Homology, Homotopy and Applications, vol.3, No.8, 2001, pp.165–192

A HOMOTOPY LIE-RINEHART RESOLUTION AND CLASSICAL BRST COHOMOLOGY

LARS KJESETH

(communicated by Larry Lambe)

Abstract We use an interlaced inductive procedure reminiscent of the integration process from traditional deformation theory to construct a homotopy Lie-Rinehart resolution for the Lie-Rinehart pair which arises as an exercise in Poisson reduction in the context of the BFV construction of classical BRST cohomology. We show that the associated homotopy Rinehart algebra and the BRST algebra are isomorphic as graded commutative algebras. In the irreducible case, the two have the same cohomology.

1. Introduction

In this paper, we utilize a strategy akin to the process of integration found in traditional deformation theory (see, for example, [GS90]) to construct a homotopy Lie-Rinehart resolution (K_{A/I}, K_{I/I^{2}}) for the Lie-Rinehart pair (A/I, I/I^{2}) that appears in the BFV formulation of classical BRST cohomology. A Lie-Rinehart pair is a couple (B, s𝔤) which admits a structure analogous to that shared by the associative commutative algebra C^{∞}(M) of smooth functions and the Lie algebra Γ(T M) of smooth vector fields on a smooth manifold M. The term Lie-Rinehart pair is not widely used. More often, s𝔤 has been called a (B, k)-Lie algebra, [Rin63] [Pal61] and [Her72]. More recently, Lie-Rinehart pairs have appeared as Lie algebroids (see [dSW]).

An associative algebra A is a Poisson algebra if it admits a Lie bracket { , } such that for any a ∈ A, the map {a, } is a graded derivation with respect to the multiplication, i.e., {a, bc} = {a, b}c ± b{a, c}. A multiplicative ideal I of a Poisson algebra A is coisotropic if it is closed under the Poisson bracket on A. Poisson reduction of a Poisson algebra A by a coisotropic ideal I is no more than the observation that while the quotient A/I is not a Poisson algebra (unless I is a Poisson ideal), the subset of I-invariant classes in A/I is again a Poisson algebra. The I-invariant classes comprise the zeroth cohomology of the Rinehart complex R for the Lie-Rinehart pair (A/I, I/I^{2}).

Received January 8, 2001, revised May 3, 2001; published on June 21, 2001. 2000 Mathematics Subject Classification: primary 17B55, 17B81; secondary 17B56, 17B99. Key words and phrases: strongly homotopy Lie algebras, Lie-Rinehart pairs, Rinehart cohomology, BRST cohomology. \begin{matrix}c \\ ◯ \end{matrix} 2001, Lars Kjeseth. Permission to copy for private use granted.

Mathematicians and physicists (see, for exammple [Sta88], [KS87], [FHST89], [HT92], [Kim92a], [Kim92b], [Sta92], [Kim93] and [Sta96]) recognized the classical analogue of the Batalin, Fradkin and Vilkovisky (BFV) construction of the quantum BRST complex ([BV77], [BF83], [BV83] and [BV85]) as something new and interesting because BRST cohomology performs Poisson reduction without passing first to the quotient A/I. The BFV construction of the classical BRST algebra (see, for example, [Kim93]) begins by replacing A/I with the Koszul-Tate resolution [Tat57] and adjoins formal (ghost) variables to the Koszul-Tate resolution. They then exploit a graded Poisson bracket to construct a differential. Under certain conditions, the BRST algebra (A, D) is a cohomological model for the Rinehart complex of (A/I, I/I^{2}).

In general, constructing cohomological models for the Rinehart complex (B, s𝔤) by the traditional homological means of replacing both B and s𝔤 with resolutions fails unless the Lie-Rinehart structure of the pair (B, s𝔤) is preserved. In [Kje01], we defined homotopy Lie-Rinehart pairs and the associated homotopy Rinehart algebra in the context of coalgebras. We defined homotopy Lie-Rinehart resolutions and presented conditions under which the associated homotopy Rinehart algebra is a cohomological model for the Rinehart algebra of the resolved Lie-Rinehart pair.

Summary

The coalgebra setting is used throughout this paper. First, we revisit the definitions of the Lie algebras of subordinate derivation sources, resting coderivations and shared Lie modules, which lie behind the coalgebra realizations of both homotopy and non-homotopy Lie algebras, Lie algebra modules, Lie-Rinehart pairs and Rinehart cohomology. We review homotopy Lie-Rinehart resolutions for Lie-Rinehart pairs and the conditions under which the homotopy Rinehart algebra for a homotopy Lie-Rinehart resolution is a model for the Rinehart cohomology complex for a Lie-Rinehart pair (§2). Next, using an interlaced inductive process, we construct a homotopy Lie-Rinehart resolution for the Lie-Rinehart pair in classical BRST algebra (§3). One surprising result is that the homotopy Rinehart complex associated with the homotopy Lie-Rinehart resolution is isomorphic to the BRST algebra (§4).

We have omitted all sign arguments from the proofs in this paper, as they are not generally instructive. All vector spaces, algebras and coalgebras in this paper are over a field k of characteristic zero. All tensor products are over k and all maps are at least k-linear or k-multilinear.

Note that s𝔤 is the suspension of the Lie algebra 𝔤, i.e., all elements of 𝔤 are assigned degree zero and hence, all elements of s𝔤 have degree 1. Throughout this paper, we will identify all Lie algebras (and strongly homotopy Lie algebras) with their suspensions. At first, this identification will be explicit; later, when we work with I/I^{2} and K_{I/I^{2}}, we will hide the suspension.

