Derived algebraic cobordism
01 Apr 2016-Journal of The Institute of Mathematics of Jussieu (Cambridge University Press)-Vol. 15, Iss: 2, pp 407-443
TL;DR: In this article, a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations was constructed, and the resulting theory agreed with algebraic cobordism as defined by Levine and Morel.
Abstract: We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.
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TL;DR: In this paper, it was shown that the moduli stack of finite quasi-smooth derived schemes over a perfect field is equivalent to a grouplike motivic space with finite syntomic transfers.
Abstract: We prove that the $\infty$-category of $\mathrm{MGL}$-modules over any scheme is equivalent to the $\infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbb{P}^1$-loop spaces, we deduce that very effective $\mathrm{MGL}$-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.
Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega^\infty_{\mathbb{P}^1}\mathrm{MGL}$ is the $\mathbb{A}^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $\Omega^\infty_{\mathbb{P}^1} \Sigma^n_{\mathbb{P}^1} \mathrm{MGL}$ is the $\mathbb{A}^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.
29 citations
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TL;DR: In this paper, the etale motivic Borel-Moore homology of derived Artin stacks is constructed using a derived version of the intrinsic normal cone and derived classes of quasi-smooth Artin stack and demonstrate functoriality, base change, excess intersection and Grothendieck-Riemann-Roch formulas.
Abstract: We construct the etale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base change, excess intersection, and Grothendieck-Riemann-Roch formulas. These classes also satisfy a general cohomological Bezout theorem which holds without any transversity hypotheses. The construction is new even for classical stacks and as one application we extend Gabber's proof of the absolute purity conjecture to Artin stacks.
29 citations
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TL;DR: In this article, the authors introduce several kinds of bivariant theory associated with a suitable ring spectrum and construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes.
Abstract: The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism. We introduce several kinds of bivariant theory associated with a suitable ring spectrum and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. This fundamental classes satisfies all the expected properties from classical intersection theory and lead to Gysin morphisms, Riemann-Roch formulas as well as duality statements, valid for general schemes, including singular ones and without need of a base field.
22 citations
Cites background from "Derived algebraic cobordism"
...n before.12 11More recently, this kind of axiomatic as well as the universal property of algebraic cobordism, has been extended to the framework of derived algebraic geometry by Lowrey and Schu¨rg in [LS16]. 12In Panin’s work, it was defined in cohomology for projective morphisms between smooth schemes over a field. In Levine’s work, it was defined in Borel-Moore homology for smooth morphisms between 10 FR...
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TL;DR: For a quasi-projective scheme M which carries a perfect obstruction theory, this paper showed that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of virtual tangent bundle.
Abstract: For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.
18 citations
TL;DR: In this paper, the derived algebraic bordism of Lowrey and Schurg is extended to a bivariant theory in the sense of Fulton and MacPherson, and some of its basic properties are established.
Abstract: We extend the derived Algebraic bordism of Lowrey and Schurg to a bivariant theory in the sense of Fulton and MacPherson, and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic $K^0$ analogously to Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.
18 citations
Cites background or methods from "Derived algebraic cobordism"
...Hence also the pushforwards of the fundamental classes of W0 and W∞ to B∗(X) agree so the homotopy fibre relation holds on fundamental classes in B∗, giving us a morphism dΩ naive ∗ → B∗ (in the notation of [16])....
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...2 Derived algebraic bordism dΩ∗ and its universality The purpose of this section is to recall the construction of derived algebraic bordism dΩ∗ from [16], and to provide it with a convenient universal property....
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...Although not explicitly stated in [16], dΩ∗ is also the universal oriented Borel-Moore homology on quasi-projective derived schemes....
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...We begin by observing that the induced homology groups M∗(X) := M (X → pt) coincide with the cobordism cycle groups M∗ (X) of [16]....
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...Like the construction of dΩ∗ in [16], the construction of our bivariant algebraic cobordism Ω ∗ makes perfect sense without any restrictions on the characteristic....
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References
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TL;DR: In this article, the notion of (perfect) obstruction theory for algebraic stacks was introduced, and it was shown how to construct, given a perfect obstruction theory, a pure-dimensional virtual fundamental class in the Chow group of ��$X$¯¯¯¯.
Abstract: Let
$X$
be an algebraic stack in the sense of Deligne-Mumford. We construct a purely
$0$
-dimensional algebraic stack over
$X$
(in the sense of Artin), the intrinsic normal cone
${\frak C}_X$
. The notion of (perfect) obstruction theory for
$X$
is introduced, and it is shown how to construct, given a perfect obstruction theory for
$X$
, a pure-dimensional virtual fundamental class in the Chow group of
$X$
. We then prove some properties of such classes, both in the absolute and in the relative context. Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.
962 citations
TL;DR: In this paper, a construction of virtual fundamental classes of certain types of moduli spaces is proposed, where the fundamental classes can be expressed as a set of classes of modulus spaces.
Abstract: We suggest a construction of virtual fundamental classes of certain types of moduli spaces.
702 citations
TL;DR: In this article, the authors introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex and apply this method to construct the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety.
Abstract: We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.
610 citations
"Derived algebraic cobordism" refers background in this paper
...This result also provides a conceptual explanation why the virtual fundamental classes of Behrend and Fantechi [1] and Li and Tian [8] exist....
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TL;DR: In this paper, the authors introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex and apply this method to construct the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety.
Abstract: We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.
461 citations
TL;DR: In this article, it was shown that the reduced complex cobordism o*(X) of a finite complex is generated by its elements of positive degree as a module over the complex ring.
364 citations
"Derived algebraic cobordism" refers background in this paper
...In his treatise on the universality of the formal group law of complex oriented cobordism [13], Quillen introduced a geometric set of generators and relations for complex oriented cobordism....
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