SMOOTH FUNCTIONS ON ALGEBRAIC K-THEORY

J.P.PRIDHAM

Abstract. For any complex scheme X or any dg category, there is an associated Ktheory presheaf on the category of complex affine schemes. We study real C^{∞} functions on this presheaf, defined by Kan extension, and show that they are closely related to real Deligne cohomology. When X is quasi-compact and semi-separated, and for various non-commutative derived schemes, these smooth functions on K-theory are dual to the homotopy fibre of the Chern character from Blanc’s semi-topological Ktheory to cyclic homology.

Introduction

The idea of looking at smooth functions on algebraic K-theory presheaves was introduced in [Pri]. There, we looked at presheaves on a category of Fréchet manifolds, and showed that for the Fréchet algebra of complex analytic functions on a polydisc, the smooth functions on connective algebraic K-theory are the linear dual of a form of real Deligne cohomology.

The motivation behind this paper was the desire to adapt this characterisation to more general dg categories and to non-connective K-theory, in particular to obtain results for smooth proper complex varieties without having to impose descent by hand after the fact as in [Pri].

Rather than working with presheaves on Fréchet manifolds, our primary focus is to look at functors on C^{∞}-rings. This is justified because for any Fréchet manifold M and proper dg category A, we have C^{∞}(M, A) ≃ C^{∞}(M, ℝ) ⊗_{ℝ} A, and C^{∞}(M, ℝ) is a C^{∞}-ring. Since we concentrate on dg categories A without extra topological structure, the adjunction between C^{∞}-rings and commutative ℂ-algebras then allows us to frame our constructions in terms of the K-theory presheaf

\underline{K}(A)(Z) := K(A ⊗_{ℂ} Γ(Z, 𝒪_{Z}))

on complex affines as considered by Blanc in [Bla].

Our main result is Corollary 3.15, which shows that for any quasi-compact and semiseparated derived scheme X, and for various non-commutative derived schemes as well, the smooth functions

RC^{∞}(\underline{K}(X), ℝ)

on the non-connective K-theory presheaf \underline{K}(X) are given by ℝ-linear functions on the homotopy fibre of the Chern character

ch: K^{Blanc,st}(X)_{ℝ} → HC^{ℂ}(X)_{[−2]},

where K^{Blanc,st} is Blanc’s semi-topological K-theory. As explained in Remark 3.16, this homotopy fibre only differs from real Deligne cohomology by a β-torsion module for the Bott element β.

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/2].

Our approach to proving this is very different from that in [Pri], appealing directly to Goodwillie’s comparison [Goo] between K-theory and cyclic homology instead of resorting to symmetric space calculations. This appeal is made possible by considering the de Rham presheaf

\underline{K}(A)_{dR}(B) := \underline{K}(A)(B^{red}),

an idea which can be traced back to [Gro, Sim]. In §2, we show that smooth functions on \underline{K}(A)_{dR} are just given by real linear functions on K^{Blanc,st}(A).

de Rham presheaves then allow us to use Goodwillie’s comparison in a systematic fashion, and we reformulate it in Corollary 3.6 to say that for many complex dg categories A of geometric origin, the homotopy fibres of \underline{K}(A)_{ℚ} → \underline{K}(A)_{dR,ℚ} and \underline{HC}^{ℚ}(A) → \underline{HC}^{ℚ}(A)_{dR} are equivalent. The key calculations of the paper are in §3.2, showing that smooth functions on smash products of Eilenberg–MacLane spectra of complex affine spaces are just given by real multilinear maps. As a consequence, Theorem 3.14 shows that smooth functions on the rational cyclic homology presheaf \underline{HC}^{ℚ}(A) are the same as linear functions on real cyclic homology HC^{ℝ}(A), with the main result Corollary 3.15 as a consequence.

Finally, in §3.4 we discuss various related constructions and generalisations of these results. These include an analytic setting extending [Pri] by considering dg categories enriched in topological vector spaces. An analogue of real Deligne cohomology is then given by compactly supported distributions on the algebraic K-theory presheaf on formal neighbourhoods of infinite-dimensional manifolds.

Contents

Introduction 1 1. Smooth functions on algebraic K-theory 2 1.1. Preliminaries on C^{∞}-rings 2 1.2. Presheaves and homotopy Kan extensions 3 1.3. K-theory and cyclic homology presheaves 6 2. de Rham presheaves and Blanc’s semi-topological K-theory 7 3. Cyclic homology and Goodwillie’s comparison 9 3.1. Goodwillie’s comparison 9 3.2. Smooth functions on smash products of Eilenberg–MacLane spectra 11 3.3. Smooth functions on cyclic homology 13 3.4. Related constructions and generalisations 14 References 16

1. Smooth functions on algebraic K-theory

1.1. Preliminaries on C^{∞}-rings.

1.1.1. Complex affines.

Definition 1.1. Write CAlg(ℂ) for the category of commutative ℂ-algebras. We denote the opposite category by Aff_{ℂ}, the category of complex affine schemes.

Definition 1.2. Let CAlg(ℂ)_{fin} ⊂ CAlg(ℂ) be the full subcategory of finitely generated ℂ-algebras, and write Aff_{ℂ,fin} := (CAlg(ℂ)_{fin})^{opp}.

Observe that because ℂ is Noetherian, finitely generated commutative ℂ-algebras are finitely presented, so the colimit functor ind(CAlg(ℂ)_{fin}) → CAlg(ℂ) is an equivalence.

1.1.2. C^{∞}-rings. We now consider the C^{∞}-rings of [Joy, §2.2].

Definition 1.3. Recall that a C^{∞}-ring on A is a set equipped with compatible operations A^{n} → A for every C^{∞}-morphism f: ℝ^{n} → ℝ. There is a forgetful functor from C^{∞}-rings to commutative ℝ-algebras.

Write C^{∞}Alg for the category of C^{∞}-rings, and denote the opposite category by C^{∞}Aff. Given Z ∈ C^{∞}Aff, denote the corresponding object of C^{∞}Alg by O^{∞}(Z); given A ∈ C^{∞}Alg, denote the corresponding object of C^{∞}Aff by Spec_{∞}A.

Definition 1.4. Let C^{∞}Alg_{fin} ⊂ C^{∞}Alg be the full subcategory on finitely presented objects, and write C^{∞}Aff_{fin} := (C^{∞}Alg_{fin})^{opp}.

Observe that the colimit functor ind(C^{∞}Alg_{fin}) → C^{∞}Alg is an equivalence.

1.1.3. Smooth functions on complex affines.

Lemma 1.5. The functor u: C^{∞}Alg → CAlg(ℂ) given by u(A) = A ⊗_{ℝ} ℂ has a left adjoint u^{∗}, which has the property that for any smooth ℂ-algebra S, u^{∗}S is the ring of real C^{∞}-functions on the complex manifold (Spec S)(ℂ).

Proof. We can write u as the composition of the forgetful functor C^{∞}Alg → CAlg(ℝ) with the tensor product functor ⊗_{ℝ}ℂ: C^{∞}Alg(ℝ) → C^{∞}Alg(ℂ).

The categories C^{∞}Alg and CAlg(ℝ) are monadic over the category of real vector spaces, with natural maps Symm_{ℝ}V → C^{∞}(V^{∨}, ℝ) of the associated monads. Here, we regard V^{∨} as a pro-finite-dimensional vector space, so C^{∞}(V^{∨}, ℝ) = \underrightarrow{lim}_{α} C^{∞}(V_{α}^{∨}, ℝ) for finite-dimensional subspaces V_{α} of V. The forgetful functor thus has a left adjoint, which sends Symm_{ℝ}V to C^{∞}(V^{∨}, ℝ), and extends to all objects by passing to coequalisers.