2. Necessary Background

Review of Chavalley-Eilenberg and Rinehart cohomologies

For any Lie algebra s𝔤 and s𝔤-module B, an n-multilinear function f_{n} : (s𝔤)^{×n} → B is alternating if f_{n}(sx_{1}, ..., sx_{i}, sx_{i+1}, ..., sx_{n}) = −f_{n}(sx_{1}, ..., sx_{i+1}, sx_{i}, ..., sx_{n}). The Chevalley-Eilenberg complex is the set of all alternating multilinear functions Alt_{k}(s𝔤, B), graded by n, and equipped with a degree +1 differential δ_{CE} : Alt_{k}^{n}(s𝔤, B) → Alt_{k}^{n+1}(s𝔤, B) given by

\begin{matrix}δ_{CE}f_{n}(sx_{0}, ..., sx_{n}) = ∑_{i=0}^{n}(−1)^{i}ω(sx_{i} ⊗ f_{n}(sx_{0}, ..., ŝx_{i}, ..., sx_{n}) \\ −∑_{i<j}(−1)^{i+j−1}f_{n}([sx_{i}, sx_{j}], sx_{0}, ..., ŝx_{i}, ..., ŝx_{j}, ..., sx_{n}), \end{matrix}

(1)

where ŝx_{k} indicates that sx_{k} should be omitted. The map ω is the s𝔤-module action of s𝔤 on B. Any element b ∈ B is considered an element of Alt_{k}^{0}(s𝔤, B). The image of b under δ_{CE} is defined by setting δ_{CE}b(sx) = ω(sx ⊗ b). When B is an algebra, the Chevalley-Eilenberg complex is a differential graded commutative algebra. For f_{n} and g_{m} in Alt_{k}(s𝔤, B), the product f_{n} ⌣ g_{m} is given by

(f_{n} ⌣ g_{m})(sx_{1}, ..., sx_{n+m}) = ∑_{unshuffles^{(n,m)}^{σ}} f_{n}(sx_{σ(1)}, ..., sx_{σ(n)})g_{m}(sx_{σ(n+1)}, ..., sx_{σ(n+m)}).

An (n, m)−unshuffle is any permutation σ in the symmetric group Σ_{n+m} such that

σ(1) < ⋯ < σ(n)

first σ hand

and σ(n + 1) < ⋯ < σ(n + m),

second σ hand

where σ(j) is the element of the set {1, ..., n + m} moved to the j^{th} position under σ. The differential δ_{CE} acts as a derivation with respect to this multiplication. The cohomology of this complex with respect to δ_{CE} is the Chevalley-Eilenberg cohomology of 𝔤 with coefficients in B [CE48].

Definition 2.1. [Rin63] Let B is an algebra and s𝔤 be a Lie algebra, both modules over an algebra A over a field k of characteristic zero and modules over each other. We denote the left B-module action μ on s𝔤 by μ(a⊗sα) := asα. Let ω : s𝔤⊗B ⟶ B (or, alternatively, ω : s𝔤 → Der(B)) denote the s𝔤-module action on B. The pair (B, s𝔤) is a Lie-Rinehart pair, provided the Lie-Rinehart relations (LRa) and (LRb) are satisfied for all a, b ∈ B and sx, sy ∈ s𝔤:

LRa: ω(asx ⊗ b) = a · ω(sx ⊗ b), where · indicates the multiplication on B.

LRb: [sx, asy] = a[sx, sy] + ω(sx ⊗ a)sy.

Suppose (B, s𝔤) is a Lie-Rinehart pair and the alternating function f_{n} is Bmultilinear, i.e., f_{n}(a_{1}sx_{1}, ..., a_{n}sx_{n}) = a_{1} ⋯ a_{n}f_{n}(sx_{1}, ..., sx_{n}). Because the Lie action map ω maps s𝔤 into the derivations of B and as a result of the Lie-Rinehart relations, the image of f_{n} under the Chevalley-Eilenberg differential δ_{CE} is again B-multilinear, despite the fact that the bracket is not B-multilinear. The Rinehart algebra R = Alt_{B}(s𝔤, B) with differential δ_{R} = δ_{CE} is a subcomplex of the Chevalley-Eilenberg algebra and the cohomology with respect to δ_{R} is the Rinehart cohomology of s𝔤 with coefficients in B [Rin63].

Coalgebras and Subcoalgebras

A graded coassociative coalgebra is a pair (C, Δ), where C is a graded module over k together with a 0-degree coassociative comultiplication Δ : C → C ⊗ C. A function f with degree |f| = r is a coderivation on C if (f ⊗ 1 + 1 ⊗ f)Δ = Δf. The set of all coderivations on a coalgebra C, denoted Coder(C), is a graded Lie algebra under the graded commutator bracket, that is to say, [f, g] = fg − (−1)^{|f||g|}gf for all f and g ∈ Coder(C), where |f| and |g| are the degrees of f and g.

