In this section we state and prove our main theorem, which extends the Lagrangian intersection theorem (see [PTVV], Theorem 2.9) in the context of shifted Poisson structures. We start by recalling the following result, which is Theorem 1.9 in [Sa].
Theorem 4.2.1(Safronov). Let A, B1, B2 be three commutative algebras inM, and letf :A→B1
and g : A → B2 two algebra morphisms. Let π ∈ Pois(A, n) be a Pn+1-structure on A, and let γ1 ∈Cois(f, n) andγ2 ∈Cois(g, n) be coisotropic structures on f andg respectively, in the sense of Definition 4.1.2. Then the (derived) tensor product B1 ⊗AB2 carries a natural Pn-structure such that the projection B1op⊗B2→B1⊗AB2 is a Poisson morphism, whereB1op is the algebraB1 taken with opposite Poisson structure.
Notice that this theorem recovers in particular the constructions in [BG] for affine schemes. More generally, derived algebraic geometry provides a suited general context to interpret the results of Baranovsky and Ginzburg: we will extend Theorem4.2.1for general derived stacks, giving a general conceptual explanation for the Gerstenhaber algebra structure constructed in [BG]. Concretely, we will prove the following result.
Theorem 4.2.2. Let X, L1 and L2 be derived Artin stacks, locally of finite presentation, and let π ∈ Pois(X, n) be a n-shifted Poisson structure on X. Let f : L1 → X and g : L2 → X be morphisms of derived stacks, and let γ1 and γ2 be coisotropic structures on f and g respectively.
The derived intersection Y =L1×X L2 naturally carries a (n−1)-shifted Poisson structure, such that the map Y → L1×L2 is a morphism of (n−1)-Poisson stacks, where L1 is taken with the opposite Poisson structure.
Proof. The cartesian diagram of stacks
Y j //
i
L1
f
L2 g //X induces a commutative square ofDYDR(∞)-algebras
j∗f∗PX(∞)∼=i∗g∗PX(∞) //
j∗PL1(∞)
i∗PL2(∞) //PY(∞)
By definition, the two coisotropic structuresγ1 andγ2 produce twoPnn+1-structures on the maps j∗f∗PX(∞)→j∗PL1(∞) and i∗g∗PX(∞)→i∗PL2(∞)
so that by Theorem 4.2.1we obtain a natural Pn-structure on the coproduct j∗PL1(∞)⊗i∗g∗PX(∞)i∗PL2(∞) .
Our goal is now to show that this coproduct is actually equivalent toPY(∞), which would immedi-ately conclude the proof. Notice that the twist byk(∞)commutes with colimits, so that is enough to show that
j∗PL1⊗i∗g∗PX i∗PL2 ∼=PY asDYDR-algebras.
Let SpecA → YDR an A-point of YDR. We want to prove that j∗PL1 ⊗i∗g∗PX i∗PL2 and PY coincide on the point SpecA →YDR. By definition, the value ofPY on this point is D(YA), where YA is the perfect formal derived stack over SpecA constructed as the fiber product
YA //
Y
SpecA //YDR
Since the (−)DR construction is defined as a right adjoint, it automatically commutes with limits, so that YDR ∼=L1DR×XDRL2DR. In particular any A-point of YDR has corresponding A-points of L1DR, L2DR andXDR, for which one can define fibersL1A, L2A andXA. Therefore, we need to show that
D(YA)∼=D(L1A)⊗D(XA)D(L2A) as graded mixed dg algebras.
We start by remarking that the fiber square YA //
L1A
L2A //XA
induces a map of graded mixed dg algebras
D(L1A)⊗D(XA)D(L2A)→D(YA)
by universal property of the coproduct. In order to prove that this map is an equivalence, it is enough to check it at the level of algebras, forgetting the graded mixed structures. But the forgetful functor
CAlg(−dgModgr)→CAlg(dgMod) comes by definition from the forgetful functor
B−codgC(k)→C(k)
whereB is the Hopf algebra B=k[t, t−1]⊗kk[]andB−codgC(k) is the category ofB-comodules inC(k). This means in particular that forgetting the graded mixed structure preserves colimits, so that the underlying algebra of the pushout of
D(XA) //
D(L1A)
D(L2A)
is exactly the tensor product of algebrasD(L1A)⊗D(XA)D(L2A).