The left adjoint to the functor ⊗_{ℝ}ℂ is given by Weil restriction of scalars, so sends Symm_{ℂ}U to Symm_{ℝ}U for any complex vector space U (via either of the identifications Hom_{ℝ}(ℂ, ℝ) ≅ ℂ or by V ≅ (V.Gal(ℂ/ℝ))^{Gal(ℂ/ℝ)}). The left adjoint to the forgetful functor sends Symm_{ℝ}V to C^{∞}(V^{∨}, ℝ).

The composite u^{∗} thus satisfies u^{∗}ℂ[z_{1}, …, z_{n}] = C^{∞}(ℂ^{n}, ℝ). If S = ℂ[z_{1}, …, z_{n}]/(f_{1}, …, f_{m}), we have u^{∗}S = C^{∞}(ℂ^{n}, ℝ)/(ℜf_{1}, ℑf_{1}, …, ℜf_{m}, ℑf_{m}), which is precisely C^{∞}((Spec S)(ℂ), ℝ) when S is smooth. □

The lemma motivates the following choice of notation:

Definition 1.6. Given Z = Spec O(Z) ∈ Aff_{ℂ}, write C^{∞}(Z, ℝ) := u^{∗}O(Z) and Z_{C^{∞}} := Spec_{∞}u^{∗}O(Z).

Beware that we cannot interpret C^{∞}(Z, ℝ) as a ring of functions on Z(ℂ) when Z is not reduced. For instance, when Z = Spec ℂ[z]/z^{2} we have C^{∞}(Z, ℝ) = ℝ[x, y]/(x^{2} − y^{2}, xy).

1.2. Presheaves and homotopy Kan extensions.

Definition 1.7. For any ring k, let Ch_{k} be the category of (unbounded) chain complexes of k-modules. For any category I, write Ch_{k}(I) for the category of presheaves in real chain complexes on I (i.e. functors I^{opp} → Ch_{k}).

As in [Pri, Proposition 1.9], the projective model structure of [BK] adapts to give:

Proposition 1.8. For any small category I, there is a cofibrantly generated model structure on the category Ch_{k}(I), with a morphism f: A → B being a fibration (resp. weak equivalence) whenever the maps f_{i}: A(i) → B(i) are surjections (resp. quasiisomorphisms) for all i ∈ I.

Definition 1.9. For any object X ∈ I, we write k.X ∈ Ch_{k}(I) for the presheaf sending Y to the free k-module k.Hom(Y, X) generated by the set Hom(Y, X).

Given a simplicial object X ∈ I^{Δ^{opp}}, we write Nk.X ∈ Ch_{k}(I) for the Dold–Kan normalisation of the simplicial presheaf k.X in k-modules.

Note that Nk.X is cofibrant because it is a direct summand of a complex k.X_{0} ← k.X_{1} ← k.X_{2} ← … which can be constructed as a composition of pushouts of generating cofibrations k.X_{i}[−i − 1] → cone(k.X_{i})[−i − 1].

Since we want to consider presheaves on the categories Aff_{ℂ} and C^{∞}Aff, we will now take the lazy, but notationally more convenient, route of assuming that Ch_{k} lives in a larger universe, so contains limits and colimits indexed by categories such as these. The following lemmas show that that assumption is unnecessary, and that all our presheaves can be defined on genuinely small categories:

Lemma 1.10. For any small category I, the restriction functor ι^{∗}: Ch_{k}(pro(I)) → Ch_{k}(I) coming from the inclusion functor ι: I → pro(I) has a left adjoint ι_{!}. The functor ι_{!} gives an equivalence between Ch_{k}(I) and the full subcategory of Ch_{k}(ind(I)) consisting of presheaves F which are locally of finite presentation in the sense that

\underrightarrow{lim}_{α} F (X_{α}) → F (\underleftarrow{lim}_{α} X_{α})

is an isomorphism for all filtered inverse systems {X_{α}} in I.

Proof. Because F ∈ Ch_{k}(I) is given by \underrightarrow{lim}_{Y ∈(I↓F)} k.Y, where the colimit is indexed by the comma category of morphisms k.Y → F in Ch_{k}(I), the functor ι_{!} can be defined by setting

ι_{!}F = \underrightarrow{lim}_{Y ∈(I↓F)} k.ι(Y).

For a filtered inverse system X = {X_{α}}_{α} in pro(I) and Y ∈ I, we necessarily have ι_{!}k.Y = k.(ιY), and so

(ι_{!}k.Y)(X) = \underrightarrow{lim}_{α}(ι_{!}k.Y)(X_{α}) = \underrightarrow{lim}_{α} k.Y (X_{α}).

We now just observe that any F ∈ Ch_{k}(I) can be written as a colimit of objects of the form k.Y, so the maps

\underrightarrow{lim}_{α} F (X_{α}) ← \underrightarrow{lim}_{α}(ι_{!}F)(X_{α}) → (ι_{!}F)(X)

are isomorphisms.

Thus we have (ι_{!}F)(\{X_{α}\}_{α}) ≅ \underrightarrow{lim}_{α} F (X_{α}), so ι^{∗}ι_{!} is the identity, while the counit ι_{!}ι^{∗}G → G of the adjunction is an isomorphism if and only if G is locally of finite presentation. □

1.2.1. Presheaves on complex and C^{∞}-affines. Using the equivalences pro(Aff_{ℂ,fin}) ≃ Aff_{ℂ} and pro(C^{∞}Aff_{fin}) ≃ C^{∞}Aff, and noting that filtered colimits preserve quasiisomorphisms of chain complexes, the lemma above gives:

Lemma 1.11. There are left Quillen functors

Ch_{k}(dAff_{ℂ,fin}) ⟶^{ι_{!}} Ch_{k}(Aff_{ℂ})
Ch_{k}(C^{∞}Aff_{fin}) ⟶^{ι_{!}} Ch_{k}(C^{∞}Aff)

whose essential images in the homotopy category consist of presheaves F which are locally of finite presentation in the sense that

\underrightarrow{lim}_{α} F (X_{α}) → F (\underleftarrow{lim}_{α} X_{α})

is a quasi-isomorphism for all filtered inverse systems {X_{α}}. The functor ι_{!} preserves weak equivalences and is full and faithful on the associated ∞-categories. The right adjoint of ι_{!} is given by restriction ι^{∗}.

Now observe that the categories Aff_{ℂ,fin} and C^{∞}Aff_{fin} are equivalent to small categories, since each object has a finite set of generators and relations. We can therefore make sense of Ch_{k}(Aff_{ℂ,fin}) and Ch_{k}(C^{∞}Aff) without worrying about universes, as any construction we consider can be reduced to a construction in the original universe.

Lemma 1.12. For the functor (−)_{C^{∞}} of Definition 1.6, and any F ∈ Ch_{k}(Aff_{ℂ,fin}), the presheaf L_{!}F_{C^{∞}} := L((−)_{C^{∞}})_{!}F ∈ Ch_{k}(C^{∞}Aff_{fin}) is given (up to canonical weak equivalence) by

ι_{!}L_{!}F_{C^{∞}} ≃ U^{∗}ι_{!}F ∈ Ch_{k}(C^{∞}Aff),

for the forgetful functor U: C^{∞}Aff → Aff_{ℂ} given by USpec_{∞}A := Spec (A ⊗_{ℝ} ℂ).