We will work with the tensor coalgebra T^{c}(sV) = ⨂(sV), where sV is the suspension a graded module V over a k-algebra A, where k is a field of characteristic 0. We will let sv_{[1 to n]} denote the element sv_{1} ⊗ ⋯ ⊗ sv_{n} ∈ (sV)^{⊗n}. The internal graded action ρ_{n}^{∧} of the symmetric group Σ_{n} on (sV)^{⊗n} is given by σ · (sv_{[1 to n]}) = �_{id}(σ)sv_{σ[1 to n]} for all n, σ ∈ Σ_{n} and sv_{[1 to n]} ∈ (sV)^{⊗n}. The symbol sv_{σ[1 to n]} is shorthand for sv_{σ(1)} ⊗ ⋯ ⊗ sv_{σ(n)}, where σ(i) is the element of the ordered set {1, ..., k} which moves to the i^{th} position under σ. The factor �_{id}(σ) is the sign produced by rearranging the sv_{i}’s into the σ order, following the Koszul sign convention, which states that exchanging two objects of homogeneous degrees p and q (whether elements or maps) introduces a factor of (−1)^{pq}. The action ρ^{∧} = {ρ_{n}^{∧}}_{n=1}^{∞} of {Σ_{n}}_{n=1}^{∞} on T^{c}(sV) is defined in the obvious way. The ρ^{∧}-invariant subcoalgebra is well-known as the graded commutative coalgebra ⋀(sV) and is generated by elements of the form

∑_{σ∈Σ_{n}} �_{id}(σ)sv_{σ[1 to n]},

which we will denote by sv_{[1 to n]}^{∧}. The coassociative comultiplication on ⋀(sV) is given by

Δ(sv_{[1 to n]}^{∧}) = ∑_{j=0}^{n} ∑_{(j,n−j)^{ρ}} �_{id}(ρ) (sv_{ρ[1 to j]}^{∧}) ⊗ (sv_{ρ[j+1 to n]}^{∧}) .

where it is understood that the second sum is over all (j, n − j)-unshuffles.

Shared Lie modules, Subordinate and resting coderivations

Three algebraic objects are vital for what follows. The graded Lie algebra Coder_{W}^{W} (⋀(sV)) is generated by the set of coderivations l_{n} on ⋀(sV), each of which rests on a specific subordinate coderivation m_{n} in the Lie algebra Coder_{W}^{W} (⋀(sV) ⊗ W). For a graded commutative algebra W, the Lie algebra Coder_{W}^{W} (⋀(sV) ⊗ W) is generated by the set of all coderivations m_{n}, each of which is both subordinate to a specific resting coderivation l_{n} and a W -derivation source. Finally, the graded commutative algebra Hom_{W} (⋀(sV), ⋀(sV)⊗W) is generated by the set of all W -linear maps f_{n} from ⋀(sV) into ⋀(sV) ⊗ W. This algebra admits a shared Lie module structure over both Coder_{W}^{W} (⋀(sV) ⊗ W) and Coder_{W}^{W} (⋀(sV)). We review what the terms subordinate coderivation, W -derivation source, resting coderivation and shared Lie module mean. For a more detailed development, see [Kje01].

We extend any map l_{n} : (sV)^{∧n} → sV to a coderivation on the coalgebra ⋀(sV) by setting

l_{n}(sv_{[1 to k]}^{∧}) = ∑_{(n,k−n)^{ρ}} �_{id}(ρ)l_{n}(sv_{ρ[1 to n]}^{∧}) ∧ sv_{ρ[n+1 to k]}^{∧}.  (2)

Definition 2.2. A map m_{n} on ⋀(sV) ⊗ W is an l_{n}-subordinate coderivation if the degree of m_{n} and l_{n} agree and m_{n}, as the extension of a map m_{n} : (sV)^{∧n−1} ⊗ W → W as a coderivation on ⋀(sV) ⊗ W, is defined on (sV)^{k} ⊗ W by setting m_{n}(sv_{[1 to k]}^{∧} ⊗ w) = l_{n}(sv_{ρ[1 to k]}^{∧}) ⊗ w +

∑_{(k−n+1,n−1)^{σ}} �_{id}(σ) �_{id}(m_{n}; sv_{σ[1 to k−n+1]}^{∧}) · sv_{σ[1 to k−n+1]}^{∧} ⊗ m_{n}(sv_{σ[k−n+2 to k]}^{∧} ⊗ w).

(3)

when k ⩾ n − 1 and setting m_{n} = 0 when k < n − 1.

The set of all subordinate coderivations Coder(⋀(sV) ⊗ W) forms a graded Lie algebra under the graded commutator bracket. The bracket respects subordination, that is to say, if m_{i} and m_{j} are l_{i} and l_{j}-subordinate coderivations respectively, then [m_{i}, m_{j}] is a coderivation subordinate to [l_{i}, l_{j}].

Definition 2.3. An l_{n}-subordinate map m_{n} is a W -derivation source if for every sv_{[1 to n−1]}^{∧} in (sV)^{∧n−1}, the map m_{n}(sv_{[1 to n−1]}^{∧} ⊗ ( )) : W → W is a graded derivation on W.

The subset of all W -derivation sources in Coder(⋀(sV) ⊗ W) forms a graded Lie subalgebra denoted by Coder^{W} (⋀(sV) ⊗ W). The subset of all coderivations l_{n} ∈ Coder(⋀(sV)) which admit a subordinate W -derivation source m_{n} forms a graded Lie subalgebra which we will denote by Coder^{W} (⋀(sV)).

