Algebraic Cobordism and Oriented Spectra¶

Job: develop the algebraic cobordism spectrum $\mathrm{MGL}$ (Voevodsky), the universality (Levine–Morel), the formal group laws of oriented motivic spectra, the Conner–Floyd theorem (cobordism-to-$K$), the Hopkins–Morel slice theorem, and the connection to prismatic / $\mathrm{TC}$ — with the universality / Conner–Floyd for $\mathbb{P}^n$ as the worked example.

Core¶

Algebraic cobordism $\mathrm{MGL}$¶

The algebraic Thom spectrum $\mathrm{MGL}=\mathrm{MGL}_S$ (Voevodsky) is the motivic spectrum whose $n$-th space is the Thom space of the tautological rank-$n$ bundle over the classifying space $\mathrm{BGL}_{n,S}$ (the ind-scheme $\mathrm{BGL}_{\infty,S}=\varinjlim_n\mathrm{BGL}_{n,S}$) — the algebraic analog of the complex cobordism $\mathrm{MU}$: $$ \mathrm{MGL}n=\mathrm{Th}(\gamma_n),\qquad \gamma_n\to\mathrm{BGL}, $$ with the structure maps $\Sigma^{2,1}\mathrm{MGL}_n\to\mathrm{MGL}_{n+1}$ (the Thom stabilization, the $\mathbb{P}^1$-suspension). The coefficient ring $\mathrm{MGL}_{*,*}(S)=\pi_{*,*}\mathrm{MGL}$ is the algebraic cobordism ring (the Levine–Morel / Voevodsky algebraic cobordism, the analog of the complex cobordism ring $\mathrm{MU}_*=\mathbb{Z}[x_1,x_2,\dots]$ the Lazard ring).

Theorem (universality, Levine–Morel). $\mathrm{MGL}$ is the universal oriented cohomology theory on smooth varieties: for every oriented cohomology theory $A^*$ on $\mathbf{Sm}/S$ (a multiplicative cohomology theory with Chern classes $c_1^A(L)$ satisfying the projective bundle formula), there is a unique natural transformation $\mathrm{MGL}^{*,*}\to A^*$ compatible with the orientations. This is the algebraic analog of the Conner–Floyd / Quillen universality of $\mathrm{MU}$.

Oriented spectra and formal group laws¶

An oriented motivic spectrum $E\in\mathrm{SH}(S)$ is a commutative ring spectrum equipped with an orientation $c_1^E:L\mapsto c_1^E(L)\in\tilde E^{2,1}(X)$ (assigning a "first Chern class" to line bundles on $X\in\mathbf{Sm}/S$), satisfying: - naturality (pullback compatibility); - the projective bundle formula (the $E^{*,*}(\mathbb{P}^n)$ is a free $E^{*,*}(\mathrm{pt})$-module on $1,c_1,\dots,c_1^n$); - the Whitney sum formula (up to the formal group law).

The orientation gives a formal group law $F_E(x,y)\in E^{*,*}[[x,y]]$ via the tensor-product formula $$ c_1^E(L\otimes M)=F_E(c_1^E(L),c_1^E(M)), $$ a commutative one-dimensional formal group law over $E^{*,*}(\mathrm{pt})$.

Theorem (universality of $\mathrm{MGL}$ — Lazard / Quillen motivic). $\mathrm{MGL}$ is the universal oriented spectrum: the orientation of any $E$ factors through $\mathrm{MGL}\to E$, and the formal group law $F_{\mathrm{MGL}}$ is the universal (Lazard) formal group law. The coefficient ring $\mathrm{MGL}_{*,*}(\mathrm{pt})$ is (conjecturally / proven over a field by Levine–Morel / Hopkins–Morel) the Lazard ring $L=\mathbb{Z}[x_1,x_2,\dots]$.

Oriented spectrum	Formal group law	Specialization
$\mathrm{MGL}$ (algebraic cobordism)	universal (Lazard) $F_{\mathrm{MGL}}$	the universal FGL, $L=\mathbb{Z}[x_i]$
$\mathrm{KGL}$ (algebraic $K$-theory)	multiplicative $F(x,y)=x+y+xy$ (the periodic multiplicative FGL)	$c_1(L\otimes M)=c_1(L)+c_1(M)+c_1(L)c_1(M)$
$\mathbf{H}\mathbb{Z}$ (motivic cohomology)	additive $F(x,y)=x+y$	$c_1(L\otimes M)=c_1(L)+c_1(M)$
$\mathrm{MGL}\wedge_{\mathrm{MGL}}\mathbb{Z}$ (cobordism to $K$ / Chern character)	the Chern character $ch:\mathrm{MGL}\to\mathrm{KGL}$	the Conner–Floyd / Riemann–Roch
$\mathrm{H}\mathbb{F}_\ell$ (mod-$\ell$ cohomology)	additive (over $\mathbb{F}_\ell$)	the Steenrod operations

