Algebraic Cobordism Research Articles

Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable ∞ \infty -category of non- A 1 \mathbb {A}^1 -invariant motivic spectra, which turns out to be equivalent to the ∞ \infty -category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this ∞ \infty -category satisfies P 1 \mathbb {P}^1 -homotopy invariance and weighted A 1 \mathbb {A}^1 -homotopy invariance, which we use in place of A 1 \mathbb {A}^1 -homotopy invariance to obtain analogues of several key results from A 1 \mathbb {A}^1 -homotopy theory. These allow us in particular to define a universal oriented motivic E ∞ \mathbb {E}_\infty -ring spectrum M G L \mathrm {MGL} . We then prove that the algebraic K-theory of a qcqs derived scheme X X can be recovered from its M G L \mathrm {MGL} -cohomology via a Conner–Floyd isomorphism \[ M G L ∗ ∗ ( X ) ⊗ L Z [ β ± 1 ] ≃ K ∗ ∗ ( X ) , \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] where L \mathrm {L}{} is the Lazard ring and K p , q ( X ) = K 2 q − p ( X ) \mathrm {K}{}^{p,q}(X)=\mathrm {K}{}_{2q-p}(X) . Finally, we prove a Snaith theorem for the periodized version of M G L \mathrm {MGL} .

On the algebraic cobordism ring of involutions

On the algebraic cobordism ring of involutions

Bivariant algebraic cobordism with bundles

Bivariant algebraic cobordism with bundles

On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩

The algebraic cobordism spectraMSL\mathbf {MSL}andMSp\mathbf {MSp}are constructed. They are commutative monoids in the category of symmetricT∧2T^{\wedge 2}-spectra. The spectrumMSp\mathbf {MSp}comes with a natural symplectic orientation given either by a tautological Thom classthMSp∈MSp4,2(MSp2)th^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(\mathbf {MSp}_2), or a tautological Borel classb1MSp∈MSp4,2(HP∞)b_{1}^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(HP^{\infty }), or any of six other equivalent structures. For a commutative monoidEEin the categorySH(S){SH}(S), it is proved that the assignmentφ↦φ(thMSp)\varphi \mapsto \varphi (th^{\mathbf {MSp}})identifies the set of homomorphisms of monoidsφ:MSp→E\varphi \colon \mathbf {MSp}\to Ein the motivic stable homotopy categorySH(S)SH(S)with the set of tautological Thom elements of symplectic orientations ofEE. A weaker universality result is obtained forMSL\mathbf {MSL}and special linear orientations. The universality ofMSp\mathbf {MSp}has been used by the authors to prove a Conner–Floyed type theorem. The weak universality ofMSL\mathbf {MSL}has been used by A. Ananyevskiy to prove another version of the Conner–Floyed type theorem.

Algebraic cobordism and étale cohomology

Thomason's \'{e}tale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog of Thomason's theorem for Weibel's homotopy $K$-theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at \'etale motivic cohomology is the universal way to impose \'etale descent for these theories. As applications, we describe the \'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an \'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the \'{e}tale versions of these motivic spectra.

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.

Precobordism and cobordism

The purpose of this article is to compare several versions of bivariant algebraic cobordism constructed previously by the author and others. In particular, we show that a simple construction based on the universal precobordism theory of Annala--Yokura agrees with the more complicated theory of bivariant derived algebraic cobordism constructed earlier by the author, and that both of these theories admit a Grothendieck transformation to operational cobordism constructed by Luis Gonz\'alez--Karu over fields of characteristic 0. The proofs are partly based on convenient universal characterizations of several cobordism theories, which should be of independent interest. Using similar techniques, we also strengthen a result of Vezzosi on operational derived $K$-theory. In the appendix, we give a detailed construction of virtual pullbacks in algebraic bordism, filling the gaps in the construction of Lowrey--Sch\"urg.

Norms in motivic homotopy theory

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

Virtual intersection theories

We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.

On the infinite loop spaces of algebraic cobordism and the motivic sphere

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.

Bivariant derived algebraic cobordism

We extend the derived algebraic bordism of Lowrey and Schürg 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 algebraicK0K^0analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.

