ISOMORPHISM OF DE RHAM COHOMOLOGY AND RELATIVE HOCHSCHILD COHOMOLOGY

OF DIFFERENTIAL OPERATORS

F. J. Bloore and T. J. Harding

Abstract. Any de Rham p-form α on a manifold M may be extended to become a Hochschild p-cochain α_{S} on the associative algebra D of differential operators on C^{∞}(M, IR). The map α ↦ α_{S} depends on a choice of “allocation”, S, which is a rule for filling in any (p + 1)- tuple of sufficiently nearby points x_{0}, …, x_{p} of M with a p-simplex S(x_{0}, …, x_{p}) having these points as vertices. We show that (dα)_{S} = δ(α_{S}) so that the map α ↦ α_{S} passes to cohomology. We indicate the pattern of the proof that the map [α] ↦ [α_{S}] sending H_{DR}^{p}(M, IR) → H^{p}(D, C^{∞}(M, IR); D) is independent of S and is in fact an isomorphism. Here the latter group is the relative Hochschild cohomology group of D relative to C^{∞}(M, IR), with coefficients in D.

Keywords. De Rham cohomology, Hochschild cohomology, differential operators.

MS classification. 16E40, 16S32, 53C99.

1 Allocations: from de Rham to Alexander-Spanier

An allocation allows us to make an Alexander-Spanier (A-S) p-cochain ϕ_{α} on M [5] from a p-form α on M.

Let dim M = n. For 0 ≤ p ≤ n, let Δ_{p} ⊂ IR^{p} be the standard p-simplex. Let V_{p} ⊂ M^{p+1} be a neighbourhood of the diagonal M_{D} = {(x, x, …, x) | x ∈ M} ⊂ M^{p+1} which can be retracted to M_{D}. An allocation is a collection of maps S_{p},

S_{p} : Δ_{p} × V_{p} → M, 0 ≤ p ≤ n

This paper is in final form and no version of it will be submitted for publication elsewhere.

such that for each given (x_{0}, …, x_{p}) ∈ V_{p} the image of Δ_{p} in M is a p-simplex having vertices x_{0}, …, x_{p}. We denote this simplex S(x_{0}, …, x_{p}), and impose the condition that S ∘ ∂ = ∂ ∘ S, i.e.

∑_{k=0}^{p+1}(−1)^{k}S(x_{0}, …, x̂_{k}, …, x_{p+1}) = ∂S(x_{0}, …, x_{p+1}).  (1)

We may now define the A-S cochain ϕ_{α}^{S} : V_{p} → IR associated with a p-form α by

ϕ_{α}^{S}(x_{0}, …, x_{p}) = ∫_{S(x_{0},...,x_{p})} α.  (2)

Note that (1) implies that

ϕ_{dα}^{S} = δ(ϕ_{α}^{S}),

where δ is the A-S coboundary.

Thus the map α ↦ ϕ_{α}^{S} induces a map from de Rham cohomology to A-S cohomology. Any two allocations S_{1}, S_{2} are homotopic. This implies that the induced map between cohomology groups does not depend on the choice of allocation S.

2 From Alexander-Spanier to Hochschild

To each A-S p-cochain ϕ : V_{p} → IR we may associate a Hochschild p-cochain F_{ϕ} ∈ C^{p}(D; D) on the associative algebra D of differential operators on C^{∞}(M, IR). Information about Hochschild cohomology may be found in [4, Chapter 10]. Let H_{j} ∈ D, j = 1, …, p. We shall use the following notation. If f ∈ C^{∞}(V_{1}, IR) and

H(x) = a^{ij}(x)\frac{∂}{∂x^{i}} \frac{∂}{∂x^{j}} + a(x)

then

[H(y)f(x, y)]_{y=x} = a^{ij}(x)[\frac{∂}{∂y^{i}} \frac{∂}{∂y^{j}} f(x, y)]_{y=x} + a(x)f(x, x).