The Conner–Floyd theorem and cobordism-to-$K$¶

Theorem (algebraic Conner–Floyd, Levine–Morel). The algebraic cobordism, tensored with the $K$-theory coefficients, recovers the algebraic $K$-theory: $$ \mathrm{MGL}^{,}(X)\otimes_{\mathrm{MGL}_{,}}\mathbb{Z}[u,u^{-1}]\cong K^*(X), $$ where $\mathbb{Z}[u,u^{-1}]$ is the $K$-theory coefficient ring ($|u|=(2,1)$ the Bott element), via the multiplicative formal group law (the orientation of $\mathrm{KGL}$). This is the algebraic analog of the Conner–Floyd theorem $\mathrm{MU}^*(X)\otimes_{\mathrm{MU}_*}\mathbb{Z}[u^{\pm}]\cong K^*(X)$.

The Chern character from $\mathrm{MGL}$ to $\mathrm{KGL}$ (the Conner–Floyd map, the orientation change from the universal to the multiplicative FGL) and the Riemann–Roch theorem in the motivic setting (the Grothendieck–Riemann–Roch the cobordism version: $f_!$ and $f_*$ commute with the Chern character up to the Todd class, the algebraic analog of the Ochanine–Riemann–Roch).

The Hopkins–Morel theorem and the slice of $\mathrm{MGL}$¶

Theorem (Hopkins–Morel; Levine). The zeroth slice of $\mathrm{MGL}$ is the motivic Eilenberg–MacLane: $$ s_0\mathrm{MGL}\cong\mathbf{H}\mathbb{Z}. $$ The higher slices are the motivic Eilenberg–MacLane spectra of the Lazard ring: $s_q\mathrm{MGL}\cong\bigvee_i\Sigma^{2q+i,q}\mathbf{H}\mathbb{Z}\{x_i\}$ with $x_i$ a basis of $L$ in degree $i$ (the Voevodsky / Levine slice computation).

This is the slice computation of $\mathrm{MGL}$ (Levine, Spitzweck): the slices of $\mathrm{MGL}$ are the $\mathbf{H}\mathbb{Z}$-modules of the algebraic cobordism coefficients (the Lazard ring), and the slice spectral sequence for $\mathrm{MGL}$ converges to the algebraic cobordism groups. The relation to the Landweber–Novikov operations (the algebraic analog of the Landweber–Novikov operations on $\mathrm{MU}$): the $\mathrm{MGL}_{*,*}$-cooperations act on $\mathrm{MGL}$, the motivic analog of the Landweber–Novikov Hopf algebroid.

Periodic / connective variants and the chromatic picture¶

Connective algebraic cobordism $\mathrm{mgl}$: the connective cover, the non-periodic version (the non-negative slice part). The coefficient ring $\mathrm{mgl}_{*,*}$ is the connective part of $L$.
Periodic $\mathrm{MGL}$: the $\mathbb{P}^1$-periodic version (the $\mathrm{MGL}$ itself is the periodic, the $\mathrm{mgl}$ the connective).
Bott periodicity for $\mathrm{KGL}$: $\mathrm{KGL}\simeq\Sigma^{2,1}\mathrm{KGL}$ (the Bott periodicity, the $\mathbb{P}^1$-periodicity), vs the $\mathrm{MGL}$ which is not periodic in the same way (the periodicity is the Landweber / the chromatic one, not the Bott one).

Spectrum	Periodicity	Coefficients
$\mathrm{MGL}$	not $\mathbb{P}^1$-periodic (Landweber / chromatic)	$L=\mathbb{Z}[x_i]$ the Lazard ring
$\mathrm{mgl}$	connective (no negative slices)	connective part of $L$
$\mathrm{KGL}$	$\mathbb{P}^1$-periodic ($\mathrm{KGL}\simeq\Sigma^{2,1}\mathrm{KGL}$)	$\mathbb{Z}[u^{\pm}]$ the Bott-periodic
$\mathbf{H}\mathbb{Z}$	none (the Eilenberg–MacLane)	$\mathbb{Z}$

The $\mathrm{MGL}$-based chromatic picture: the motivic analog of the chromatic filtration — the $v_n$-periodic spectra, the Johnson–Wilson theories $E(n)$ in the motivic setting (the motivic $E(n)$, the motivic Morava $E$-theory), the Kervaire-invariant / the Hill–Hopkins–Ravenel theorem in the motivic setting (the $v_n$-periodic layers via the $\mathrm{MGL}$-based chromatic spectral sequence).