We can extend any map f_{n} : (sV)^{∧n} → W to function from ⋀(sV) into ⋀(sV) ⊗ W by setting f_{n} = 0 for k < n and, for k ⩾ n, setting f_{n}(sv_{[1 to k]}^{∧}) =

∑_{(k−n,n)^{σ}} �_{id}(σ) �_{id}(f_{n}; sv_{σ[1 to k−n]}^{∧})sv_{σ[1 to k−n]}^{∧} ⊗ f_{n}(sv_{σ[k−n+1 to k]}^{∧}). (4)

The set of all such maps is denoted by Hom(⋀(sV), ⋀(sV) ⊗ W). The algebra structure on W extends to a graded commutative algebra structure on ⋀(sV) ⊗ W, which provides Hom(⋀(sV), ⋀(sV) ⊗ W) with a cup product. If f_{n} and g_{s} are maps in Hom(⋀(sV), ⋀(sV) ⊗ W). Then f_{n} ⌣ g_{s} is given by setting (f_{n} ⌣ g_{s})(sv_{[1 to n+s]}^{∧}) =

∑_{(n,s)^{ρ}} �_{id}(ρ) �_{id}(g_{s}; sv_{ρ[1 to n]}^{∧})f_{n}(sv_{ρ[1 to n]}^{∧}) · g_{s}(sv_{ρ[n+1 to n+s]}^{∧})

on (sV)^{∧n+s}, where · represents the the multiplication on W (henceforth, the · will be suppressed). The map f_{n} ⌣ g_{s} is then extended to all of ⋀(sV) as in equation

If the graded module sV is a module over the graded commutative algebra W, then ⋀(sV) is a module over the tensor algebra TW. We will follow our general convention and denote w_{1}sv_{1} ∧ ⋯ ∧ w_{n}sv_{n} by wsv_{[1 to n]}^{∧}. The unraveling map

U : ⋀(sV) → W ⊗ ⋀(sV)
wsv_{[1 to n]} ↦ u([1 to n])w_{[1 to n]}^{·} ⊗ sv_{[1 to n]},

where u([1 to n]) is the sign produced when moving the w_{i}’s past the sv_{j}’s. The map U respects the coalgebra structure of ⋀(sV).

Definition 2.4. A map f_{n} : (sV)^{∧n} → W is W -linear if

f_{n}(wsv_{[1 to n]}^{∧}) = u([1 to n]) �_{id}(f_{n}; w_{[1 to n]}^{·})w_{[1 to n]}^{·}f_{n}(sv_{[1 to n]}^{∧}).

Likewise, a map m_{n} : (sV)^{∧n−1} ⊗ W → W is W -linear if m_{n}(wsv_{[1 to n−1]}^{∧} ⊗ w) =

u([1 to n − 1]) �_{id}(m_{n}; w_{[1 to n−1]}^{·})w_{[1 to n−1]}^{·}m_{n}(sv_{[1 to n−1]}^{∧} ⊗ w).

For a map m_{n} : (sV)^{∧n−1} ⊗ W → W, we define the V -extension map m_{n}^{e} : (sV)^{∧n} → V by setting

m_{n}^{e}(wsv_{[1 to n]}^{∧}) = ∑_{(n−1,1)^{ρ}} �_{id}(ρ)m_{n}(wsv_{ρ[1 to n−1]}^{∧} ⊗ w_{ρ(n)})sv_{ρ(n)}.

Definition 2.5. Given a W -linear, W -derivation source m_{n} ∈ Coder^{W} (⋀(sV) ⊗ W) which is subordinate to l_{n} ∈ Coder^{W} (⋀(sV)), the map l_{n} is said to rest on m_{n} if l_{n}(wsv_{[1 to n]}^{∧}) =

u([1 to n]) �_{id}(l_{n}; w_{[1 to n]}^{·})w_{[1 to n]}^{·}l_{n}(sv_{[1 to n]}^{∧}) + m_{n}^{e}(wsv_{[1 to n]}^{∧}).

The subset of Coder^{W} (⋀(sV)) of all coderivations l_{p} which rest on W -linear W - derivation sources m_{p} forms Coder_{W}^{W} (⋀(sV)). Similarly, the subset of Coder^{W} (⋀(sV) ⊗ W) consisting of all W -linear W -derivation sources m_{p} subordinate to maps l_{p}, which in turn rest on m_{p}, forms Coder_{W}^{W} (⋀(sV) ⊗ W).

Suppose m_{p} ∈ Coder_{W}^{W} (⋀(sV) ⊗ W) is subordinate to l_{p} ∈ Coder(⋀(sV)). Remarkably, the map

⟨m_{p}, ⟩ : Hom_{W} (⋀(sV), ⋀(sV) ⊗ W) → Hom_{W} (⋀(sV), ⋀(sV) ⊗ W),

given by setting

⟨m_{p}, f_{n}⟩ = m_{p}f_{n} − �_{id}(m_{p}; f_{n})f_{n}l_{p}

on (sV)^{∧n+p−1} and extending ⟨m_{p}, f_{n}⟩ to all of ⋀(sV) as in equation (4) is welldefined [Kje01]. This map exposes the shared Lie module structure on Hom_{W} (⋀(sV), ⋀(sV) ⊗ W) [Kje01]. In a sense, it is simultaneously a Lie-module over both Coder_{W}^{W} (⋀(sV) ⊗ W) and Coder_{W}^{W} (⋀(sV)) in that ⟨[m_{i}, m_{j}], ⟩ = ⟨m_{i}, ⟨m_{j}, ⟩⟩ − �_{id}(m_{i}; m_{j}) ⟨m_{j}, ⟨m_{i}, ⟩⟩ . The map ⟨m_{p}, ⟩ acts as a derivation with respect to the cup product.