Proof. Since (−)_{C^{∞}} is right adjoint to U and LU^{∗} = U^{∗}, we have U^{∗} = L((−)_{C^{∞}})_{!}. Because ι_{!} preserves weak equivalences, we have ι_{!} = Lι_{!}. We then just note that (−)_{C^{∞}} restricts to a functor from Aff_{ℂ,fin} to C^{∞}Aff_{fin}, and commutes with ι. □

1.2.2. Smooth completions.

Definition 1.13. Let pro(FDVect_{ℝ}) be the category of pro-finite-dimensional real vector spaces. This is equivalent to the opposite category Vect_{ℝ}^{opp} of the category of real vector spaces, by dualisation. We write Ch(pro(FDVect_{ℝ})) for the category of chain complexes in pro(FDVect_{ℝ}), and give it the model structure coming from the equivalence Ch(pro(FDVect_{ℝ})) ≃ Ch_{ℝ}^{opp}.

Given V = \{V_{α}\}_{α} ∈ pro(FDVect_{ℝ}) and U ∈ Ch_{ℝ}, we write V ˆ⊗_{ℝ}U := \underleftarrow{lim}_{α}(V_{α} ⊗_{ℝ} U); if V^{∨} ∈ Ch_{ℝ} is the continuous dual of V, this means that V ˆ⊗_{ℝ}U is given by the Homcomplex \underline{Hom}_{ℝ}(V^{∨}, U) with \underline{Hom}_{ℝ}(V^{∨}, U)_{n} = ∏_{i} Hom_{ℝ}(V_{i}^{∨}, U_{n−i}).

Definition 1.14. Define Sm: Ch(pro(FDVect_{ℝ})) → Ch_{ℚ}(C^{∞}Aff) to be the functor given by

Sm(V)(Z) := V ˆ⊗_{ℝ}O^{∞}(Z).

We also write Sm for the functor (−)_{C^{∞}}^{∗}Sm: Ch(pro(FDVect_{ℝ})) → Ch_{ℚ}(Aff_{ℂ}) given by

Sm(V)(Z) := V ˆ⊗_{ℝ}C^{∞}(Z, ℝ).

Lemma 1.15. The functors Sm: Ch(pro(FDVect_{ℝ})) → Ch_{ℚ}(C^{∞}Aff) and Sm: Ch(pro(FDVect_{ℝ})) → Ch_{ℚ}(Aff_{ℂ}) are right Quillen.

Proof. It is immediate that both functors preserve fibrations and trivial fibrations. Left adjoints Sm^{∗} send presheaves F to the coends

Z ↦ ∫ ^{Z∈C^{∞}Aff} \underline{Hom}_{ℚ}(F (Z), O^{∞}(Z))^{∨}.
Z ↦ ∫ ^{Z∈Aff_{ℂ}} \underline{Hom}_{ℚ}(F (Z), C^{∞}(Z, ℝ))^{∨}.
□

Lemmas 1.12 and 1.11 imply that for F ∈ Ch_{ℚ}(Aff_{ℂ}) locally of finite presentation, the complexes

LSm^{∗}F, LSm^{∗}(F |_{dAff_{ℂ,fin}},), LSm^{∗}(F ∘ U), and LSm^{∗}((F ∘ U)|_{dC^{∞}Aff_{fin}})

are all equivalent. In particular, LSm^{∗}F is determined by the functor F (− ⊗_{ℝ} ℂ) on C^{∞}-rings.

1.3. K-theory and cyclic homology presheaves.

1.3.1. K-theory presheaves.

Definition 1.16. Given a spectrum Y = {Y^{n}} and a ring k, write Y_{k} for the chain complex

Y_{k} := \underrightarrow{lim}_{n} NC̄_{•}(Y^{n}, k))[n]

of k-modules, where C̄ denotes reduced chains and N the Dold–Kan normalisation, and the maps

NC̄_{•}(Y^{n}, k) → NC̄_{•}(Y^{n+1}, k)[1]

combine the structure map

(N^{−1}k[−1]) ⊗_{k} C̄_{•}(Y^{n}, k) ≅ C̄_{•}(S^{1} ∧ Y^{n}, k) → C̄_{•}(Y^{n+1}, k)

with the Eilenberg–Zilber shuffle map

k[−1] ⊗_{k} NC̄_{•}(Y^{n}, k) → N((N^{−1}k[−1]) ⊗_{k} C̄_{•}(Y^{n}, k)).

This construction is left adjoint to the Eilenberg–MacLane functor H from Ch_{k} to spectra (cf. Definition 3.10).

The following definition is taken from [Bla]:

Definition 1.17. Given a dg category A over ℂ, define the functor \underline{K}(A) from CAlg_{ℂ} to spectra by

\underline{K}(A)(B) := K(A ⊗_{ℂ} B),

the non-connective K-theory of A⊗_{ℂ}B. Given a derived ℂ-scheme X, we write \underline{K}(X) := \underline{K}(per_{dg}(X)), for per_{dg}(X) the dg category of perfect complexes of 𝒪_{X}-modules on X.

Combining these definitions, we obtain a presheaf \underline{K}(A)_{ℚ} ∈ Ch_{ℚ}(dAff_{ℂ}) which is locally of finite presentation.

Definition 1.18. Given a functor F from CAlg_{ℂ} to spectra, locally of finite presentation, define

RC^{∞}(F, ℝ) := (LSm^{∗}(F_{ℚ}))^{∨},

for F_{ℚ} ∈ Ch_{ℚ}(Aff_{ℂ}) as above.

Given a functor F from CAlg_{ℂ} to simplicial sets, locally of finite presentation, define

RC^{∞}(F, ℝ) := RC^{∞}(Σ^{∞}F_{+}, ℝ) = (LSm^{∗}(Nℚ.F))^{∨},

for Nℚ.F as in Definition 1.9.

We refer to these as the complexes of smooth functions on the presheaves F.

Definition 1.19. Given a functor F from C^{∞}Alg to spectra, locally of finite presentation, define

RO^{∞}(F) := (LSm^{∗}(F_{ℚ}))^{∨},

for F_{ℚ} ∈ Ch_{ℚ}(C^{∞}Aff) as above.

Given a functor F from C^{∞}Alg to simplicial sets, locally of finite presentation, define

RO^{∞}(F) := RO^{∞}(Σ^{∞}F_{+}) = (LSm^{∗}(Nℚ.F))^{∨}.

We refer to these as the complexes of functions on the presheaves F.

Equivalently, the dual functors LO^{∞}(−)^{∨}: Ch_{ℚ}(C^{∞}Aff) → Ch(pro(FDVect_{ℝ})) and LC^{∞}(−, ℝ)^{∨}: Ch_{ℚ}(Aff_{ℂ}) → Ch(pro(FDVect_{ℝ})) are the derived enriched left Kan extensions of O^{∞}(−)^{∨} and C^{∞}(−, ℝ)^{∨} along the spectral Yoneda embeddings of C^{∞}Aff and Aff_{ℂ}.

Our primary object of study will be RC^{∞}(\underline{K}(A), ℝ), the complex of smooth functions on the K-theory presheaf of A.

1.3.2. Cyclic homology presheaves.

Definition 1.20. Given a commutative ring k and a dg category A over k, write HC^{k}(A) for the chain complex associated to cyclic homology of A over k.