Operations and poly-operations in algebraic cobordism

In the case of a field of characteristic zero, we describe all operations (including non-additive ones) from a theory A⁎ obtained from Algebraic Cobordism Ω⁎ of M. Levine-F. Morel by change of coefficients to any oriented cohomology theory B⁎ (in the sense of Definition 2.1). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done in [24], as well as the topological one obtained by T. Kashiwabara - see [6]. The key new ingredients which permit us to treat the non-additive operations are: the use of poly-operations and the “Discrete Taylor expansion”. As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p - see [23], which permits to sharpen results on the structure of Algebraic Cobordism - see [22]. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the case of multiplicative operations proved by I. Panin in [13].

A counter-example by Yagita

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

On the Relation of Symplectic Algebraic Cobordism to Hermitian $K$-Theory

Эрмитова $K$-теория восстанавливается посредством алгебраических симплектических кобордизмов. В мотивной стабильной гомотопической категории $\mathrm {SH}(S)$ имеется единственный морфизм $\varphi \colon \mathbf {MSp}\to \mathbf {BO}$ коммутативных кольцевых $T$-спектров, посылающий класс Тома $\mathrm {th}^{\mathbf {MSp}}$ в класс Тома $\mathrm {th}^{\mathbf {BO}}$. С помощью $\varphi $ строится изоморфизм биградуированных кольцевых теорий когомологий на категории $\mathcal Sm\mathcal Op/S$ вида $\overline \varphi \colon \mathbf {MSp}^{*,*}(X,U)\otimes _{\mathbf {MSp}^{4*,2*}(\mathrm {pt})} \mathbf {BO}^{4*,2*}(\mathrm {pt}) \cong \mathbf {BO}^{*,*}(X,U)$. Этот результат представляет собой алгебраический аналог теоремы Коннера и Флойда, которая восстанавливает вещественную $K$-теорию по симплектическим кобордизмам. При переписывании бииндексов таким образом, что $\mathbf {MSp}^{p,q}=\mathbf {MSp}^{[q]}_{2q-p}$, получается изоморфизм $\overline \varphi \colon \mathbf {MSp}^{[*]}_*(X,U)\otimes _{\mathbf {MSp}^{[2*]}_0(\mathrm {pt})} \mathrm {KO}^{[2*]}_0(\mathrm {pt}) \cong \mathrm {KO}^{[*]}_*(X,U)$, в котором $\mathrm {KO}^{[n]}_i(X,U)$ - эрмитовы $K$-группы Шлихтинга.

Relative Equivariant Motives and Modules

AbstractWe introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant and Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives of versal flag varieties.

Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism

In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles.

On the Relation of Symplectic Algebraic Cobordism to Hermitian K-Theory

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class thMSp to the Thom class thBO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category $${\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)$$. The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSpp,q = MSp1[q], we have an isomorphism $$\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)$$, where the KOi[](X,U) are Schlichting’s hermitian K-theory groups.

Algebraic elliptic cohomology theory and flops I

We define the algebraic elliptic cohomology theory coming from Krichever’s elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever’s elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting.

Oriented bivariant theory, II: Algebraic cobordism of S-schemes

This is a sequel to our previous paper “Oriented bivariant theory, I”. In 2001, Levine and Morel constructed algebraic cobordism for (reduced) schemes [Formula: see text] of finite type over a base field [Formula: see text] in an abstract way and later Levine and Pandharipande reconstructed it more geometrically, using “double point degeneration”. In this paper in a similar manner to the construction of Levine–Morel, we construct an algebraic cobordism for a scheme [Formula: see text] over a fixed scheme [Formula: see text] in such a way that if the target scheme [Formula: see text] is the point [Formula: see text], then our algebraic cobordism is isomorphic to Levine–Morel’s algebraic cobordism. Our algebraic cobordism can be interpreted as “a family of algebraic cobordism” parametrized by the base scheme [Formula: see text].

Segre classes and Damon–Kempf–Laksov formula in algebraic cobordism

In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle.