Differential calculus

We now concentrate on algebra objects inside an∞-category Mas in the previous section. Given such an A ∈ M, we can define an ∞-category of A-modules in M, which can be realized as the localization along weak equivalences of the model category of A-modules in M, where we have considered Aas an algebra in M. Notice that sinceA is commutative, the category A−ModM of A-modules in Mis itself a symmetric monoidal stable∞-category.

Our next goal is to define the cotangent complex and the de Rham algebra of an algebra A ∈ CAlg_M. As usual, given an A-module N, we can consider the trivial square-zero extension A⊕N; this is again an object inCAlg_M, and comes moreover with a canonical projectionA⊕N →A, which is a morphism of commutative algebras in M.

We can the define the space Der(A, N) of derivations from A to N as in [HAG-II]: it is the space of dotted maps making the following diagram

A⊕N

A

;;

id //A commute incdga_M. This defines an∞-functor

Der(A,−) : A−ModM −→ sSet N 7−→ Der(A, N) This functor is corepresentable by an object, denotedL^int_A .

Definition 0.2.1. Given an algebra A ∈ CAlg_M, the object L^int_A ∈ M is the internal cotangent complex ofA. Its realization LA:=|L^int_A |is simply called the cotangent complex of A.

Remark that by general properties of the realization functor, LA is naturally an |A|-module in C(k). Moreover, in the simple case whereM =C(k), then the above definition coincides with the usual one.

We now pass to de Rham algebras. Given any graded mixed algebra B ∈ −cdga^gr_M, it is immediate to check that its weight zero partB(0)naturally inherits the structure of a commutative algebra inM. This defines an∞-functor

(−)(0) : −cdga^gr_M −→ cdga_M

B 7−→ B(0)

It turns out this functor has a left adjoint, which will be denoted DR^int: cdga_M −→−cdga^gr_M. We refer to [CPTVV], Section 1.3.2 for more details.

Definition 0.2.2. Given a commutative algebra A ∈cdga_M, the object DR^int(A) ∈−cdga^gr_M is the internal de Rham algebra ofA. Its realization DR(A) :=|DR^int(A)| ∈−cdga^gr_k will be simply called the de Rham algebra ofA.

The relation between the (internal) cotangent complex and the de Rham algebra is given by the following result, which is Proposition 1.3.12 in [CPTVV].

Proposition 0.2.3. For every A∈cdga_M, there is a natural equivalence Sym_A(L^intA [−1])−→DR^int(A)

in the ∞-category cdga^gr_M, which is moreover natural in A.

Notice that this is not an equivalence of graded mixed algebras, simply because a priori there is no mixed structure on Sym_A(L^int_A [−1]). One can actually look at this proposition as a way to induce a (weak) mixed structure on the left hand side, giving an abstract construction of the de Rham differential.

A similar construction of the internal cotangent complex and of the internal de Rham algebra applies to the relative context: starting with a morphism A→ B incdga_M, one can construct an object L^int_B/A which lives in B−ModM, together with a graded mixed algebra DR^int(B/A) ∈ − cdga^gr_M. Both graded algebrasSym_B(L^int_B/A[−1])andDR^int(B/A)comes equipped with a morphism fromA, considered as concentrated in weight 0. Then we have an equivalence

Sym_B(L^int_B/A[−1])−→DR^int(B/A)

in the comma∞-categoryA/cdga^gr_M. Again, we refer to Section 1.3.2 in [CPTVV] for more details on the relative version ofDR^int, which is however completely analogous to the absolute case.

0.2.1 Differential forms and polyvectors

Using the above formalism, we can define differential forms and polyvector fields for general com-mutative algebras in M.

Definition 0.2.4. Let A∈CAlg_M be a commutative algebra in M. Then the space of p-forms of degree nonA is the mapping space

A^p(A, n) := Map_M(1M[−n],∧^p_AL^intA ).

We define moreover the space of closedp-forms of degree nto be the mapping space A^p,cl(A, n) := Map_−Mgr(1M(p)[−n−p],DR^int(A)),

where 1M(p)[−n−p] is the monoidal unit of M viewed as a graded mixed object concentrated in weight p, with trivial mixed structure.