Definition 1.21. Given a dg category A over ℂ, define the functor \underline{HC}^{ℚ}(A) ∈ Ch_{ℚ}(Aff_{ℂ}) by \underline{HC}^{ℚ}(A)(Z) := HC^{ℚ}(A ⊗_{ℂ} O(Z)). Given a derived ℂ-scheme X, we write \underline{HC}^{ℚ}(X) := \underline{HC}^{ℚ}(per_{dg}(X)), for per_{dg}(X) the dg category of perfect complexes of 𝒪_{X}-modules on X.

2. de Rham presheaves and Blanc’s semi-topological K-theory

Definition 2.1. Given B ∈ CAlg_{ℂ}, define B^{red} to be the quotient of B by its nilradical.

The following definition is due to Carlos Simpson in [Sim]:

Definition 2.2. Given a presheaf F on Aff_{ℂ}, define the de Rham presheaf F_{dR} of F by setting

F_{dR}(B) := F (B^{red}).

Proposition 2.3. For any affine ℂ-scheme Z of finite type, the complex RC^{∞}(Z_{dR}, ℝ) is naturally quasi-isomorphic to the derived global sections RΓ(Z(ℂ)_{an}, ℝ) of the space Z(ℂ) with the analytic topology.

Proof. We first need to find a cofibrant replacement for the presheaf (ℚ.Z)_{dR}. Choose a closed immersion Z ↪ Y into a smooth ℂ-scheme Y, and write Ŷ_{Z} for the formal completion of Y along Z — this is an object of ind(Aff_{ℂ,fin}). Now consider the simplicial ind-scheme Ỹ_{Z} given by (Ỹ_{Z})_{n} := { \atop Y^{n+1}}_{Z}. For any B ∈ CAlg(ℂ), we thus have

(Ỹ_{Z})_{n}(B) ≅ Y (B)^{n+1} ×_{Y (B^{red})^{n+1}} Z(B^{red}),

so Ỹ_{Z}(B) is the Čech nerve of π: Y (B) ×_{Y (B^{red})} Z(B^{red}) → Z(B^{red}), and thus weakly equivalent to the image of π. Since Y is smooth, Y (B) → Y (B^{red}) is surjective, so Ỹ_{Z}(B) → Z(B^{red}) is a weak equivalence.

We therefore have a weak equivalence Nℚ.Ỹ_{Z} → (ℚ.Z)_{dR} in Ch_{ℚ}(Aff_{ℂ}), with Nℚ.Ỹ_{Z} cofibrant, so

RC^{∞}(Z_{dR}, ℝ) ≃ N_{c}C^{∞}(Ỹ_{Z}, ℝ),

where N_{c} denotes cosimplicial conormalisation. The construction of [Gro] applied to the pro-ring C^{∞}(Ỹ_{Z}) gives a quasi-isomorphism from N_{c}C^{∞}(Ỹ_{Z}, ℝ) to { \atop A^{•}(Y (ℂ), ℝ)}_{Z}, the completion of the real de Rham complex of Y (ℂ) with respect to ker(C^{∞}(Y, ℝ) → C^{∞}(Z, ℝ). If I is the kernel of C^{∞}(Ỹ_{Z}, ℝ) → C^{∞}(Ŷ_{Z}, ℝ), then filtration by powers of I maps quasi-isomorphically to the brutal truncation filtration on { \atop A^{•}(Y (ℂ), ℝ)}_{Z}.

Writing { \atop 𝒜^{•}(Y (ℂ), ℝ)} _{Z} for the natural complex of sheaves on Y (ℂ)_{an} whose global sections are { \atop A^{•}(Y (ℂ), ℝ)}_{Z}, the proof of [Har, Theorem IV.1.1] generalises from the holomorphic setting to the smooth setting to show that ℝ_{Z(ℂ)} → { \atop 𝒜^{•}(Y (ℂ), ℝ)} _{Z} is a

sheaf quasi-isomorphism on Y (ℂ)_{an}. Because the sheaves { \atop 𝒜^{n}(Y (ℂ), ℝ)} _{Z} are flabby, we thus have

RΓ(Y (ℂ)_{an}, ℝ_{Z}) ≃ { \atop A^{•}(Y (ℂ), ℝ)}_{Z};

since RΓ(Z(ℂ)_{an}, ℝ) ≃ RΓ(Y (ℂ)_{an}, ℝ_{Z}), this gives

RΓ(Z(ℂ)_{an}, ℝ) ≃ RC^{∞}(Z_{dR}, ℝ).

In order to make this quasi-isomorphism natural in Z, we would need to choose Y functorially. However, passing to filtered colimits allows the construction to work for any pro-smooth affine scheme Y, and we can just take Y = Spec Symm_{ℂ}O(Z). □

Corollary 2.4. For any spectrum-valued presheaf F on Aff_{ℂ} which is locally of finite presentation, we have

RC^{∞}(F_{dR}, ℝ) ≃ \underline{Hom}_{ℝ}((|F |_{𝕊})_{ℝ}, ℝ),

for Blanc’s functor | − |_{𝕊} from [Bla, Definition 3.13].

Proof. The functor | − |_{𝕊} is defined as the derived enriched left Kan extension of the singular space functor ssp: Aff_{ℂ} → sSet along the Yoneda embedding of Aff_{ℂ} in spectral presheaves. Since F is locally of finite presentation, we can calculate |F |_{𝕊} as the enriched left Kan extension along the Yoneda embedding of Aff_{ℂ,fin}, either by the reasoning of Lemma 1.10 or because the comma category (Aff_{ℂ} ↓F) is equivalent to pro(Aff_{ℂ,fin} ↓F), so has (Aff_{ℂ,fin} ↓F) as a final subcategory.

Now, observe that F ↦ F_{dR} is the enriched derived left Kan extension of Z ↦ Z_{dR} along the Yoneda embedding, so Proposition 2.3 combines with the definition of RC^{∞} to show that

RC^{∞}(F_{dR}, ℝ) ≃ ∫_{Z∈Aff_{ℂ,fin}}^{h} \underline{Hom}_{ℝ}(F (Z)_{ℝ}, RC^{∞}(Z_{dR}, ℝ))

≃ ∫_{Z∈Aff_{ℂ,fin}}^{h} \underline{Hom}_{ℝ}(F (Z)_{ℝ} ⊗_{ℝ} NC_{•}(ssp(Z), ℝ), ℝ)
≃ \underline{Hom}_{ℚ}((|F |_{𝕊})_{ℝ}, ℝ),

where ∫^{h} denotes homotopy end in the category of real chain complexes. □

Corollary 2.5. For any complex dg category A, there is a canonical zigzag

RC^{∞}(\underline{K}(A)_{dR}, ℝ) ≃ \underline{Hom}_{ℝ}(K^{Blanc,st}(A)_{ℝ}, ℝ)

of quasi-isomorphisms, where K^{Blanc,st}(A) is Blanc’s semi-topological K-theory.

Proof. The K-theory presheaf \underline{K}(A) is locally of finite presentation, and by [Bla, Definition 4.1], |\underline{K}(A)|_{𝕊} = K^{Blanc,st}(A). □

Remark 2.6. Beware that Blanc’s semi-topological K-theory is not the same as that of Friedlander and Walker. However, it shares the property [Bla, Definition 4.13 and Proposition 4.32] that inverting the Bott element β gives a weak equivalence between K^{Blanc,st}(X)[β^{−1}] and the topological K-theory of X(ℂ)_{an} for any separated ℂ-scheme of finite type.