Relation to prismatic / crystalline cohomology¶

The connection to the prismatic cohomology (Bhatt–Scholze / Bhatt–Morrow–Scholze, see prismatic cohomology) via the cyclotomic structure: the algebraic cobordism $\mathrm{MGL}$ maps to the topological periodic / cyclic homology $\mathrm{TP}$ / $\mathrm{TC}$ (the cyclotomic spectrum, the BMS theory) — the cyclotomic structure on the algebraic cobordism is the input to the prismatic / the $A_{\inf}$-cohomology.

Theorem (Bhatt–Morrow–Scholze connection). The $\mathrm{MGL}$ / the algebraic cobordism (via the cyclotomic spectrum $\mathrm{TC}$) is the input to the prismatic / $A_{\inf}$-cohomology, the syntomic / the prismatic comparison: the prismatic cohomology of a smooth scheme over a prism is recovered from the $\mathrm{TC}$ / the cyclotomic $\mathrm{MGL}$ via the BMS spectral sequence. This is the deep connection between the algebraic cobordism and the $p$-adic Hodge theory.

$\mathrm{MGL}$ relates to	The connection
$\mathrm{KGL}$ (algebraic $K$)	Conner–Floyd: $\mathrm{MGL}\otimes\mathbb{Z}[u^{\pm}]\cong\mathrm{KGL}$ (the multiplicative FGL)
$\mathbf{H}\mathbb{Z}$ (motivic cohomology)	the slice $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$ (Hopkins–Morel) — the additive FGL
$\mathrm{TC}$ / $\mathrm{TP}$ (cyclotomic)	the Bhatt–Morrow–Scholze — $\mathrm{MGL}$ as the input to prismatic / $A_{\inf}$
$\mathrm{MGL}_{,}$ (Lazard ring)	the universal formal group law, the chromatic $v_n$-periodic layers

See it¶

Universality of $\mathrm{MGL}$ and the Conner–Floyd theorem for $\mathbb{P}^n$. The algebraic cobordism of the projective space: $$ \mathrm{MGL}^{,}(\mathbb{P}^n)\cong\mathrm{MGL}_{,}[\beta]/(\beta^{n+1}), $$ the polynomial ring in the cobordism class $\beta=c_1^{\mathrm{MGL}}(\mathcal{O}(1))$ (the hyperplane section, the orientation of the tautological line bundle), truncated by $\beta^{n+1}=0$ (the projective bundle formula). The formal group law $F_{\mathrm{MGL}}$ determines the truncation: the relation $\beta^{n+1}=F_{\mathrm{MGL}}$-combination of the lower powers (the formal group law of $\mathrm{MGL}$ is the universal / Lazard, so the truncation is the universal one).

The Conner–Floyd theorem specializes to: $$ \mathrm{MGL}^{,}(\mathbb{P}^n)\otimes_{\mathrm{MGL}_{,}}\mathbb{Z}[u^{\pm}]\cong K^*(\mathbb{P}^n)=\mathbb{Z}[t^{\pm}]/(t^{n+1}), $$ where $t=c_1^{\mathrm{KGL}}(\mathcal{O}(1))=1-\beta^{\vee}$ (the $K$-theory first Chern class, the multiplicative FGL $F(x,y)=x+y+xy$ giving $c_1(\mathcal{O}(1)\otimes\mathcal{O}(1))=c_1+c_1+c_1^2$ — the multiplicative Chern class). The Conner–Floyd map sends the Lazard FGL to the multiplicative FGL, recovering the $K$-theory of $\mathbb{P}^n$.

Contrast with the additive FGL for $\mathbf{H}\mathbb{Z}$: the additive formal group law $F(x,y)=x+y$ gives $c_1(L\otimes M)=c_1(L)+c_1(M)$, and the ordinary cohomology $H^{*,*}(\mathbb{P}^n)=\mathbb{Z}[x]/(x^{n+1})$ with $x=c_1^{\mathbf{H}\mathbb{Z}}(\mathcal{O}(1))$. The universality of $\mathrm{MGL}$ is the one-formula unifying the additive (motivic cohomology) and the multiplicative ($K$-theory) formal group laws via the specialization of the Lazard FGL: setting the FGL to additive gives $\mathbf{H}\mathbb{Z}$, setting it to multiplicative gives $\mathrm{KGL}$, and the general Lazard FGL gives $\mathrm{MGL}$ itself.