Notice that there is an induced map A^p,cl(A, n)→ A^p(A, n), which sends a closedp-form to the underlying p-form. By definition of realization functors, one has natural equivalences

A^p(A, n)'Map_C(k)(k[−n],∧^p_|A|LA)

A^p,cl(A, n)'Map_−dgMod^gr(k(p)[−n−p],DR(A))

We now pass to symplectic structures. LetA∈CAlg_M, and consider the∞-categoryA−Mod_M. This is a closed symmetric monoidal∞-category, so that in particular we can take dual ofA-modules.

The dual of an A-module N will be denoted byN^∨.

Definition 0.2.5. For A ∈ CAlg_M, the internal tangent complex of A is the A-module T^int_A :=

(L^int_A )^∨. Its realization TA=|T^int_A | ∈ |A| −Mod will be simply called the tangent complex of A.

In particular, if L^int_A is dualizable, any 2-form ω of degree n on A induces by adjunction a morphism

ω^]:T^int_A −→L^int_A [n].

Definition 0.2.6. Let A∈CAlg_M, and suppose that L^int_A is dualizable. Then we say that a closed 2-form ω ∈ A^2,cl(A, n) is non-degenerate if the morphism induced by the underlying 2-form gives an equivalences

T^intA 'L^intA [n]

of A-modules. We define the space of n-shifted symplectic structures on A Symp(A, n) to be the subspace of A^2,cl(A, n)given by the union of connected components of non-degenerate closed forms.

The definition of polyvector fields, which is in some sense the dual notion of differential forms, is a bit more delicate in this context. In particular, the ordinary lack of functoriality of the algebra of polyvector fields makes the definition in the∞-categorical world not entirely trivial.

Let us us thus start with an algebra Ain the model categoryM. We can defineT^(p)(A, n), the n-shiftedp-multiderivations onAas in section 1.4.2 of [CPTVV]. Note that this is in particular an object ofM. Moreover, the symmetric groupS_p acts naturally onT^(p):= (A, n), and let us denote by T^(p)(A, n)^Sp the object of M ofSp-invariants multiderivations.

Definition 0.2.7. Let A ∈ CAlg_M. The object of symmetric internal n-shifted multiderivations on A is

P ol^int(A, n) :=M

T^(p)(A, n)^Sp

Note that Pol^int(A, n) is in a natural way an object of M^gr. In addiction to this, there is a canonical multiplication of weight0inPol^int(A, n), and the composition by insertion gives rise to a shifted Lie structure, which has to be interpreted as an analogue of the classical Schouten-Nijenhuis bracket. These algebraic structures are compatible, so thatP ol^int(A, n)is a gradedPn+1-algebra in M: by this we mean that the commutative product has weight 0, while the Lie bracket has weight

−1.

Even if the association A7→P ol^int(A, n)is not completely functorial in A, it is still possible to define a restricted functoriality, as in section 1.4.2 of [CPTVV].

Recall that a morphism of f :A→ B insideCAlg_M is said to be formally étale if the induced map

L^intA ⊗_AB−→L^intB

is an equivalence ofB-modules.

Proposition 0.2.8. Let CAlg^{f et}_M be the sub-∞-category ofCAlg_M consisting of formally étale mor-phisms. There is a well defined ∞-functor

Pol^int(−, n) : CAlg^{f et}_M −→Pn+1−alg_M

such that ifA∈CAlg_M andB is a fibrant-cofibrant replacement ofA, thenPol^int(A, n)'P ol^int(B, n).

Definition 0.2.9. The object Pol^int(A, n) of the previous proposition is called the internal algebra of polyvectors fields on A. Its realization Pol(A, n) := |Pol^int(A, n)| is simply called algebra of polyvector fields on A.

The relation between polyvectors and the tangent complex is as expected: ifA∈CAlg_M is such thatL^int_A is dualizable, then we have a natural equivalence

Pol^int(A, n)'Sym_A(T^intA [−n]) in the∞-categoryCAlg^gr_M of graded algebras inM.