Remark 2.7. For any affine ℂ-scheme Z, a model for the associated simplicial set ssp(Z) is given by n ↦ C^{∞}(Δ_{ℝ}^{n}, Z(ℂ)) = Z(C^{∞}(Δ_{ℝ}^{n}, ℂ)), where we write Δ_{ℝ}^{n} for the hyperplane \{∑ x_{i} = 1\} in ℝ^{n+1}. In other words, ssp(Z) ≃ ho\underrightarrow{lim}_{n∈Δ^{opp}} Z(C^{∞}(Δ_{ℝ}^{n}, ℂ)). For any spectral presheaf F on Aff_{ℂ}, we thus have

|F |_{𝕊} ≃ ho \underrightarrow{lim}_{n∈Δ^{opp}} F (C^{∞}(Δ_{ℝ}^{n}, ℂ))

by left Kan extension, and in particular

K^{Blanc,st}(A) ≃ ho \underrightarrow{lim}_{n∈Δ^{opp}} K(A ⊗_{ℝ} C^{∞}(Δ_{ℝ}^{n}, ℝ)).

There is thus a close analogy between Blanc’s semi-topological K-theory and Weibel’s homotopy K-theory [Wei1], the latter being

KH(A) ≃ ho \underrightarrow{lim}_{n∈Δ^{opp}} K(A ⊗_{ℤ} ℤ[t_{0}, …, t_{n}]/(1 − ∑ t_{i})).

Thus K^{Blanc,st} should perhaps be thought of as the stabilisation of K with respect to homotopies for the manifold ℝ, in the same way that KH is the stabilisation of K with respect to 𝔸^{1}-homotopies.

3. Cyclic homology and Goodwillie’s comparison

3.1. Goodwillie’s comparison.

Definition 3.1. Given a commutative ring k, write dAlg_{k} for the category of associative, not necessarily commutative dg algebras concentrated in non-negative chain degrees. Given A ∈ dAlg_{k}, write dAlg_{A} for the comma category (A↓dAlg_{k}).

Definition 3.2. Given a commutative ring k and a dg category A over k, write HC^{k}(A) for the chain complex associated to cyclic homology of A over k.

Definition 3.3. Given a functor F from dg categories to chain complexes, and a dg functor f: A → B, write F (f) for the homotopy fibre of F (A) → F (B).

Theorem 3.4. If f: A → B is a morphism in dAlg_{ℤ} for which H_{0}A → H_{0}B is a surjection with nilpotent kernel, then there is a canonical zigzag K(f)_{ℚ} ≃ HC^{ℚ}(f ⊗_{ℤ} ℚ)_{[−1]} of weak equivalences between chain complexes given by the Chern character.

Proof. Using the Quillen equivalence between simplicial algebras and dg algebras, the main theorem of [Goo] gives K̃(f)_{ℚ} ≃ HC^{ℚ}(f ⊗_{ℤ} ℚ)_{[−1]} for connective K-theory K̃.

Since the morphisms A[t] → B[t] and A[t, t^{−1}] → B[t, t^{−1}] are also nilpotent surjections on H_{0}, iterated Bass delooping then extends the comparison to non-connective K-theory via [Sch, Theorem 7.1].

More explicitly, a resolution of the good truncation (τ_{≥n−1}K(f)_{ℚ})_{[−1]} is given by the total complex of

τ_{≥n}K(f)_{ℚ} → τ_{≥n}K(f[t])_{ℚ} ⊕ τ_{≥n}K(f[t^{−1})_{ℚ} → τ_{≥n}K(f[t, t^{−1}])_{ℚ},

and K̃ = τ_{≥0}K. Meanwhile, the total complex of

HC^{ℚ}(f ⊗_{ℤ} ℚ) → HC^{ℚ}(f ⊗_{ℤ} ℚ[t]) ⊕ HC^{ℚ}(f ⊗_{ℤ} ℚ[t^{−1}]) → HC^{ℚ}(f[t, t^{−1}] ⊗_{ℤ} ℚ)

is quasi-isomorphic to HC^{ℚ}(f ⊗_{ℤ} ℚ)_{[−1]}, because cohomology of projective space gives HC^{ℚ}(f ⊗_{ℤ} ℙ_{ℚ}^{1}) ≃ HC^{ℚ}(f ⊗_{ℤ} ℚ) ⊕ HC^{ℚ}(f ⊗_{ℤ} ℚ). Inductively comparing these total complexes then gives

τ_{≥n}K(f)_{ℚ} ≃ HC^{ℚ}(f ⊗_{ℤ} ℚ)_{[−1]}

for all n ≤ 0. □

Corollary 3.5. For any A ∈ dAlg_{ℂ}, there is a canonical zigzag of weak equivalences

cone(\underline{K}(A)_{ℚ} → \underline{K}(A)_{dR,ℚ}) ≃ cone(\underline{HC}^{ℚ}(A) → \underline{HC}^{ℚ}(A)_{dR})_{[−1]}

of presheaves in Ch_{ℚ}(Aff_{ℂ}) given by the Chern character.

Proof. For any B ∈ CAlg(ℂ)_{fin}, the morphism B → B^{red} has nilpotent kernel. Thus A ⊗_{ℂ} B → A ⊗_{ℂ} B^{red} satisfies the conditions of Theorem 3.4, giving

cone(\underline{K}(A)(B)_{ℚ} → \underline{K}(A)_{dR}(B)_{ℚ}) ≃ cone(\underline{HC}^{ℚ}(A)(B) → \underline{HC}^{ℚ}(A)_{dR}(B))_{[−1]}.

The result then extends to the whole of CAlg(ℂ) by passing to filtered colimits. □

Zariski descent allows us to extend this far more generally:

Corollary 3.6. If X is a quasi-compact semi-separated derived scheme over ℂ, with ℰ ∈ dAlg_{𝒪_{X}} perfect as an 𝒪_{X}-module, and A is a semi-orthogonal summand of per_{dg}(ℰ), the Chern character gives

cone(\underline{K}(A)_{ℚ} → \underline{K}(A)_{dR,ℚ}) ≃ cone(\underline{HC}^{ℚ}(A) → \underline{HC}^{ℚ}(A)_{dR})_{[−1]}.

Proof. A derived scheme X is a derived n-geometric stack whose underived truncation π^{0}X is a scheme. The quasi-compactness hypothesis for π^{0}X means that there is a Zariski cover U_{1}, …, U_{n} for X by derived affines, so each O(U_{i}) is a commutative object of dAlg_{ℂ}. Now take the reduced Čech resolution X̌ for X, given by

… ∐_{i<j<k} U_{i} ×_{X}^{h} U_{j}× ⇛ ∐_{i<j} U_{i} ×_{X}^{h} U_{j} ⟹ ∐_{i} U_{i};

this diagram terminates at U_{1} ×_{X}^{h} U_{2} ×_{X}^{h} … ×_{X}^{h} U_{n} and since X is semi-separated, it is a diagram of derived affines.

Zariski descent for K then gives a resolution of \underline{K}(per_{dg}(ℰ)) by

\underline{K}(ℰ (X̌_{0})) → \underline{K}(ℰ (X̌_{1})) → … → \underline{K}(ℰ (X̌_{n})),

so Corollary 3.5 implies the required result when A = per_{dg}(ℰ).

Finally, for a A a semi-orthogonal summand of per_{dg}(ℰ) as in [Orl, §1.2], additivity of K gives \underline{K}(A)_{ℚ} as a summand of \underline{K}(per_{dg}(ℰ)); likewise, additivity of HC gives \underline{HC}^{ℚ}(A) as a summand of \underline{HC}^{ℚ}(per_{dg}(ℰ)), so the equivalence extends to A. □

Remarks 3.7. Beware that the hypothesis ℰ ∈ dAlg_{𝒪_{X}} implies that ℋ_{i}(ℰ) = 0 for i < 0, so is stronger than the customary cohomological boundedness condition for derived non-commutative schemes in [Orl, Definition 3.3]. However, Corollary 3.6 applies to all quasi-compact semi-separated derived schemes X (taking ℰ = 𝒪_{X}) and to all of Orlov’s geometric noncommutative schemes of [Orl, Definition 4.3].