Define the differential operator F_{ϕ}(H_{1}, …, H_{p}) ∈ D as follows. For ψ ∈ C^{∞}(M, IR),

(F_{ϕ}(H_{1}, …, H_{p})ψ)(x_{0}) =
[H_{1}(x_{1})[H_{2}(x_{2})[… H_{p}(x_{p})[ϕ(x_{0}, …, x_{p})ψ(x_{p})]]_{x_{p}=x_{p−1}}]_{x_{p−1}=x_{p−2}} …]_{x_{1}=x_{0}}.  (3)

One may verify directly that

F_{δϕ} = δ(F_{ϕ})

where the right hand side is the Hochschild codifferential.

For example, for an A-S 0-cochain ϕ : V_{0} → IR,

(F_{δϕ}(H)ψ)(x_{0}) = [H(x_{1})[δϕ(x_{0}, x_{1})ψ(x_{1})]]_{x_{1}=x_{0}}
= [H(x_{1})[(ϕ(x_{1}) − ϕ(x_{0}))ψ(x_{1})]]_{x_{1}=x_{0}} = ([H, ϕ]ψ)(x_{0}) = (δF_{ϕ}(H))ψ(x_{0})

where F_{ϕ} ∈ D is the 0-cochain ϕ, regarded as a zero order differential operator on

C^{∞}(M, IR). Thus the map α ↦ ϕ_{α}^{S} ↦ F_{ϕ_{α}^{S}} is a cochain map, giving a map Ď : H_{DR}^{p}(M, IR) → H_{Hoch}^{p}(D, D). We write α_{S} for the Hochschild p-cochain F_{ϕ_{α}^{S}} on D obtained by substituting (2) in (3). One can verify that α_{S} has the following properties.

(i) Regarding vector fields X_{1}, …, X_{p} as first order differential operators,

α_{S}(X_{1}, …, X_{p}) = \frac{1}{p!}α(X_{1}, …, X_{p}).  (4)

(ii) For f_{j} ∈ C^{∞}(M, IR) and H_{j} ∈ D,

α_{S}(f_{1}∘H_{1}, …, H_{j−1}, f_{j} ∘H_{j}, …, H_{p}∘f_{p}) = f_{1}∘α_{S}(H_{1}, …, H_{j−1}∘f_{j}, H_{j}, …, H_{p})∘f_{p}.

(iii) α_{S}(H_{1}, …, H_{p}) = 0 if any H_{j} is a function.

(iv) The order of the operator α_{S}(H_{1}, …, H_{p}) is ∑_{j=1}^{p}(ord(H_{j}) − 1).

Thus α_{S} : ⊗IR^{p}D → D factors through ⊗_{F}^{p}(D/F) where F = C^{∞}(M, IR), and is therefore an F-relative p-cochain [3]. The Hochschild coboundary operation preserves F-relativity and the cohomology of the F-relative complex is denoted H^{p}(D, F; D).

The map α ↦ α_{S} for p = 1 also gives a construction of the Poincaré-Cartan form Θ for a class of Lagrangians in the multivariate higher order calculus of variations. The dependence of the map upon choice of allocation and the independence of Ď relate to known properties of Θ, [1].

We end with a sketch of the proof that the de Rham cohomology of M and the relative Hochschild cohomology of D relative to F are isomorphic.

3 The isomorphism H_{DR}^{p} ≃ H^{p}(D, F; D)

The map D_{S} : Ω^{p} → C_{Hoch}^{p}(D, F; D), α ↦ α_{S} defined above satisfies D_{S} ∘ d = δ ∘ D_{S} so passes to cohomology as the map Ď : H_{DR}^{p} → H^{p}(D, F; D). There is a map ǎ in the reverse direction. Define a : C_{Hoch}^{p}(D, F; D) → Ω^{p}, A ↦ aA, by

(aA)(X_{1}, …, X_{p}) = ∑_{σ∈S_{p}}(−1)^{σ}A(X_{σ_{1}}, …, X_{σ_{p}}).

