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fundamental theorem of dg-algebraic rational homotopy theory

Contents

Contents

Idea

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

Preliminaries

Definition

(nilpotent and finite rational homotopy types)

Write

(1)Ho(SimplicialSets Qu) 1,nil fin AAAHo(SimplicialSets Qu) 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 XX which are

and

(2)Ho(SimplicialSets Qu) 1,nil ,fin AAAHo(SimplicialSets Qu) 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(DiffGradedCommAlgebras 0) fin 1AAAHo(DiffGradedCommAlgebras 0) 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 AA over the rational numbers which are

(Bousfield-Gugenheim 76, 9.2)

Statement

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

Ho((DiffGradedCommAlgebras 0) proj op)exp PL𝕃Ω PLdR Ho(SimplicialSets Qu) 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 XX (1) the derived adjunction unit is rationalization

    Ho(SimplicialSets Qu) 1,nil fin Ho(SimplicialSets Qu) 1,nil ,fin X exp PLΩ PLdR (X) \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) }
    Xη X derrationalizationexp PLΩ 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((DiffGradedCommAlgebras 0) proj op) fin 1exp PL𝕃Ω PLdR Ho(SimplicialSets Qu) 1,nil ,fin 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 Ω PkLdR exp PkL\Omega^\bullet_{\mathrm{P}\!k\!\mathrm{LdR}} \dashv \exp_{\mathrm{P}\!k\!\mathrm{L}} (“piecewise kk-linear”) exists over all ground fields kk of characteristic zero, with induced derived adjunction

Ho((DGCAlgebras k 0) proj op)exp PkL𝕃Ω PkLdR Ho(SimplicialSets Qu), 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 kk-localization of spaces unless k=k = \mathbb{Q}. However, it does still relate to rationalization under extension of scalars, given by the derived adjunction (via this Prop.)

Ho((DGCAlgebras k 0) proj op)(() )𝕃res Ho((DGCAlgebras 0) proj op), 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 kk 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 kk of characteristic zero, abbreviate

L kexp PkL𝕃Ω PkLdR , 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=k = \mathbb{Q}.

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

(4)ch A k(X):H(X;A)H(X;𝔻η PkL A)H(X;L kA). 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)H(X;L A) () k H(X;L kA) ()˜ ()˜ H(𝔻Ω PLdR (X);𝔻Ω PLdR (A)) 𝔻(() k) H(𝔻Ω PkLdR (X);𝔻Ω PkLdR (A)) \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(;)Ho(,)H(-;\,-) \coloneqq Ho(-,\,-) denotes hom-sets in the respective homotopy category)

of:

  1. the hom-isomorphisms of the derived PL de Rham adjunction;

  2. 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 kk-Chern-Dold character (4) factors through the rational Chern-Dold character via the extension-of-scalars-transformation (5).

ch A k(X)=(() k)ch A (X). 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=k = \mathbb{R}, just for definiteness):

(diagram from FSS 20)

Here:

Together this implies that the top rectangle commutes, which is the claim to be shown.

References

The full-blown equivalence first appears in

Concise review (without model category-theory, but discussion as an sSet-enriched adjunction nonetheless) and generalization both to Borel-equivariant rational homotopy theory (of covering spaces of non-nilpotent spaces), as well as to real homotopy theory:

Further related discussion over the real numbers:

Re-derivation in a context of derived algebraic geometry:

Review and interpretation in terms of non-abelian Chern-Dold character-theory:

Last revised on September 26, 2021 at 11:51:30. See the history of this page for a list of all contributions to it.