Check¶

Show that the Conner–Floyd theorem for $\mathbb{P}^n$ recovers $K^*(\mathbb{P}^n)=\mathbb{Z}[t^{\pm}]/(t^{n+1})$ from $\mathrm{MGL}^{*,*}(\mathbb{P}^n)$, and identify the role of the multiplicative FGL.

$\mathrm{MGL}^{*,*}(\mathbb{P}^n)=\mathrm{MGL}_{*,*}[\beta]/(\beta^{n+1})$ by the projective bundle formula (the orientation gives the basis $1,\beta,\dots,\beta^n$). The Conner–Floyd map $\mathrm{MGL}\to\mathrm{KGL}$ sends the Lazard FGL $F_{\mathrm{MGL}}$ to the multiplicative FGL $F(x,y)=x+y+xy$ (the tensor-product formula $c_1(L\otimes M)=c_1(L)+c_1(M)+c_1(L)c_1(M)$ in $K$-theory). Setting $t=1+\beta^{\vee}\in K^0(\mathbb{P}^n)$ (the $K$-theory Chern class, the multiplicative one) and using the periodicity $t=u^{-1}\beta$ (the Bott element), the specialization $\mathrm{MGL}_{*,*}\to\mathbb{Z}[u^{\pm}]$ sends $\beta\mapsto t$, and the truncation $\beta^{n+1}=0$ becomes $t^{n+1}=0$, giving $K^*(\mathbb{P}^n)=\mathbb{Z}[t^{\pm}]/(t^{n+1})$. The multiplicative FGL is essential: it is the relation $c_1(\mathcal{O}(1)^{\otimes m})=1-(1-t)^m$ that gives the $K$-theory Chern classes, and the Conner–Floyd map is the FGL specialization.

Verify that the slice $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$ (Hopkins–Morel) is consistent with the additive FGL of $\mathbf{H}\mathbb{Z}$.

The slice $s_0\mathrm{MGL}$ is the "weight-zero piece" of the cobordism, the part surviving the slice filtration at $q=0$. By Hopkins–Morel, $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$ — the motivic cohomology, which is the additive FGL ($c_1(L\otimes M)=c_1(L)+c_1(M)$). The consistency: the additive FGL is the "linearization" of the universal Lazard FGL (the first-order / the coefficient of the linear term), and the slice $s_0$ is exactly the linear / additive part of the cobordism (the higher slices $s_q$ for $q>0$ contain the non-linear / higher-order FGL terms, the $x_i$ generators of the Lazard ring). So $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$ (the additive FGL) and the higher slices contain the Lazard generators (the higher-order FGL terms) — the slice filtration separates the additive (linear) part from the higher-order (non-linear) parts of the universal FGL.

Pitfalls¶

The universality of $\mathrm{MGL}$ (Levine–Morel) is for oriented cohomology theories on smooth varieties (not arbitrary cohomology theories) — the orientation is essential (the Chern classes satisfying the projective bundle formula); non-oriented theories (e.g. Chow without the multiplicative structure) are not in the universal category.
The Conner–Floyd theorem relates $\mathrm{MGL}$ and $\mathrm{KGL}$ but is NOT an equivalence of categories — the $\mathrm{MGL}\otimes\mathbb{Z}[u^{\pm}]\to K$ is a ring map (the coefficient-level identification), not a categorical equivalence of the spectra. The spectrum $\mathrm{MGL}\otimes_{\mathrm{MGL}_{*,*}}\mathbb{Z}[u^{\pm}]$ is the $K$-theory as a ring spectrum only after inverting / completing carefully (the Ravenel / the Landweber exactness).
The formal group law of $\mathrm{MGL}$ is the Lazard FGL (universal, not multiplicative or additive) — do not conflate with $\mathrm{KGL}$ (multiplicative) or $\mathbf{H}\mathbb{Z}$ (additive). The Lazard FGL $F(x,y)=x+y+\sum a_{ij}x^iy^j$ has the universal coefficients $a_{ij}\in L$.
The relation of $\mathrm{MGL}$ to prismatic / $\mathrm{TC}$ is the Bhatt–Morrow–Scholze connection — a deep theorem (the cyclotomic structure on the cobordism, the BMS spectral sequence), not a formal consequence. The prismatic cohomology is the $p$-adic / cyclotomic avatar, not a direct special case of $\mathrm{MGL}$.
The slice $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$ (Hopkins–Morel) is a deep theorem, not a formal consequence — it identifies the linear / additive part of the cobordism with the motivic cohomology (the additive FGL). The higher slices (the Lazard ring) are the non-linear parts.
The $\mathrm{MGL}$ coefficients $\mathrm{MGL}_{*,*}=L$ is proven over a field (Levine–Morel, Hopkins–Morel) but the integral identification over a general base is more subtle (the Hopkins–Morel theorem holds over a perfect field; the general base requires care with the slice convergence).
The chromatic picture (the $v_n$-periodic layers, the motivic Johnson–Wilson $E(n)$) is partly conjectural / under construction — the motivic Morava $E$-theory and the motivic chromatic filtration are active research, not a fully developed analog of the classical chromatic picture.