Note that aA is indeed a de Rham p-form, and that (4) implies that

a ∘ D_{S} = id_{Ω^{p}}.

One may check directly that a ∘ δ = d ∘ a, so that a passes to cohomology,

ǎ : H^{p}(D, F; D) → H_{DR}^{p}(M, IR),

and

ǎ ∘ Ď = id_{H_{DR}^{p}}.

Thus Ď is injective, ker Ď=0.

It remains to prove that ǎ is injective also. Let A ∈ C_{Hoch}^{p}(D, F; D), δA = 0 and aA = 0. We seek B ∈ C_{Hoch}^{p−1}(D, F; D) such that A = δB. We indicate the pattern of the argument by sketching it for the case p=3. We construct B(H_{1}, H_{2}) for increasing order of the operator H_{1}H_{2}. If either H_{j} is of zero order, then B=0. Suppose each H_{j} = X_{j}, a vector field. From A = −δB_{1}, ord(H_{1}H_{2}H_{3}) ≤ 3, (the minus sign is for convenience), we have

A(X_{1}, X_{2}, X_{3}) = −δB_{1}(X_{1}, X_{2}, X_{3}) =
= −X_{1}B_{1}(X_{2}, X_{3}) + B_{1}(X_{1}X_{2}, X_{3}) − B_{1}(X_{1}, X_{2}X_{3}) + B_{1}(X_{1}, X_{2})X_{3}.  (5)

For fixed X_{1}, X_{2}, X_{3} (5) together with 5 similar equations obtained by permuting X_{1}, X_{2}, X_{3} are linear equations in the unknowns B_{1}(X_{i}, X_{j}), B_{1}(X_{i}X_{j}, X_{k}), B_{1}(X_{i}, X_{j}X_{k}). Let us choose B_{1}(X_{i}, X_{j}) = 0, i.e. B_{1}(H_{1}, H_{2}) = 0, for ord(H_{1}H_{2}) ≤ 2. Then

B_{1}(X_{1}X_{2}, X_{3}) = B_{1}(X_{2}X_{1}, X_{3}) ≡ B_{1}({1, 2}, 3).

Write A(X_{1}, X_{2}, X_{3}) as A(1, 2, 3). Then equations (5) reduce to

A(1, 2, 3) = B_{1}({1, 2}, 3) − B_{1}(1, {2, 3})  (6)

and its five permutations. The cochains in these equations may be regarded as functions of the ordered partitions of the set {1, 2, 3} into 2 or 3 subsets. The condition aA = 0 is clearly necessary for a solution B_{1} to exist. That it is also sufficient may be seen from figure 1 showing the barycentric decomposition of the circle.

Figure 1: Barycentric decomposition of S^{1}

Label the vertices of the triangle (1, {2, 3}), (2, {1, 3}), (3, {1, 2}); label the midpoints of the sides 12, 23, 31 as ({1, 2}, 3), ({2, 3}, 1), ({3, 1}, 2). Thus the points (vertices and barycentres) are ordered partitions of the set {1, 2, 3} into 2 subsets. Label the half-side from the vertex (2, {1, 3}) to the midpoint ({1, 2}, 3) as (2,1,3), and similarly label the other 5 half-sides. A now takes values on half-sides and B_{1} on vertices and midpoints. The condition that aA = 0 means that the sum of the A-values taken round the triangle vanishes. This is the sufficient condition that there are B_{1}-values satisfying (6).

We may take

B_{1}(1, {2, 3}) = 0,
B_{1}({1, 2}, 3) = A(1, 2, 3),
B_{1}(2, {1, 3}) = A(1, 2, 3) − A(2, 1, 3),

etc. We thank Alastair King and Johan Dupont for this argument.

This defines B_{1}(H_{1}, H_{2}) for ord(H_{1}H_{2}) ≤ 3. Extend the definition of B_{1} to higher orders of H_{1}H_{2} arbitrarily, consistent with this. Now let

A_{1} = A + δB_{1}.

