nLab fundamental theorem of dg-algebraic rational homotopy theory

Contents

Context

Rational homotopy theory

Idea
Preliminaries
Statement
Change of scalars
Related concepts
References

Idea

The fundamental theorem of rational homotopy theory modeled by dgc-algebras.

Preliminaries

Definition

(nilpotent and finite rational homotopy types)

Write

(1)

Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)

for the full subcategory of the classical homotopy category (homotopy category of the classical model structure on simplicial sets) on those homotopy types $X$ which are

connected: $\pi_0(X) = \ast$ ;
nilpotent: the fundamental group $\pi_1(X)$ is a nilpotent group and the higher homotopy groups are nilpotent modules over it;
rationally of finite type: the rational cohomology-groups are finite-dimensional, $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$ , for all $n \in \mathbb{N}$ .

and

(2)

Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)

for the further full subcategory on those homotopy types that are already rational.

Similarly, write

(3)

Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)_{fin}^{\geq 1} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)

for the full subcategory of the homotopy category of the projective model structure on connective dgc-algebras on those dgc-algebras $A$ over the rational numbers which are

connected: $H^0(A) \simeq \mathbb{Q}$
finite type: the cochain cohomology groups are finite dimensional, $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$ , for all $n \in \mathbb{N}$ .

(Bousfield-Gugenheim 76, 9.2)

Statement

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)^{op}_{proj} \right) \underoverset { \underset {\;\;\; \mathbb{R} exp_{PL} \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot} Ho \big( SimplicialSets_{Qu} \big)

of the Quillen adjunction between simplicial sets and connective dgc-algebras (whose left adjoint is the PL de Rham complex-functor) has the following properties:

on connected, nilpotent rationally finite homotopy types $X$ (1) the derived adjunction unit is rationalization

$\array{ Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} & \overset{ }{\longrightarrow} & Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \\ X &\mapsto& \mathbb{R}\exp_{PL} \circ \Omega^\bullet_{PLdR}(X) }$
$X \underoverset {\eta_X^{der}} {rationalization} {\longrightarrow} \mathbb{R}\exp_{PL} \circ \Omega^\bullet_{PLdR}(X)$
on the full subcategories of nilpotent and finite rational homotopy types from Def. it restricts to an equivalence of categories:

$Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp_{PL} \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}$

(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)

Change of scalars

Often it is desireable to work with dgc-algebras not over the rational numbers but over the real numbers, because these relate to de Rham theory (e.g.: the PL de Rham complex of a smooth manifold is equivalent to the de Rham complex). While a PL de Rham complex-Quillen adjunction $\Omega^\bullet_{\mathrm{P}\!k\!\mathrm{LdR}} \dashv \exp_{\mathrm{P}\!k\!\mathrm{L}}$ (“piecewise $k$ -linear”) exists over all ground fields $k$ of characteristic zero, with induced derived adjunction

Ho \Big( \big( DGCAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \Big) \underoverset {\underset{\mathbb{R} exp_{\mathrm{P}\!k\!\mathrm{L}}}{\longrightarrow}} {\overset{\mathbb{L} \Omega^\bullet_{\mathrm{P}\!k\!\mathrm{LdR}}}{\longleftarrow}} {\bot} Ho \big( SimplicialSets_{Qu} \big) \,,

this does not model $k$ -localization of spaces unless $k = \mathbb{Q}$ . However, it does still relate to rationalization under extension of scalars, given by the derived adjunction (via this Prop.)

Ho \Big( \big( DGCAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \Big) \underoverset { \underset{ \mathbb{R}\big( (-)\otimes_{\mathbb{Q}}\mathbb{R} \big) }{ \longrightarrow } } { \overset{ \mathbb{L} res_{\mathbb{Q}} }{ \longleftarrow } } {\bot} Ho \Big( \big( DGCAlgebras^{\geq 0}_{\mathbb{Q}} \big)^{op}_{proj} \Big) \,,

in that the following holds:

Proposition

For $k$ be a field of characteristic zero, the following diagram of derived functors commutes up to natural isomorphism:

This is effectivley the statement of Bousfield&Gugenheim 1976, Lem. 11.7.

This state of affairs may be recast as follows (FSS 2020):

For any field $k$ of characteristic zero, abbreviate

L_k \;\coloneqq\; \mathbb{R} exp_{\mathrm{P}k\mathrm{L}} \circ \mathbb{L} \Omega^\bullet_{\mathrm{P}kLdR} \,,

keeping in mind that this is a localization of spaces only if $k = \mathbb{Q}$ .

Then for $X, A \in Ho(sSet)^{fin_{\mathbb{Q}}}_{\geq 1, nil}$ a pair of connected nilpotent ℚ-finite homotopy types, define the $k$ -Chern-Dold character on the non-abelian $A$ -cohomology of $X$ to be the cohomology operation induced by the derived adjunction unit of the PL de Rham adjunction (this Prop.):

(4)

ch^k_A(X) \;\colon\; H(X;\, A) \;\xrightarrow{ \;\; H(X;\, \mathbb{D}\eta^{\mathrm{P}k\mathrm{L}_A}) \;\; }\; H(X;\, L_k A) \,.

Moreover, define extension of scalars on non-abelian rational cohomology to be the comoposite

(5)

\array{ H(X;\, L_{\mathbb{Q}}A) &\xrightarrow{ (-) \otimes_{{}_{\mathbb{Q}}} k }& H(X;\, L_{k}A) \\ {}^{\mathllap{ \widetilde{(-)} }} \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{\simeq}} \big\uparrow {}^{\mathrlap{ \widetilde{(-)} }} \\ H \big( \mathbb{D}\Omega^\bullet_{\mathrm{P}\mathbb{Q}\mathrm{LdR}}(X); \, \mathbb{D}\Omega^\bullet_{\mathrm{P}\mathbb{Q}\mathrm{LdR}}(A) \big) & \xrightarrow{ \mathbb{D} \big( (-) \otimes_{{}_{\mathbb{Q}}} k \big) } & H \big( \mathbb{D}\Omega^\bullet_{\mathrm{P}k\mathrm{LdR}}(X); \, \mathbb{D}\Omega^\bullet_{\mathrm{P}k\mathrm{LdR}}(A) \big) }

(where $H(-;\,-) \coloneqq Ho(-,\,-)$ denotes hom-sets in the respective homotopy category)

of:

the hom-isomorphisms of the derived PL de Rham adjunction;
the corresponding hom-component of the derived functor of extension of scalars (this Prop.).

(This is essentially the construction of “tensoring a homotopy type with $\mathbb{R}$ ” that is mentioned in DGMS 1975, Footnote 5.)

Then:

Proposition

The $k$ -Chern-Dold character (4) factors through the rational Chern-Dold character via the extension-of-scalars-transformation (5).

ch^k_A(X) \;=\; \big( (-)\otimes_{{}_{\mathbb{Q}}} k \big) \,\circ\, ch^{\mathbb{Q}}_A(X) \,.

Proof

Consider the following diagram of hom-sets (shown for $k = \mathbb{R}$ , just for definiteness):

Here:

the commutativity of the bottom part is Prop. ;
the triangle on the left as well as as the outer diagram commute by basic properties of adjoint functors (this Lemma);
the square on the right commutes by definition (5);