Refs¶

Voevodsky, "$\mathbb{A}^1$-homotopy theory" (1998); the $\mathrm{MGL}$ definition
Levine–Morel, Algebraic Cobordism (Springer 2007) — the universality, the Conner–Floyd
Hopkins–Morel, "Slices of the motivic sphere and the Lazard ring" (unpublished, the $s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}$); Levine, "Slices of the motivic sphere" (2012)
Spitzweck, "A commutative $\mathbb{P}^1$-spectrum representing motivic cohomology over $\mathbb{Z}$" (2014)
Bhatt–Morrow–Scholze, "Integral $p$-adic Hodge theory" (Publ. Math. IHÉS 2019); the $\mathrm{TC}$ / prismatic connection
Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres (the classical analog)

Next¶

MH index · Foundations · Six Functors & Base Change · Slice Filtration · Motivic Adams & Stable Stems · K-theory · Prismatic cohomology · Spectral AG · SAG / chromatic — the chromatic / Morava $E$-theory as the spectral-geometric realization of the $\mathrm{MGL}$ chromatic picture

Oriented spectrum	Formal group law	Specialization
\(\mathrm{MGL}\) (algebraic cobordism)	universal (Lazard) \(F_{\mathrm{MGL}}\)	the universal FGL, \(L=\mathbb{Z}[x_i]\)
\(\mathrm{KGL}\) (algebraic \(K\)-theory)	multiplicative \(F(x,y)=x+y+xy\) (the periodic multiplicative FGL)	\(c_1(L\otimes M)=c_1(L)+c_1(M)+c_1(L)c_1(M)\)
\(\mathbf{H}\mathbb{Z}\) (motivic cohomology)	additive \(F(x,y)=x+y\)	\(c_1(L\otimes M)=c_1(L)+c_1(M)\)
\(\mathrm{MGL}\wedge_{\mathrm{MGL}}\mathbb{Z}\) (cobordism to \(K\) / Chern character)	the Chern character \(ch:\mathrm{MGL}\to\mathrm{KGL}\)	the Conner–Floyd / Riemann–Roch
\(\mathrm{H}\mathbb{F}_\ell\) (mod-\(\ell\) cohomology)	additive (over \(\mathbb{F}_\ell\))	the Steenrod operations

Spectrum	Periodicity	Coefficients
\(\mathrm{MGL}\)	not \(\mathbb{P}^1\)-periodic (Landweber / chromatic)	\(L=\mathbb{Z}[x_i]\) the Lazard ring
\(\mathrm{mgl}\)	connective (no negative slices)	connective part of \(L\)
\(\mathrm{KGL}\)	\(\mathbb{P}^1\)-periodic (\(\mathrm{KGL}\simeq\Sigma^{2,1}\mathrm{KGL}\))	\(\mathbb{Z}[u^{\pm}]\) the Bott-periodic
\(\mathbf{H}\mathbb{Z}\)	none (the Eilenberg–MacLane)	\(\mathbb{Z}\)

\(\mathrm{MGL}\) relates to	The connection
\(\mathrm{KGL}\) (algebraic \(K\))	Conner–Floyd: \(\mathrm{MGL}\otimes\mathbb{Z}[u^{\pm}]\cong\mathrm{KGL}\) (the multiplicative FGL)
\(\mathbf{H}\mathbb{Z}\) (motivic cohomology)	the slice \(s_0\mathrm{MGL}=\mathbf{H}\mathbb{Z}\) (Hopkins–Morel) — the additive FGL
\(\mathrm{TC}\) / \(\mathrm{TP}\) (cyclotomic)	the Bhatt–Morrow–Scholze — \(\mathrm{MGL}\) as the input to prismatic / \(A_{\inf}\)
\(\mathrm{MGL}_{,}\) (Lazard ring)	the universal formal group law, the chromatic \(v_n\)-periodic layers