Then δA_{1} = 0 and A_{1}(H_{1}, H_{2}, H_{3}) = 0 for ord(H_{1}H_{2}H_{3}) ≤ 3.

Fix vector fields X_{1}, X_{2}, X_{3}, X_{4}. We have

A_{1}(X_{1}X_{2}, X_{3}, X_{4}) = A_{1}(X_{2}X_{1}, X_{3}, X_{4}) ≡ A_{1}({1, 2}, 3, 4).

Let B_{2}(H_{1}, H_{2}) = 0 for ord(H_{1}H_{2}) ≤ 3. Then

B_{2}(X_{1}X_{2}X_{3}, X_{4}) = B_{2}(X_{σ_{1}}X_{σ_{2}}X_{σ_{3}}, X_{4}) ≡ B_{2}({1, 2, 3}, 4).

Similarly

B_{2}(X_{1}X_{2}, X_{3}X_{4}) ≡ B_{2}({1, 2}, {3, 4}).

We wish to solve the equations A_{1} = −δB_{2}. These state

A_{1}({1, 2}, 3, 4) = B_{2}({1, 2, 3}, 4) − B_{2}({1, 2}, {3, 4})  (7)

and 35 similar equations.

The appropriate diagram this time is the barycentric decomposition of S^{2}, (figure 2). A_{1}-values are attached to line segments, which are ordered partitions of the set {1, 2, 3, 4}

into 3 non-empty subsets, and B_{2}-values are attached to vertices and barycentres, which are ordered partitions into 2 non-empty subsets. The subfaces are permutations of (1,2,3,4). Again the equations (7) have the form A_{1}(σ) = B_{2}(∂σ) where σ is any line segment of the decomposition, and the “integrability conditions” for the existence of a solution B_{2} have the form

δA_{1}(1, 2, 3, 4) = −A_{1}({1, 2}, 3, 4) + A_{1}(1, {2, 3}, 4) − A_{1}(1, 2, {3, 4}) = 0.

We thus obtain B_{2}(H_{1}, H_{2}) for ord(H_{1}H_{2}) ≤ 4, and extend B_{2} arbitrarily to higher orders of H_{1}H_{2}. Then

A(H_{1}, H_{2}, H_{3}) = −δ(B_{1} + B_{2})(H_{1}, H_{2}, H_{3})

for ord(H_{1}H_{2}H_{3}) ≤ 4. Continuing in this way, one may define B_{n} for any n. The details are in a forthcoming paper [2]. Then for ord(H_{1}H_{2}H_{3}) = n,

A(H_{1}, H_{2}, H_{3}) = −δB(H_{1}, H_{2}, H_{3})

where B(H_{1}, H_{2}) = ∑_{r=1}^{n−2} B_{r}(H_{1}, H_{2}).

Figure 2: Barycentric decomposition of S^{2}

References

[1] T. J. Harding and F. J. Bloore, A fresh approach to the Poincaré-Cartan form for a linear p.d.e. and a map between cohomologies, Differential Geometry and its Applications, 1989, Brno, Czechoslovakia (J. Janys̆ka and D. Krupka, eds.), World Scientific, 1990, pp. 220–229.

[2] , Isomorphism of de Rham cohomology and relative Hochschild cohomology of differential operators, 1992, submitted to Ann. Inst. Henri Poincaré.

[3] G. Hochschild, Relative homological algebra, Trans. Am. Math. Soc. 82 (1956), 246– 269.

[4] S. MacLane, Homology (3rd corrected printing), Springer-Verlag, 1975.

[5] E. H. Spanier, Algebraic topology, McGraw-Hill, 1966.

F. J. Bloore, D.A.M.T.P., The University of Liverpool, Liverpool L69 3BX, England.

T. J. Harding, E.P.C.C., The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland.

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DE RHAM AND RELATIVE HOCHSCHILD COHOMOLOGY