Homotopy (and non-homotopy) Lie algebras and their modules

Strongly homotopy Lie algebras (shLie algebras) first appeared implicitly in [Sul77] and explicitly in [SS] in the context of deformation theory. A concise introduction to shLie algebras is found in [LM95]. The definitions below were lifted directly from [LM95] and [LS93] and then modified in [Kje01] to fit the language of this paper. The machinery used in deformation theory (see, for example, [GS90]) will prove useful in the process of constructing an shLie structure and an shLie module structure on the resolution of the Lie-Rinehart pair in §3, although we will not use the language in the traditional way.

Definition 2.6. An L(m)-structure on a graded module sL consists of a system of coderivations {l_{k} : ⋀(sL) → ⋀(sL) : 1 ⩽ k ⩽ m ⩽ ∞, k ≠ ∞} which are extensions of maps l_{k} : (sL)^{∧k} → sL. Each l_{k} has degree −1. Moreover, the following generalized form of the Jacobi identity is satisfied for n ⩽ m:

J_{ID}_{n} = \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}[l_{j}, l_{i}] = 0  (5)

on ⋀(sL). If L admits an L(∞)-structure, then L is a strongly homotopy Lie algebra (an shLie algebra).

For this paper, we will view a standard Lie algebra s𝔤 as a degenerate shLie algebra ⋀(s𝔤) where all coderivations l̂_{i} are zero except l̂_{2}. We decorate the bracket l̂_{2} on s𝔤 to distinguish it from l_{2} on the shLie algebra which resolves ⋀(s𝔤).

The following maps will be useful in §3, where we will use a deformation theoretic approach to construct an shLie algebra together with an shLie module.

Definition 2.7. For n > 1, the n^{th} Jacobi coderivation J_{n} : ⋀(sL) → ⋀(sL) is given by

J_{n} = −\frac{1}{2} ∑_{i,j>1^{i+j=n+1}}[l_{j}, l_{i}].

For an L(m)-algebra, J_{n} = [l_{1}, l_{n}] for every 2 < n < m. Since [l_{1}, l_{n}] = l_{1}l_{n} + (−1)^{n−1}l_{n}l_{1} and l_{1} is a differential on L, it will be useful to have a name for the following map even though it is not a coderivation on ⋀(sL).

Definition 2.8. For n > 1, the n^{th} Jacobi obstruction map J_{OBST}_{n} : (sL)^{∧n} → sL is given by

J_{OBST}_{n} = J_{n} − l_{n}l_{1} = − ∑_{i>0 , j>1^{i+j=n+1}} l_{j}l_{i}

In the language of deformation theory (see, among others, [Ger63], [Ger64], [Ger66], [Ger68], [GS90], [GS88], [FGV95] and [FGV]), an L(m)-structure on L may be extended to an L(m+1)-structure if there is a linear map l_{m+1} : L^{∧m+1} → L such that J_{ID}_{m+1} = 0 or, equivalently, such that \frac{1}{2}[l_{1}, l_{m+1}] = J_{m+1}. What we are deforming here is a bit obscure and less important than the process of extending an L(m)-structure. The graded Lie algebra Coder(⋀ L), together with the differential given by [l_{1}, ], governs the extension theory of L(m)-structures on ⋀ L. If we were viewing the construction of an L(∞)-structure strictly as a deformation theory problem rather than simply borrowing the machinery, the map J_{n} would be called the n^{th} obstruction. However, in the context of our construction in §3, the use of the map J_{OBST}_{n} is more consistent with the common notion of an obstruction. The following lemmas concerning J_{n} and J_{OBST}_{n} will also be useful in §3.

Lemma 2.9. The coderivation J_{n} is a cocycle, i.e., [l_{1}, J_{n}] = 0.

Proof of Lemma 2.9. We will show that [l_{1}, −4J_{n}] = 0.

[l_{1}, −4J_{n}] = [l_{1}, ∑_{i,j⩾1^{i+j=n+1}} 2[l_{j}, l_{i}]] = [l_{1}, − ∑_{i,j⩾1^{i+j=n+1}} ([l_{j}, l_{i}] + [l_{i}, l_{j}])]

= ∑_{i,j⩾1^{i+j=n+1}} ([l_{1}, [l_{j}, l_{i}]] + [l_{1}, [l_{i}, l_{j}]]) .

Since [l_{1}, [l_{j}, l_{i}]] = [[l_{1}, l_{j}], l_{i}] − [[l_{1}, l_{i}], l_{j}], it follows that

[l_{1}, [l_{j}, l_{i}]] + [l_{1}, [l_{i}, l_{j}]]

= [[l_{1}, l_{j}], l_{i}]] − [[l_{1}, l_{i}], l_{j}]] + [[l_{1}, l_{i}], l_{j}]] − [[l_{1}, l_{j}], l_{i}]]
= 0.

Therefore, [l_{1}, J_{n}] = −\frac{1}{4}[l_{1}, −4J_{n}] = 0. □

Lemma 2.10. If sL admits an L(n)-structure, then l_{1}J_{OBST}_{n} = 0 on (sL)^{∧n}.

