Another proof of the main result

In this last section we will give a more explicit description of our results: we take a particular resolution of the graded dg-Lie algebrak[−1](2)and study the induced resolution of theLieoperad.

We check that its algebras are justLie∞-algebras in the standard sense, see for example [HS]. These concrete computations also give an alternative proof of Theorem1.3.2.

The graded dgLie algebra k[−1](2)has a cofibrant resolution L₀ given by the free Lie algebra generated by elements p_i fori= 2,3, . . ., such thatp_i sits in weight i and in cohomological degree 1 ; the differential in L0 is defined as to satisfy

dp_n=−1 2

X

i+j=n+1

[p_i, p_j]

Notice that in particular that we havedp2 = 0. The map L0 →k[−1](2)sends p2 to the generator of k[−1](2)and the otherp_i to zero.

By definition, the space of n-shifted Poisson structures onA is Map_dgLiegr(k[−1](2),Sym_A(T_A[−n−1])[n+ 1])

and we can use the explicit resolutionL0 to compute its n-simplices: these are just elements in Hom_dgLie^gr(L0,Sym_A(TA[−n−1])[n+ 1]⊗Ωn) .

In particular, the points of the space ofn-shifted structures onA can be identified with Hom_dgLie^gr(L₀,Sym_A(T_A[−n−1])[n+ 1]) .

If A is cofibrant, then the dg module of derivations ofA is a model for T_A. In this case the same argument used in the proof of Theorem1.3.1 proves that Sym_A(T_A[−n−1])[n+ 1] maps into the dg Lie algebra of symmetric (shifted) multi-derivationsL(sⁿ⁺¹MD(A))^S, which in turn sits inside L(sⁿ⁺¹EndA)^S∼=L(End_A[n+1])^S, the dg Lie of all symmetric multilinear maps ofA[n+ 1].

Putting all together, we get a map from the points of Pois(A, n) to Hom_dgLie^gr(L₀,M

i∈N

Hom_k(Symⁱ_k(A[n+ 1]), A[n+ 1])).

So at the level of the vertices, a n-Poisson structure on A gives a sequence of symmetric mul-tilinear maps q_i (the images of the p_i) on A[n+ 1], such that every q_i is an i-linear map of degree 1.

One of the possible definitions (see for example [Man]) of aL∞-structures is the following.

Definition 1.4.1. If V is a graded vector space, an L∞-structure on V is a sequence of symmetric maps of degree 1

l_n: SymⁿV[1]→V[1] , n >0 such that for every n >0 we have

X

i+j=n+1

[l_i, l_j] = 0, where the bracket is the Lie bracket we defined before on M

i∈N

Hom_k(Symⁱ_k(V[1]), V[1]).

So if we want to prove that (still at the level of the vertices) an n-Poisson structure gives us an L∞-structure on A[n], we could try to find such l_n on A[n+ 1]. Natural candidates are the q_i that come directly from the shifted Poisson structure; these are given for i > 1. Notice that our brackets satisfy the graded antisymmetry relation [x, y] = −(−1)^|x||y|[y, x] ; in particular, this relation does not involve the weights of x and y. In our case |p_i| = |q_i| = 1, and so it follows [p_i, p_j] = [q_i, q_j] = [p_j, p_i] = [q_j, q_i]. Let us takeq₁=d, the differential of A[n+ 1]. We should now verify that the symmetric mapsqi satisfy

X

i+j=n+1

[q_i, q_j] = 0.

The other observation we need to make is that for every multilinear mapf ∈Homk(Symⁱ_k(A[n+ 1]), A[n+ 1])), we have[q₁, f] = [f, q₁] =d(f), where dhere is the differential of multilinear maps on A[n+ 1].

So using these facts we have X

i+j=n+1

[qi, qj] = 2d(qn) + X

i+j=n+1 i,j>1

[qi, qj] = 0,

which is what we wanted. To summarize, ann-Poisson structure induces anL∞-structure onA[n].

Now we need to show that the induced L∞-structure on A[n] is compatible with the algebra structure onA, that is to say thatAbecomes a semi-strictP_n+1-algebra. But theq_i we constructed in the previous step are (by definition) derivations of the given commutative product on A ; this gives A precisely the structure of a semi-strictPn+1-algebra.

The upshot of this discussion is the fact that we got a map

Hom_dgLie^gr(L₀,Sym_A(T_A[−n−1])[n+ 1])−→Hom_dgOp(Pb_n+1,End_A)

for which the image is contained in the Pb_n+1-structures whose commutative product is the one given on A. Equivalently, we get a function from Hom_dgLie^gr(L₀,Sym_A(T_A[−n−1])[n+ 1]) to the (non-homotopical) fiber product of the following diagram of sets

Hom_dgOp(Pb_n+1,End_A)

pt ^µA Hom_dgOp(Comm,End_A)

where µA denotes the given commutative product of A. From here one can proceed in the exact same way as done towards the end of Section 2: namely, we can use Theorem1.2.11(and the explicit descriptions of the simplicial framings in dgOp and dgLie^gr) in order to prove that we have a map of simplicial sets from Hom_dgLie^gr(L0,Sym_A(TA[−n−1])[n+ 1]) to the (strict) fiber of the natural map Hom_dgOp(Pb_n+1,End_A)→Hom_dgOp(Comm,End_A), taken atµ_A.

Now the same arguments used at the end of Section 3 allow to obtain a map in the homotopy category of simplicial sets

Pois(X, n)−→P_n+1^h (A) giving a more concrete proof of Theorem 1.3.2.

