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algebraic cobordism

Idea

Algebraic cobordism is the bigraded generalized cohomology theory represented by the motivic Thom spectrum $MGL$. Hence it is the algebraic or motivic analogue of complex cobordism. The $(2n,n)$-graded part has a geometric description via cobordism classes, at least over fields of characteristic zero.

Definition

Let $S$ be a scheme and $MGL_S$ the motivic Thom spectrum over $S$. Algebraic cobordism is the generalized motivic cohomology theory? $MGL_S^{*,*}$ represented by $MGL_S$:

… formula here …

Properties

The (2n,n)-graded part

Let $S = Spec(k)$ where $k$ is a field of characteristic zero. A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by Marc Levine and Fabien Morel. More precisely, Levine-Morel constructed the universal oriented cohomology theory $\Omega^* : \Sm_k \to CRing^*$. Here oriented signifies the existence of direct image or Gysin homomorphisms for proper morphisms of schemes. This implies the existence ofChern classes for vector bundles.

Theorem

(Levine-Morel). There is a canonical isomorphism of graded rings

$$\mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k))$$

where $\mathbf{L}^*$ denotes the Lazard ring with an appropriate grading.

Theorem

(Levine-Morel). Let $i : Z \hookrightarrow X$ be a closed immersion of smooth $k$-schemes and $j : U \hookrightarrow X$ the complementary open immersion. There is a canonical exact sequence of graded abelian groups

$$\Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0,$$

where $d = \codim(Z, X)$.

Theorem

(Levine-Morel). Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings

$$\Omega^*(k) \longrightarrow MU^{2*}(pt)$$

is invertible.

Theorem

(Levine 2008). The canonical homomorphisms of graded rings

$$\Omega^*(X) \longrightarrow MGL^{2*,*}(X)$$

are invertible for all $X \in \Sm_k$.

Related concepts

• motivic Thom spectrum
• complex cobordism
• motivic homotopy theory
• motivic cohomology
• algebraic K-theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

There are two notions of “algebraic cobordism”, not closely related, one due to Snaith 77, and one due to Levine-Morel 01.

Snaith’s construction

• Victor Snaith, Towards algebraic cobordism, Bull. Amer. Math. Soc. 83 (1977), 384-385 (doi:10.1090/S0002-9904-1977-14281-X)

• Victor Snaith, Algebraic Cobordism and K-theory, Mem. Amer. Math. Soc. no 221 (1979)

• David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Documenta Math. 2008 (arXiv:0712.2817)

The construction in Snaith 77, motivated from the Conner-Floyd isomorphism, uses a variant of his general construction (Snaith's theorem) of a periodic multiplicative cohomology theory $X(b)^*(-)$ out of a pair consisting of a homotopy commutative H-monoid $X$ and a class $b\in \pi_n(X)$:

When $X = B S^1$ (the classifying space of the circle group) and $b$ is a generator of $\pi_2(BS^1)\cong\mathbb{Z}$ then $X(b)^*(-)$ is isomorphic with 2-periodic complex K-theory.

When $X = B U$ and $b$ a generator of $\pi_2(BU)\cong\mathbb{Z}$ one obtains $MU^*[u_2,u_2^{-1}]$ where MU is the (topological) complex cobordism cohomology and $u_2$ is the periodicity element.

Then Snaith introduces a variant of such constructions with a more general ring $A$ replacing the complex numbers; and uses the Quillen’s description of algebraic K-theory of a ring $A$ in terms of the classifying space $B GL(A)$; this way he obtains an algebraic cobordism theory.

Later, Gepner-Snaith 08 returned to the question of algebraic cobordism this time using the motivic version of algebraic cobordism of Voevodsky, namely the motivic spectrum $M GL$ representing universal oriented motivic cohomology theory (which is different from Morel-Voevodsky algebraic cobordism), and to the motivic version of Conner-Floyd isomorphism for which they give a comparably short proof.

Morel-Levine’s construction

• Marc Levine, Fabien Morel, Cobordisme algébrique I, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723–728, 2001 (doi:10.1016/S0764-4442(01)01833-X); Cobordisme algébrique II, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815–820, 2001 (doi:10.1016/S0764-4442(01)01832-8).

• Marc Levine, Algebraic cobordism, Proceedings of the ICM, Beijing 2002, vol. 2, 57–66, math.KT/0304206

• Marc Levine, Fabien Morel, Algebraic cobordism, Springer 2007, pdf.

• Marc Levine, A survey of algebraic cobordism (pdf)

• Marc Levine, Three lectures on algebraic cobordism, University of Western Ontario Mathematics Department, 2005, Lecture I, Lecture II, Lecture III.

• Marc Levine, Fabien Morel, Oberwolfach Arbeitsgemeinschaft mit aktuellem Thema, April 2005 report, notes

• Ivan Panin, K. Pimenov, Oliver Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology, Homotopy and Applications, 2008, 10(2), 211-226 (arXiv:0709.4116)

• Ivan Panin, K. Pimenov, Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Inventiones mathematicae, 2009, 175(2), 435-451, doi:10.1.1.244.7301, arXiv:0709.4124.

• Marc Hoyois, From algebraic cobordism to motivic cohomology, pdf, arXiv.

• Markus Spitzweck, Algebraic cobordism in mixed characteristic (arXiv:1404.2542)

• Marc Levine, Girja Shanker Tripathi?, Quotients of MGL, their slices and their geometric parts, arXiv:1501.02436.

More chat about the relation to motivic homotopy theory:

• Interdependence between A^1-homotopy theory and algebraic cobordism, MO/36659.

A simpler construction was given in

• M. Levine, R. Pandharipande, Algebraic cobordism revisited (math.AG/0605196)

A Borel-Moore homology version of $MGL^{*,*}$ is considered in

• Marc Levine, Oriented cohomology, Borel-Moore homology and algebraic cobordism, arXiv.

The comparison with $MGL^{2*,*}$ is in

• Marc Levine, Comparison of cobordism theories, Journal of Algebra, 322(9), 3291-3317, 2009, arXiv.

The construction was extended to derived schemes in the paper

• Parker Lowrey, Timo Schuerg. Derived algebraic bordism, 2012, arXiv:1211.7023.

The close connection of algebraic cobordism with K-theory is discussed in

• José Luis González, Kalle Karu. Universality of K-theory. 2013. arXiv:1301.3815.

An algebraic analogue of h-cobordism:

• Aravind Asok, Fabien Morel, Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms, arXiv:0810.0324