The conclusion of Corollary 3.6 will also hold for any complex dg category A Morita equivalent to one satisfying the conditions, and to any filtered colimit of such. This applies for instance if A is a dg category with H_{i}A(x, y) = 0 for all i < 0 and all objects x, y; it is a filtered colimit of its dg subcategories A|_{S} on finite subsets S of objects, and each A|_{S} is Morita equivalent to τ_{≥0}(⨁_{x,y∈S} A(x, y)) ∈ dAlg_{ℂ}. By Morita invariance, the conclusions thus hold for any complex dg category A for which per_{dg}(A) has a set of generators with no positive Ext-groups between them; this amounts to having what Bondarko calls a weight structure (whereas a t-structure means having no negative Ext-groups between generators).

3.2. Smooth functions on smash products of Eilenberg–MacLane spectra.

Proposition 3.8. Given a cosimplicial real vector space V with H^{0}V = 0, finitedimensional in each level, the morphism

Symm_{ℝ}V → C^{∞}(V^{∨}, ℝ)

of cosimplicial vector spaces is a quasi-isomorphism.

Proof. Let Q be the quotient of Symm_{ℝ}V → C^{∞}(V^{∨}, ℝ); we wish to show that this is acyclic. Now consider the algebraic and smooth de Rham complexes respectively, giving morphisms

Ω_{ℝ}^{•}(Symm_{ℝ}V) → A^{•}(V^{∨}, ℝ)

of cosimplicial cochain complexes. Each of these cochain complexes is a resolution of ℝ, so the quotient T is levelwise acyclic as a cosimplicial cochain complex, and in particular H^{∗}(Tot T) = 0.

Taking the filtration on T given by brutal truncations in the cochain direction, we get a convergent spectral sequence

H^{i}(Q ⊗ Λ^{j}V) ⟹ H^{i+j}(Tot T) = 0.

If Q is not acyclic, let m be least such that H^{m}(Q) ≠ 0. Since H^{0}V = 0, the Eilenberg– Zilber theorem [Wei2, Theorem 8.5.1] gives H^{<m+j}(Q ⊗ Λ^{j}V) = 0; therefore we must have H^{m}(Tot T) ≠ 0, which is a contradiction. □

Definition 3.9. Given V ∈ Ch_{ℂ}, define \underline{V} ∈ Ch_{ℚ}(Aff_{ℂ}) by \underline{V} := V ⊗_{ℂ} 𝔾_{a}. Explicitly, this presheaf is given by \underline{V} (Z) := V ⊗_{ℂ} O(Z).

Definition 3.10. For any chain complex V, we may form the Eilenberg–MacLane spectrum HV by (HV)_{n} := N^{−1}τ_{≥0}(V_{[−n]}), where N^{−1} is Dold–Kan denormalisation. For V ∈ Ch_{ℂ}, we therefore obtain a spectrum \underline{HV} in simplicial ind-affine ℂ-schemes, which we can regard as a spectral presheaf on Aff_{ℂ}.

Corollary 3.11. For V^{(1)}, …, V^{(r)} ∈ Ch_{ℂ}, there is a canonical zigzag of quasiisomorphisms

RC^{∞}(\underline{HV}^{(1)} ∧ … ∧ \underline{HV}^{(r)}, ℝ) ≃ \underline{Hom}_{ℝ}(V^{(1)} ⊗_{ℝ} … ⊗_{ℝ} V^{(r)}, ℝ)

of real chain complexes.

Proof. Since RC^{∞}(−, ℝ) sends filtered colimits to filtered limits, we may reduce to the case where the complexes V^{(1)}, …, V^{(r)} all have finite total dimension. Now, the spectra HV and H′V := {N^{−1}τ_{≥1}(V_{[−n]})}_{n} are stably equivalent, and the cosimplicial vector spaces (H′V^{(1)} × … × H′V^{(r)})_{n}^{∨} all satisfy the conditions of Proposition 3.8.

Write

C̄^{∞}(X_{1}∧…∧X_{r}, ℝ) := ker(C^{∞}(X_{1}×…×X_{r}, ℝ) → ∏_{i} C^{∞}(X_{1}×…×{x_{i}}×…×X_{r}, ℝ))

for pointed manifolds (X_{i}, x_{i}), and note that this has a cochain resolution by products of vector spaces of the form C^{∞}((∏_{j∈S} X_{j}) × ∏_{i∉S}{x_{i}}, ℝ)) for subsets S ⊂ [1, n].

Substituting in Definition 1.18 then gives

RC^{∞}(\underline{HV}^{(1)} ∧ … ∧ \underline{HV}^{(r)}, ℝ)

≃ \underleftarrow{lim}_{n} C̄^{∞}((HV^{(1)})_{n} ∧ … ∧ (HV^{(r)})_{n}, ℝ)_{[−nr]},

≃ \underleftarrow{lim}_{n}(Symm_{ℝ}^{+}τ_{≤0}(V_{[−n]}^{(1)})^{∨}) ⊗_{ℝ} … ⊗_{ℝ} (Symm_{ℝ}^{+}τ_{≤0}(V_{[−n]}^{(r)})^{∨})_{[−nr]},

≃ V (1)^{∨} ⊗_{ℝ} … ⊗_{ℝ} V (r)^{∨},

≃ \underline{Hom}_{ℝ}(V (1) ⊗_{ℝ} … ⊗_{ℝ} V (r), ℝ),

where Symm^{+}V = ⨁_{i>0} Symm^{i}V. □

Corollary 3.12. For V^{(1)}, …, V^{(r)} ∈ Ch_{ℂ}, there is a canonical zigzag of quasiisomorphisms

LSm^{∗}(\underline{V}^{(1)} ⊗_{ℚ} … ⊗_{ℚ} \underline{V}^{(r)})^{∨} ≃ \underline{Hom}_{ℝ}(V^{(1)} ⊗_{ℝ} … ⊗_{ℝ} V^{(r)}, ℝ)

of real chain complexes.

Proof. The spectral presheaf \underline{HV} on Aff_{ℂ} has the important property that \underline{HV}_{ℚ} ≃ \underline{V}; moreover we have

(\underline{HV}^{(1)} ∧ … ∧ \underline{HV}^{(r)})_{ℚ} ≃ \underline{V}^{(1)} ⊗_{ℚ} … ⊗_{ℚ} \underline{V}^{(r)},

so

LSm^{∗}(\underline{V}^{(1)} ⊗_{ℚ} … ⊗_{ℚ} \underline{V}^{(r)})^{∨} ≃ RC^{∞}(\underline{HV}^{(1)} ∧ … ∧ \underline{HV}^{(r)}, ℝ),
≃ \underline{Hom}_{ℝ}(V^{(1)} ⊗_{ℝ} … ⊗_{ℝ} V^{(r)}, ℝ).
□

We will also need to look at smooth functions on de Rham presheaves:

Proposition 3.13. For V^{(1)}, …, V^{(r)} ∈ Ch_{ℂ}, there are canonical zigzags of quasiisomorphisms

LSm^{∗}(\underline{V}^{(1)} ⊗_{ℚ} … ⊗_{ℚ} \underline{V}^{(r)})_{dR}^{∨} ≃ RC^{∞}((\underline{HV}^{(1)} ∧ … ∧ \underline{HV}^{(r)})_{dR}, ℝ) ≃ 0

of real chain complexes.