Shifted Poisson structures on general derived stacks

In this chapter we put together the results of the two previous sections, giving a definition of shifted Poisson structure on a general derived Artin stack. The material is mostly taken from [Me] and [CPTVV], and we will need it in chapter 4 to extend the results of chapter 3 to the general case.

In the first section, we remark that theorem1.3.2, the main result of chapter 1, is not perfectly suited to be used in the context arising from formal localization. More specifically, in the proof of Theorem 1.3.2 we needed a finiteness assumption on the algebra, assuring that the cotangent complex will be perfect. This is not a big issue though, since the slightly peculiar definition of polyvectors in chapter 0 does not require one to deal with the tangent complex, and thus there is no need to assume any dualizability condition on the cotangent complex. We end up by stating Theorem 2.1.2, which is the modified version of Thereom 1.3.2 that we will use in the following sections.

In section 2, we attack the problem of defining shifted Poisson structures for general derived stacks. As for the affine case, we give two possible definitions, and we show that Theorem 2.1.2 implies that they are in fact equivalent as expected. Thanks to formal localization, this looks a lot like the affine case, where we already showed this kind of result in chapter 1.

2.1 A slight modification of Theorem 1.3.2

Later in this chapter, we will try to extend the results of the previous chapter to general derived Artin stacks, not necessarily affine. In particular, we will need to prove a statement similar to Theorem1.3.2, which shows that the two possible definitions of shifted Poisson structures (the one in term of bivectors and the one in term of Poisson brackets) are in fact equivalent.

Recall that in Chapter 1 we worked in the simpler category C(k) of cochain complexes over k. But it is immediate to check that all arguments translates smoothly in the context of any symmetric monoidal ∞-category M having a well behaved C(k)-enrichment. In particular, our starting hypothesis on the model categoryM in the preliminary chapter are enough to assure that Theorem 1.3.2 stays true in the associated∞-category.

Using the general formalism of differential calculus in M presented in chapter 0, we can thus state a general categorical version of Theorem1.3.2.

Let M be a symmetric monoidal model category, satisfying our starting assumption of chapter 53

0. Recall that in particular M is C(k)-enriched. Let M be the associated symmetric monoidal

∞-category.

Theorem 2.1.1. With notations as above, take A ∈ CAlg_M to be a commutative algebra in M, and suppose moreover that L^int_A is dualizable as an object of A−ModM. Then the homotopy fiber of the map

Map_dgOp(Pn+1,End_A)−→Map_dgOp(Comm,End_A),

taken at the point given by the commutative structure on A, is naturally equivalent to Map_LieAlg^gr

M(1M[−1](2),Sym_A(T^int_A [−n−1])[n+ 1])

where 1M is the monoidal unit andLieAlg^gr_M is the ∞-category of graded Lie algebras in M.

Note that this results reduces exactly to theorem1.3.2if we start withM =C(k), and as already mentioned the hypothesis on M assures that we do get an interpretation of theorem 1.3.2 also in the general case.

There is still a small issue with Theorem 2.1.1, which prevents us to use it freely for Poisson structures on general derived stacks. Namely, we put the hypothesis that as an A-module, the cotangent complex L^int_A was dualizable. In the affine case treated in chapter 1, this was a natural assumption, since we want to work to geometric objects locally of finite presentation, and in par-ticular this assures that the cotangent complex will be perfect (hence dualizable). Notice that as explained in the proof of theorem 1.3.2, if L^int_A is not dualizable there is no hope to get the desired equivalence, and Theorem 2.1.1 is, as stated, simply false.

The situation for a general derived stack X is different though: the algebra A to which we would like to apply Theorem 2.1.1 is P_X(∞), as we saw that formal localization techniques allow to establish a deep link between P_X(∞) and the geometry of X itself. Here it is not entirely clear whetherP_X(∞)has a dualizable cotangent complex, internal to the∞-category ofDXDR (∞)-modules.

It turns out that the conclusions of Theorem 2.1.1 can be slightly modified if one drops the dualizability hypothesis on the cotangent complex.

Theorem 2.1.2. LetMbe an∞-category satisfying the hypothesis of chapter 0, and letA∈CAlg_M be a commutative algebra in M. Then the homotopy fiber of the map

Map_dgOp(Pn+1,End_A)−→Map_dgOp(Comm,End_A),

taken at the point given by the commutative structure on A, is naturally equivalent to Map_LieAlg^gr

M(1M[−1](2),Pol^int(A, n+ 1)[n+ 1]),

wherePol^int(A, n+1)is the algebra of internal(n+1)-shifted polyvectors onA, as defined in chapter 0.

Proof. The proof is basically the same exact proof of Theorem1.3.2 (and thus of Theorem2.1.1).

One just needs to notice that during the proof of Theorem 1.3.2 we passed through a canonical identification between the fiber of the morphism

Map_dgOp(Pn+1,EndA)−→Map_dgOp(Comm,EndA)

and the space of Maurer-Cartan elements of weight≥2in the completed Lie algebra of multideriva-tions, using the dual of the symmetric algebra on the cotangent complex.

Recall from chapter 0 thatPol^int(A, n)was defined precisely using multiderivations, without ever mentioning the internal tangent complex ofA, so that we immediately get the desired result.

Of course, if the internal cotangent complexL^int_A happens to be dualizable, we already observed that there is a natural equivalence

Pol^int(A, n)'Sym_A(T^int_A [−n]), so that we find back theorem2.1.1.