Recall that following [CPTVV], a formal derived stack F (in the sense of Definition 0.4.1) is calledaffine if it satisfies the following two conditions:
• its reduced stackFred is an affine derived scheme;
• F has a cotangent complexLF (in the sense of [HAG-II, Section 1.4]), such that for everyB ∈ dAff and every map u:SpecB →F, the dg B-module u∗LF is coherent and cohomologically bounded above.
Moreover, an affine formal derived stack F is calledalgebraisable if it is equivalent to the formal completion along a mapFred→G, whereG is an algebraicn-stack for some n.
SinceXA, L1A, L2A are all algebraisable, by applying Theorem 2.2.2 of [CPTVV] we have equiv-alences of algebras
D(L1A)⊗D(X
A)D(L2A)∼= SymAred(LAred/L1A[−1])⊗Sym
Ared(LAred/XA[−1])SymAred(LAred/L2A[−1]) D(YA)∼= SymAred(LAred/YA[−1])
The result now follows directly from the following lemma.
Lemma 4.2.1. Consider the following diagram of derived stacks K φ //X i //
j
Y
g
Z f //W
where the right square is cartesian. Then the following diagram ofOK-modules TK/X //
TK/Y
TK/Z //TK/W
is cartesian.
Proof. From the diagram of stacks, one immediately gets two fiber sequences of OK-modules TK/Y //TK/W //φ∗i∗TY /W
TK/Z //TK/W //φ∗j∗TZ/W
and therefore the limit of
TK/Y
TK/Z //TK/W
is precisely the fiber of the map TK/W → φ∗i∗TY /W ⊕φ∗j∗TZ/W. But by general properties of cartesian squares, TX/W ∼=i∗TY /W ⊕j∗TZ/W, and hence φ∗TX/W ∼= φ∗i∗TY /W ⊕φ∗j∗TZ/W. We now conclude by observing that the fiber of the map
TK/W →φ∗TX/W
is naturally identified withTK/X.
We can now just apply the lemma to the diagram of algebraisable stacks Spec(Ared) //YA //
L1A
L2A //XA
and get a cartesian square of Ared-modules
TAred/YA //
TAred/L1A
TAred/L2A
//TAred/XA
From this we deduce a pushout diagram ofAred-algebras SymAred(LAred/XA[−1]) //
SymAred(LAred/L1A[−1])
SymAred(LAred/L2A[−1]) //SymAred(LAred/YA[−1]) which is exactly what we wanted.
Remark. The argument in this section works in the same way if one wants to use the definition of coisotropic structures given in [CPTVV], provided one has a result similar to Theorem 4.2.1.
Namely, one needs the following statement.
Proposition 4.2.3. Let f1:A→B1 andf2 :A→B2 be morphisms of cdgas, withA equipped with a Pn+1-structure π. Suppose both f1 and f2 are endowed with a coisotropic structure relative to π, in the sense of [CPTVV], section 3.4. Then the intersection B1⊗AB2 has a natural Pn-structure, such that the map B1⊗kB2op→B1⊗AB2 is a map of Pn-algebras.
This will be an immediate corollary of the following slightly more general fact.
Proposition 4.2.4. Let A, B and C three cdgas with Pn+1-structures, and let f1 :A⊗Bop →L1
andf2 :B⊗Cop→L2 be cdgas map equipped with coisotropic structures (in the sense of [CPTVV], section 3.4). LetL12=L1⊗BL2. Then the mapA⊗Cop→L12 has a natural coisotropic structure.
Proof. By definition of coisotropic structures as given in [CPTVV], L1 is a left A-module and a rightB-module in the monoidal category of Pn-algebras, and similarlyL2 is a leftB-module and a rightC-module. From this we immediately see thatL12'L1⊗BL2 is a leftA-module and a right C-module, which is exactly what we wanted. We conlude by noticing that the canonical morphism L1⊗kL2 →L12 is a map of leftA-modules and of right C-modules.
In particular, Theorem 4.2.2 is true for all definitions of derived coisotropic structures, that is to say Definition 4.1.2, Definition 4.1.4 and [CPTVV, Definition 3.4.4].