Proof. We have (HA^{(1)} ∧ … ∧ HA^{(r)})_{ℚ} ≃ A^{(1)} ⊗_{ℤ} … ⊗_{ℤ} A^{(r)} ⊗_{ℤ} ℚ for any chain complexes A^{(i)} of abelian groups, which implies the first equivalence. As in the proof of Corollary 3.11, the second equivalence reduces to calculating spaces of the form C̄^{∞}((\underline{U}^{(1)} × … × \underline{U}^{(r)})_{dR}, ℝ) for finite-dimensional complex vector spaces U^{(i)}. But by Proposition 2.3, these are all quasi-isomorphic to ℝ, so C̄^{∞}((\underline{U}^{(1)} ∧ … ∧ \underline{U}^{(r)})_{dR}, ℝ) ≃ 0. □

3.3. Smooth functions on cyclic homology.

Theorem 3.14. Given a dg category A over ℂ, there are canonical zigzags of quasiisomorphisms

LSm^{∗}(\underline{HC}^{ℚ}(A))^{∨} ≃ \underline{Hom}_{ℝ}(HC^{ℂ}(A), ℝ)
LSm^{∗}(\underline{HC}^{ℚ}(A)_{dR})^{∨} ≃ 0.

of real chain complexes.

Proof. Since ℂ is étale over ℝ, the map HC^{ℝ}(A) → HC^{ℂ}(A) is a quasi-isomorphism. Because \underline{HC}^{ℚ}(A) is formed from homotopy colimits of rational tensor products of the complexes \underline{A(x, y)} for objects x, y ∈ A, Corollary 3.12 immediately then implies the first equivalence, and Proposition 3.13 the second. □

Corollary 3.15. If X is a quasi-compact semi-separated derived scheme over ℂ, with ℰ ∈ dAlg_{𝒪_{X}} perfect as an 𝒪_{X}-module, and A is a semi-orthogonal summand of per_{dg}(ℰ), then the complex RC^{∞}(\underline{K}(A), ℝ) of smooth functions is given by the cone of the Chern character

\underline{Hom}_{ℝ}(HC^{ℂ}(A)_{[−2]}, ℝ) ⟶^{ch^{∗}} \underline{Hom}_{ℝ}(K^{Blanc,st}(A)_{ℝ}, ℝ),

so

RC^{∞}(\underline{K}(A), ℝ) ≃ \underline{Hom}_{ℝ}(HN^{ℂ}(A) ×_{HP^{ℂ}(A)}^{h} K^{Blanc,st}(A)_{ℝ}, ℝ).

Proof. The first statement just combines Corollaries 2.5 and 3.6 with Theorem 3.14. The second statement then follows because HC_{[−2]} is the cone of the map HN → HP from negative cyclic homology to periodic cyclic homology. □

See Remarks 3.7 for examples satisfying these hypotheses.

Remark 3.16. The expression HN^{ℂ}(A) ×_{HP^{ℂ}(A)}^{h} K^{Blanc,st}(A)_{ℝ} (or equivalently the ho-

motopy fibre of ch: K^{Blanc,st}(A)_{ℝ} → HC^{ℂ}(A)_{[−2]}) is very closely related to real Deligne cohomology. If X is a separated ℂ-scheme of finite type, then K^{Blanc,top}(X) := K^{Blanc,st}(X)[β^{−1}] is the topological K-theory of X(ℂ)_{an}.

For a smooth separated ℂ-scheme X, the HKR isomorphism then identifies HN^{ℂ}(X) ×_{HP^{ℂ}(A)}^{h} (K^{Blanc,st}(A)_{ℝ}[β^{−1}]) with

∏_{p∈ℤ} cocone(RΓ(X(ℂ)_{an}, ℝ(2πi)^{p}) ⟶^{ch} RΓ(X, 𝒪_{X} → … → Ω_{X/ℂ}^{p−1}))^{[2p]}

identifying chain and cochain complexes in the obvious way; when X is smooth and proper, this is just real Deligne cohomology ∏_{p∈ℤ} RΓ_{D}(X, ℝ(p))^{[2p]}.

There does not yet seem to be a simple description of K^{Blanc,st}(X), but for smooth affines it is connective by [Bla, Theorem 4.6], and for products of projective spaces it is the connective truncation of topological K-theory by [Bla, Theorems 4.5 and 4.6] and [TT, Theorem 7.3]. Taking Gysin sequences, this means that for any complement X of a product of projective spaces by a disjoint union of products of projective spaces (the simplest interesting examples are X = 𝔸^{1}, 𝔾_{m}), we have K^{Blanc,st}(X)_{ℚ} ≃ ∏_{p≥0} W_{0}RΓ(X(ℂ)_{an}, ℚ(p))[2p], for Deligne’s weight filtration W.

Remark 3.17. For Z ∈ Aff_{ℂ}, there is a natural morphism C^{∞}(Z, ℝ) → ℝ^{Z(ℂ)} from smooth functions to discontinuous functions. The spectral derived left Kan extension of Z ↦ (ℝ^{Z(ℂ)})^{∨} is just given by F ↦ \underline{Hom}_{ℝ}(F (Spec ℂ)_{ℝ}, ℝ)^{∨}, so there is a natural map RC^{∞}(F, ℝ) → \underline{Hom}_{ℝ}(F_{ℝ}(∗), ℝ). Similarly, we have RO^{∞}(F) → \underline{Hom}_{ℝ}(F_{ℝ}(∗), ℝ) for spectral presheaves on C^{∞}Aff.

Applied to \underline{K}(A), these give RC^{∞}(\underline{K}(A), ℝ) → \underline{Hom}_{ℝ}(K(A)_{ℝ}, ℝ), which recovers Beilinson’s regulator [Bei] on duals when applied in the context of Remark 3.16.

3.4. Related constructions and generalisations.

3.4.1. Smaller diagram categories. As observed in §1.2.2, the calculation of RC^{∞}(\underline{K}(A), ℝ) can be made by looking at the restricted presheaf \underline{K}(A)|_{C}:

RC^{∞}(\underline{K}(A), ℝ) ≃ RC^{∞}(\underline{K}(A)|_{C}, ℝ),

taking C to be any of the categories Aff_{ℂ}, Aff_{ℂ,fin}, C^{∞}Aff, C^{∞}Aff_{fin}. The same is true for K^{Blanc,st}(A), which [Bla, Theorem 3.18] shows can also be recovered by restricting to the category of smooth complex affines. By Remark 2.7 we can even recover K^{Blanc,st}(A) from its restriction to the simplex category {Δ_{ℝ}^{n} ≅ ℝ^{n}}_{n} living inside C^{∞}Aff_{fin}.

However, calculations involving \underline{K}(A)_{dR} only make sense in a category having nonreduced objects. The smallest choice of category C for which Corollary 2.5, Theorem 3.14 and Corollary 3.15 would still hold is probably the full subcategory of ind(C^{∞}Aff) consisting of objects of the form ℝ^{m} × ℝ̂^{n}_{0}, with associated pro-C^{∞}-ring C^{∞}(ℝ^{m}, ℝ)[[t_{1}, …, t_{n}]]. Theorem 3.14 still holds because (ℝ^{m})_{dR} has a simplicial resolution by such spaces. Corollary 2.5 still holds because Hom_{C}(Δ_{ℝ}^{•}, −) is a resolution of the constant copresheaf ∗ on C, while C^{∞}((−)_{dR}, V) is quasi-isomorphic to the constant presheaf V.