Proof of Lemma 2.10. The map J_{OBST}_{n} = − ∑_{i>0,j>1^{i+j=n+1}}(−1)^{i(j−1)}l_{j}l_{i}, so

l_{1}J_{OBST}_{n} = l_{1} (− ∑_{i>0,j>1^{i+j=n+1}} l_{j}l_{i}) = − ∑_{i>0,j>1^{i+j=n+1}} l_{1}(l_{j}l_{i})

= − ∑_{i>0,j>1^{i+j=n+1}}(l_{1}l_{j})l_{i}.

We substitute l_{1}l_{j} with − ∑_{s>0,r>1^{s+r=j+1}} l_{r}l_{s} and reassociate so that

∑_{i>0,j>1^{i+j=n+1}} ( ∑_{s>0,r>1^{s+r=j+1}} l_{r}l_{s}) l_{i} = ∑_{i,s>0,r>1^{i+s+r=n+2}} l_{r}(l_{s}l_{i}).

If we let i + s = t + 1, the sum above becomes

∑_{t>0,r>1^{t+r=n+1}} l_{r} ( ∑_{i,s>0^{i+s=t+1}} l_{s}l_{i}) = ∑_{t>0,r>1^{t+r=n+1}} l_{r}J_{ID}_{t}.

Since J_{ID}_{t} = 0 on (sL)^{∧t} for all 1 ⩽ t ⩽ n − 1, the final sum above is zero. □

Definition 2.11. [LM95] Let (sL, l_{i}) be an L(p)-algebra (0 < p < ∞) and let M be a differential graded module with differential m_{1}. Then a left L(k)-module structure over sL on M (for k ⩽ p) is a collection of coderivations {m_{n} : ⋀(sL) ⊗ M → ⋀(sL) ⊗ M : 1 ⩽ n ⩽ k, n ≠ ∞}. Each m_{n} is l_{n}-subordinate and the extension of a map m_{n} : L^{∧n−1} ⊗ M → M such that the n^{th} action identity map

ACT_{ID}_{n} = \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}[m_{j}, m_{i}] = 0

on ⋀(sL)⊗M. The differential graded module M is a strongly homotopy Lie module over sL (or an sL-shLie module) if sL admits an L(∞)-structure and M is a module with respect to that L(∞)-structure.

Definition 2.11 implies that the differential m_{1} on M must be l_{1}-subordinate. It is simple to verify that m_{1} is a differential on ⋀ L ⊗ M.

When M is a differential graded commutative algebra, we will insist the maps m_{i} be M-derivation sources, that is to say, the map m_{i}(v_{[1 to i−1]} ⊗ ( )) : M → M is in Der(M) for every v_{[1 to i−1]}^{∧} ∈ L^{∧i−1}.

In this paper a module B over a Lie algebra 𝔤 is a degenerate shLie module ⋀(s𝔤) ⊗ B over the shLie algebra ⋀(s𝔤), where all the coderivations m_{i} are zero except for m̂_{2} (again, we decorate this particular map).

Below, we define the maps ACT_{n} and ACT_{OBST}_{n}, the analogs of J_{n} and J_{OBST}_{n}.

Definition 2.12. For n > 1, the n^{th} action map ACT_{n} : ⋀(sL)⊗M → ⋀(sL)⊗M is given by

ACT_{n} = −\frac{1}{2} ∑_{i,j>1^{i+j=n+1}}[m_{j}, m_{i}].

Definition 2.13. For n > 1, the n^{th} action obstruction map ACT_{OBST}_{n} : (sL)^{∧n−1} ⊗ M → M is given by

ACT_{OBST}_{n} = ACT_{n} − m_{n}m_{1} = − ∑_{i>0,j>1^{i+j=n+1}} m_{j}m_{i}.

If sL is an L(p)-algebra and M is an L(n)-module over sL with n < p − 1 (or n < ∞ if p = ∞), we can extend the L(n)-module structure to an L(n + 1)-module structure if an l_{n+1}-subordinate M-derivation source m_{n+1} : (sL)^{∧n} ⊗ M → M can be found such that ACT_{ID}_{n+1} = 0. Again, what exactly we are deforming here is less important than the language and machinery of deformation theory.

The following lemmas and their proofs mimic lemmas 2.9 and 2.10 and their proofs.

Lemma 2.14. The coderivation ACT_{n} is a cocycle, i.e., [m_{1}, ACT_{n}] = 0.

Lemma 2.15. If L admits an L(p)-structure and M is an L-L(n)-module for n < p − 1 (or n < ∞ if p = ∞), then m_{1}ACT_{OBST}_{n} = 0 on L^{∧n−1} ⊗ M.

Proposition 2.16. [LS93] If (sL, l_{i}) is an shLie algebra, the map D_{sL} = ∑_{i=1}^{∞} l_{i} is a differential on ⋀(sL), i.e. it is a map of degree -1 such that D_{sL} ∘ D_{sL} = 0.

Proof of Proposition 2.16. A proof can be found in [LS93]. Here the proof is essentially the same but takes advantage of the bracket of coderivations:

D_{sL} ∘ D_{sL} = (∑_{j=1}^{∞} l_{j}) ∘ (∑_{i=1}^{∞} l_{i}) = ∑_{j=1}^{∞} ∑_{i=1}^{∞} l_{j}l_{i} = ∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}} l_{j}l_{i}.

Using equation in definition 2.6, we see that

∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}} l_{j}l_{i} = \frac{1}{2} ∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}}[l_{j}, l_{i}] = \frac{1}{2} ∑_{n=1}^{∞} J_{ID}_{n} = 0.

