundamental fiber sequence of 𝜂-periodic motivic stable homotopy theory established in Bachmann and Hopkins (2020) from fields to arbitrary base schemes, and use this to determine (among other things) the 𝜂-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1∕2.

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A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

Cited by 6 publications

References 26 publications

Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Conics meeting eight lines over perfect fields

η$\eta$‐Periodic motivic stable homotopy theory over Dedekind domains

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