3.4.2. Algebraic functions on algebraic K-theory. Instead of looking at the Kan extension of smooth functions Z ↦ C^{∞}(Z, ℝ) applied to \underline{K}(A), we could look at the Kan extension of algebraic functions Z ↦ Γ(Z, 𝒪). The proof of Corollary 2.5 adapts to give RΓ(\underline{K}(A)_{dR}, 𝒪) ≃ \underline{Hom}_{ℂ}(K^{Blanc,st}(A)_{ℂ}, ℂ), while Theorem 3.14 adapts without needing to appeal to Proposition 3.8. The analogue of Corollary 3.15 then gives RΓ(\underline{K}(A)_{dR}, 𝒪) dual to the homotopy fibre of ch: K^{Blanc,st}(A)_{ℂ} → HC^{ℂ}(A)_{[−2]}.

3.4.3. Refinements for non-proper dg categories. As in §1.2.2, the smooth functions RC^{∞}(\underline{K}(A), ℝ) are recovered from the presheaf \underline{K}(A)|_{C^{∞}Aff}. This presheaf is only really a sensible object of study for proper dg categories, because for manifolds Z, the map

C^{∞}(Z, ℝ) ⊗_{ℝ} A(x, y) → C^{∞}(Z, A(x, y))

only tends to be a quasi-isomorphism when A(x, y) is a perfect complex.

When cyclic homology is finite-dimensional, it can be recovered from Theorem 3.14 by taking duals. If we wanted to refine the comparison of Theorem 3.14 to recover cyclic homology more generally, we would instead have to work with ind-objects, looking at the left Kan extension of the composition (O^{∞})^{∨}: C^{∞}Aff → Ch(pro(FDVect_{ℝ})) → ind(Ch(pro(FDVect_{ℝ}))).

A slightly more natural choice would be the left Kan extension of functor (O^{∞})^{∨}: ind(C^{∞}Aff) → ind(Ch(pro(FDVect_{ℝ}))), applied to the presheaf

\underline{K}(A)(\underleftarrow{lim}_{α} B_{α}) := K(ho\underleftarrow{lim}_{α} A ⊗_{ℝ} B_{α})

on ind(C^{∞}Aff).

However, since Deligne cohomology for non-proper schemes is not defined in terms of their cyclic homology, there is less motivation for seeking such a refinement.

3.4.4. K-theory of dg categories enriched in topological vector spaces. In [Pri], we considered certain Fréchet algebras and their associated algebraic K-theory presheaves on a category of Fréchet manifolds, and thus obtained a comparison between real Deligne cohomology and smooth functions on K-theory. The setting of this paper is inadequate to recover such a result because C^{∞}-rings only model smoothly realcompact spaces.

As in [KM], the largest class of topological vector spaces for which it makes sense to talk about C^{∞}-morphisms consists of the convenient vector spaces. There is as natural monoidal structure on the category of convenient vector spaces, given by c^{∞}-completions ˆ⊗ of bornological tensor products — by [KM, Proposition 5.8], these operations coincide with projective tensor products on the subcategory of Fréchet spaces.

For any category A enriched in chain complexes of convenient ℂ-vector spaces, and any infinite-dimensional manifold M modelled on convenient vector spaces, there is then a dg category C^{∞}(M, A) over ℂ.

To adapt the methods of this paper to such a setting, we need to incorporate nonreduced objects in order to define meaningful de Rham presheaves. The obvious generalisation of C^{∞}-rings might be the convenient coalgebras of [KM, 23.13], but these do not seem to have a suitable analogue of the nilradical ideal. Instead, we can take a lead from §3.4.1 and just consider formal completions of manifolds. Let J be the category of products T × V̂_{0} for convenient vector spaces T, V, with morphisms

Hom_{J} (S × Û_{0}, T × V̂_{0}) = C^{∞}(S, J_{0}^{∞}(U, T) × Poly^{∞}(U, V))
= C^{∞}(S, J_{0}^{∞}(U, T) × J_{0}^{∞}(U, V)_{0}),

for the jet spaces J^{∞} of [KM, 41.1]. The following reasoning will apply equally well if we allow T to be a star-shaped c^{∞}-open in a convenient vector space.

Given a convenient dg category A, we have a presheaf \underline{K}(A) defined on J by

\underline{K}(A)(T × V̂_{0}) := ho\underleftarrow{lim}_{n} K(C^{∞}(T, J_{0}^{n}(V, A))),

with \underline{K}(A)_{dR}(T × V̂_{0}) := \underline{K}(A)(T) = K(C^{∞}(T, A)).

The Poincaré lemma of [KM, 33.20] allows the proofs of Corollary 2.5 and Theorem 3.14 to adapt to this context, replacing tensor products ⊗_{ℝ} with their bornological c^{∞}- completions ˆ⊗. Moreover, the analogues of Corollary 2.5 and Theorem 3.14 hold for smooth functions C^{∞}(−, V) with coefficients in any convenient vector space V.

As in [KM, 23.13], there is a free convenient vector space functor λ from Frölicher spaces to convenient vector spaces, with the property that the space L(λX, V) of bounded linear maps is isomorphic to C^{∞}(X, V). For finite-dimensional manifolds M, λM = C^{∞}(M, ℝ)′, the space of compactly supported distributions on M. The functor

λ extends naturally to J by setting λ(T × V̂_{0}) := (λT) ˆ⊗(⨁_{i≥0} { \atop Symm}_{ℝ}^{i}V).

We can then consider the derived enriched left Kan extension Lλ of λ along the Yoneda embedding of J in spectral presheaves.

Observe that if A is a proper dg category over ℂ, the maps A ⊗_{ℝ} C^{∞}(T, J_{0}^{n}(V, ℝ)) → C^{∞}(T, J_{0}^{n}(V, A)) are quasi-equivalences, so Lλ\underline{K}(A) can be recovered from the presheaf \underline{K}(A)|_{C^{∞}Aff} considered throughout this paper, and its complex of bounded functionals satisfies Lλ\underline{K}(A)′ = L(Lλ\underline{K}(A), ℝ) ≃ RC^{∞}(\underline{K}(A)|_{Aff_{ℂ}}, ℝ).

For all convenient dg categories A over ℂ, Corollary 2.5 and Remark 2.7 adapt to give

Lλ\underline{K}(A)_{dR} ≃ Tot K(C^{∞}(Δ_{ℝ}^{•}, A))_{ℝ},

a chain homotopy equivalence of chain complexes of convenient vector spaces, where the right-hand side is given the finest ℝ-linear topology.

The analogue of Corollary 3.15 is then that for convenient dg ℂ-algebras A concentrated in non-negative chain degrees, we have a chain homotopy equivalence

Lλ\underline{K}(A) ≃ cocone(Tot K(C^{∞}(Δ_{ℝ}^{•}, A))_{ℝ} ⟶^{ch} \begin{matrix} \\ HC & ℂ \end{matrix}(A)_{[−2]}),

where \begin{matrix} \\ HC & ℂ \end{matrix} is defined by analogy with cyclic homology, but using ˆ⊗_{ℂ} instead of ⊗_{ℂ}.

Dually to Remark 3.17, the morphism T → λ(T × V̂_{0}) of sets which sends points to distributions induces a morphism

K(A)_{ℝ} → Lλ\underline{K}(A)

on Kan extensions, which we can think of as a generalisation of Beilinson’s regulator.

References

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