Note also that D_{sL} is a degree −1 coderivation, so D_{sL} ∘ D_{sL} = \frac{1}{2}[D_{sL}, D_{sL}] and the graded skew-commutativity of the bracket ensures that [D_{sL}, D_{sL}] = 0. □

For the Lie algebra ⋀(s𝔤), the differential D_{s𝔤} = l̂_{2}.

Proposition 2.17. The map D_{M} = ∑_{i=1}^{∞} m_{i} is a differential on ⋀(sL) ⊗ M, i.e., D_{M} ∘ D_{M} = 0.

Proof of Proposition 2.17. Isomorphic to the proof of proposition 2.16. Notice that D_{M} is D_{sL}-subordinate and that D_{M} ∘ D_{M} = \frac{1}{2}[D_{M}, D_{M}] = 0. □

For the Lie module ⋀(s𝔤) ⊗ B, the differential D_{B} = m̂_{2}.

Homotopy Chevalley-Eilenberg cohomology Definition 2.18. A homotopy Chevalley-Eilenberg pair (M, L) consists of an shLie algebra L and an L-shLie module M.

A multi-linear graded alternating function F_{n} : L^{×n} → M can be seen as a linear function F_{n} : (sL)^{∧n} → M. So the homotopy Chevalley-Eilenberg algebra Alt_{k}(L, M) is isomorphic to Hom_{k}(⋀(sL), ⋀(sL)⊗M) with the differential δ_{hCE} = ⟨D_{M}, ⟩

Proposition 2.19. δ_{hCE} ∘ δ_{hCE} = 0.

Proof of Proposition 2.19. δ_{hCE} ∘ δ_{hCE} = ⟨D_{M}, ⟨D_{M}, ⟩⟩ = ⟨\frac{1}{2}[D_{M}, D_{M}], ⟩ = 0. □

The cohomology with respect to δ_{hCE} is the homotopy Chevalley-Eilenberg cohomology of sL with coefficients in M. A quick check shows that the Chevalley-Eilenberg cohomology complex for the pair (B, s𝔤) is simply a degenerate form of the homotopy version where δ_{CE} = δ_{hCE} = ⟨m̂_{2}, ⟩ because m̂_{i} and l̂_{i} are zero for all i ≠ 2.

When M is a differential graded commutative algebra, the homotopy Chevalley-Eilenberg complex Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) is a differential graded commutative algebra. The differential δ_{hCE} acts as a derivation with respect to this multiplication.

Homotopy Lie-Rinehart pairs and homotopy Rinehart cohomology The homotopy Rinehart complex is a straightforward generalization of the Rinehart complex in the ungraded setting. Since the Rinehart complex is defined only for Lie-Rinehart pairs (B, s𝔤), we must define what constitutes a homotopy Lie-Rinehart pair (M, sL). The subset Alt_{M}(sL, M) of Alt_{k}(sL, M) consisting of all M-linear graded alternating functions is isomorphic to Hom_{M}(⋀(sL), ⋀(sL) ⊗ M).

Definition 2.20. A homotopy Lie-Rinehart pair (M, sL) consists of a differential graded commutative algebra (M, m_{i}) which is an shLie module over the shLie algebra (sL, l_{i}), which in turn is an M-module. Moreover, the following two homotopy Lie-Rinehart relations must be satisfied for all i ⩾ 1:

(hLRa_{i}): The shLie module structure map m_{i} is M-linear.

(hLRb_{i}): The shLie structure map l_{i} rests on m_{i} (definition 2.5). An L(p)-Lie-Rinehart pair ((M, m_{i}), (sL, l_{i})) has maps m_{i} and l_{i} which satisfy

(hLRa_{i}) and (hLRb_{i}) for 1 ⩽ i ⩽ p.

The proposition below follows from the fact that, for every i, the image of F under the map ⟨m̂_{i}, ⟩ is again a map in Hom_{M}(⋀(sL), ⋀(sL) ⊗ M) for all i [Kje01].

Proposition 2.21. If F is M-linear, then so is δ_{hCE}F.

We conclude that the subset of all M-linear functions in Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) forms a subcomplex R = Hom_{M}(⋀(sL), ⋀(sL) ⊗ M) with differential δ_{R} = ⟨D_{M}, ⟩. The differential, together with the cup product, provides the homotopy Rinehart complex R with the structure of a differential graded commutative algebra. The cohomology of R with respect to δ_{R} is the homotopy Rinehart cohomology of sL with coefficients in M.

The following two propositions for general homotopy Lie-Rinehart pairs (M, sL) are observations based on the fact that both Coder_{M}^{M}(⋀(sL)⊗M) and Coder_{M}^{M}(⋀(sL)) are graded Lie algebras.

Proposition 2.22. The maps ACT_{ID}_{n} and ACT_{n} are are J_{ID}_{n} and J_{n}-subordinate, respectively; both are M-linear and M-derivation sources.

The significance of this proposition lies in recognizing that, as a result, each of the maps ACT_{ID}_{n} and ACT_{n} are completely determined by their image on a generating set for ⋀(sL) as an M-module and a generating set for M, so long as the maps J_{ID}_{n} and J_{n} are available.

Proposition 2.23. The maps J_{ID}_{n} and J_{n} rest on ACT_{ID}_{n} and ACT_{n}, respectively.

Here again, the implication is that the maps J_{ID}_{n} and J_{n} are completely determined by their image on a generating set of ⋀(sL) as an M-module.

