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dg-geometry

Contents

Idea

This entry describes the higher geometry/derived geometry modeled on (∞,1)-sites of formal duals of dg-algebras, bounded or unbounded, over a field of characteristic 0.

The corresponding (∞,1)-topos is the context for classical rational homotopy theory, which arises by forming function algebras on ∞-stacks over constant ∞-stacks. It is also the context in which classical and higher order Hochschild homology of algebras and dg-algebras arises naturally as the function \infty-algebra on free loop space objects.

The (,1)(\infty,1)-toposes

We discuss some basic aspects of the (∞,1)-toposes over (∞,1)-sites of formal duals of cdg-algebras and of cdg-algebras of functions on its objects.

Over formal duals of non-positively graded cdg-algebras

Let kk be a field of characteristic 0, or more generally a commutative \mathbb{Q}-algebra.

Definition

Proposition/Definition

Write

  • cdgAlg kcdgAlg_k for the category of graded-commutative cochain dg-algebras (meaning: with differential of degree +1) in arbitrary degree;

  • cdgAlk k cdgAlk_k^- for the full subcategory on objects with vanishing cochain cohomology in positive degree, H 1()=0H^{\bullet \geq 1}(-) = 0.

There are the standard projective model structures on dg-algebras on these categories, whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degreewise surjections.

This is considered in (Toën-Vezzosi, 2.3.1)

Proposition/Definition

Let

C((cdgAlg k ) op) C \hookrightarrow ((cdgAlg_k^-)^{op})^\circ

be a small full sub-(∞,1)-category of the (∞,1)-category presented by this model structure, and let CC be equipped with the structure of a subcanonical (∞,1)-site.

Write

H:=Sh (,1)(C) \mathbf{H} := Sh_{(\infty,1)}(C)

for the (∞,1)-category of (∞,1)-sheaves on CC. We have a derived Isbell duality

(𝒪j):(cdgAlg k op) j𝒪H (\mathcal{O} \dashv j) : (cdgAlg_k^{op})^\circ \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} \mathbf{H}

where the left adjoint (∞,1)-functor 𝒪\mathcal{O} is the Yoneda extension of the inclusion cdgAlg k +cdgAlg kcdgAlg^+_k \hookrightarrow cdgAlg_k.

This is considered in (Ben-ZviNadler). See function algebras on ∞-stacks for details.

Properties

Proposition

The inclusion

cdgAlg k cdgAlg k cdgAlg^-_k \hookrightarrow cdgAlg_k

is a homotopical context in the sense of (ToënVezzosi, def. 1.1.0.11).

This is (ToënVezzosi, lemma 2.3.11). We record the following implications of this statement

Corollary

(cdgAlg k, k)(cdgAlg_k, \otimes_k) is a symmetric monoidal model category.

Corollary

For B(dgcAlg k) projB \in (dgcAlg_k)_{proj} a cofibrant object, the tensor product with BB preserves weak equivalences.

This follows from (ToënVezzosi, assumption 1.1.0.4).

Corollary

The inclusion

(cdgAlg k ) op(cdgAlg k) op (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

preserves homotopy limits, hence the induced inclusion

((cdgAlg k ) op) ((cdgAlg k) op) ((cdgAlg_k^-)^{op})^\circ \hookrightarrow ((cdgAlg_k)^{op})^\circ

preserves (∞,1)-limits.

This follows from (ToënVezzosi, assumption 1.1.0.6).

Definition

For AcdgAlg kA \in cdgAlg_k a dg-algebra, write

Corollary

For any AcdgAlg kA\in cdgAlg_k say a morphism in cdgAlg AcdgAlg_A is

  • a weak equivalence precisely if it is a quasi-isomorphism;

  • a fibration precisely if it is degreewise surjective.

This makes cdgAlg AcdgAlg_A into a model category that is

There is an equivalence of categories with the under category of cdg-algebras under AA

cdgAlg AA/cdgAlg k cdgAlg_A \simeq A/cdgAlg_k

which is a Quillen equivalence with respect to the standard model structure on an under category on the right.

This is (ToënVezzosi, assumption 1.1.0.4, remark on p. 18).

Corollary

For BcdgAlg AB \in cdgAlg_A cofibrant with respect to the model structure in cor 4, the tensor product (base change) functor

B A():AModBMod B \otimes_A (-) : A Mod \to B Mod

preserves weak equivalences.

This is (ToënVezzosi, assumption 1.1.0.4).

Proposition

The monoidal Dold-Kan correspondence provides a Quillen equivalence

(Γ cmonN ):cAlg k Δ opN Γ cmoncdgAlg k + (\Gamma^{cmon} \dashv N_\bullet) : cAlg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}} cdgAlg_k^+

(since kk is assumed to be of characteristic 0). Under this equivalence we have that UcAlg kcAlg k Δ opHU \in cAlg_k \hookrightarrow cAlg_k^{\Delta^{op}} \hookrightarrow \mathbf{H} is 𝒪\mathcal{O}-perfect:

𝒪(X K)K𝒪(X) \mathcal{O} (X^{K}) \simeq K \cdot \mathcal{O}(X)

and this recovers the constructions discussed above in The Hochschild chain complex of an associative algebra.

Proof

Since the (∞,1)-Yoneda embedding yy commutes with (∞,1)-limits we have that the powering (y(U)) Ky(U K)(y(U))^{K} \simeq y(U^K) is still representable. Therefore

𝒪((y(U)) K)𝒪(U K)(cdgAlg k ) op(cdgAlg k) op \mathcal{O} ((y(U))^K) \simeq \mathcal{O}(U^K) \;\; \in (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

is simply the formal dual of U KU^K, which is K𝒪(U)K \cdot \mathcal{O}(U) formed in cdgAlg kcdgAlg_k by formal duality. By the above proposition the inclusion cdgAlg k cdgAlg kcdgAlg_k^- \hookrightarrow cdgAlg_k preserves this (,1)(\infty,1)-colimit.

Over formal duals of general cdg-algebras

(…)

Applications

References

Various model category presentations of dg-geometry are presented in

The geometric ∞-function theory of perfect ∞-stacks in dg-geometry, and the corresponding Hochschild cohomology is considered in

The (𝒪Spec)(\mathcal{O} \dashv Spec)-adjunction for dg-geometry is studied in

The basic reference for the model structure on dg-algebras (see there for more details) for the commutative case over a field of characteristic 0 is

Details on the use of this model category structure for modelling dg-spaces are in

Revised on October 17, 2011 13:27:35 by Urs Schreiber (82.113.99.57)