Homotopy Lie-Rinehart resolutions of Lie-Rinehart pairs

Let (B, s𝔤) be a Lie-Rinehart pair over a k-algebra A with module structure maps μ̂ : B ⊗ s𝔤 → s𝔤 and m̂_{2} : s𝔤 ⊗ B → B. Let l̂_{2} denote the bracket on s𝔤 and π_{B} denote the multiplication on B. The basic ingredients of a homotopy Lie-Rinehart resolution for a Lie-Rinehart pair (B, s𝔤) are (sL, l_{i}) and (M, m_{i}, π_{M}) where H_{l_{1}}(sL) = s𝔤 and H_{m_{1}}(M) = B.

Definition 2.24. A homotopy Lie-Rinehart resolution of a Lie-Rinehart pair (B, s𝔤) over an algebra A is a homotopy Lie-Rinehart pair (M, sL) over A, such that

(a) the shLie algebra (sL, l_{i}), seen as the coalgebra ⋀(sL), resolves ⋀(s𝔤), i.e.,

H_{l_{1}}(⋀(sL)) = ⋀(s𝔤),  (6)

where (following the physicists’ notation) H_{l_{1}} denotes the homology with respect to the differential l_{1},

(b) the differential graded commutative algebra (M, m_{i}, π_{M}) satisfies

H_{m_{1}}(⋀(sL) ⊗ M) = ⋀(s𝔤) ⊗ B,  (7)

(c) the dgca (M, m_{i}, π_{M}) also satisfies

H_{m_{1}}(M ⊗ M) = B ⊗ B.  (8)

Furthermore, the following conditions hold:

i. H_{l_{1}}(l_{2}) = l̂_{2} on ⋀(s𝔤).

ii. H_{m_{1}}(m_{2}) = m̂_{2} on ⋀(s𝔤) ⊗ B. iii. H_{m_{1}}(π_{M}) = π_{B} on B ⊗ B.

In the proposition below, we state conditions on the homotopy Lie-Rinehart pair that guarantee it satisfies definition 2.24. The proof is found in [Kje01].

Proposition 2.25. Let (M, sL) be a homotopy Lie-Rinehart pair. Suppose the differential graded algebra (M, m_{i}, π_{M}) is a projective resolution of B over A which respects the algebra structure on B, i.e., condition iii of definition 2.24 is satisfied. Suppose (sL, l_{i}) is a projective resolution s𝔤. Furthermore,

(a) H_{l_{1}}(l_{2}) = l̂_{2} on s𝔤 ∧ s𝔤 and

(b) H_{m_{1}}(m_{2}) = m̂_{2} on s𝔤 ⊗ B. Then (M, sL) is a homotopy Lie-Rinehart resolution of (B, s𝔤).

We used a spectral sequence argument in [Kje01] to prove the following theorem.

Theorem 2.26. If the pair (M, sL) is a projective homotopy Lie-Rinehart resolution of the Lie-Rinehart pair (B, s𝔤), then (R, ⟨D_{M}, ⟩) is a cohomological model for (R, ⟨m̂_{2}, ⟩).

3. Constructing the homotopy Lie-Rinehart pair (K_{A/I}, K_{I/I^{2}}) and classical BRST cohomology

Constructing homotopy Lie-Rinehart resolutions for a Lie-Rinehart pair is not always easy, as we will see as we construct a homotopy Lie-Rinehart resolution (K_{A/I}, K_{I/I^{2}}) for the Lie-Rinehart pair (A/I, I/I^{2}) from classical BRST cohomology. Details which are not instructive (especially sign arguments) are ommitted, but can be found in [Kje96].

The Lie algebra I/I^{2} has already been suspended, i.e., every element of I/I^{2} has degree +1 and the bracket l̂_{2} is a coderivation on ⋀ I/I^{2}. However, we will suppress the suspension indicator “s” both here and in the shLie algebra K_{I/I^{2}}.

Let the set y_{α} = {y_{1}, ..., y_{s}} generate the coisotropic ideal I as an finitely presented A-module. If y_{α} forms a basis of I over the field k, there is a unique set of structure constants C_{αβ}^{δ} ∈ k, each of which is antisymmetric in its lower indices, such that {y_{α}, y_{β}} = C_{αβ}^{δ}y_{δ}. (Note: We will use Einstein’s summation convention throughout §3 and §4.) With structure constants, the Jacobi identity is a statement about coefficients, namely C_{αβ}^{δ}C_{δγ}^{ϵ} + C_{βγ}^{δ}C_{δα}^{ϵ} + C_{γα}^{δ}C_{δβ}^{ϵ} = 0 for every ϵ. When the generators do not form a basis for I over k, we can still find (not necessarily unique) structure functions C_{αβ}^{δ} in A such that {y_{α}, y_{β}} = C_{αβ}^{δ}y_{δ}. (In the BRST context, the elements of the Poisson algebra A are functions on a symplectic manifold, hence the term structure functions.) The Jacobi identity becomes (C_{αβ}^{δ}C_{δγ}^{ϵ} + {C_{αβ}^{ϵ}, y_{γ}} + c.p.)y_{ϵ} = 0, where c.p. stands for cyclic permutations of α, β and γ. For convenience we will set J^{ϵ} equal to (C_{αβ}^{δ}C_{δγ}^{ϵ} + {C_{αβ}^{ϵ}, y_{γ}} + c.p.), so that the Jacobi identity can be written as J^{ϵ}y_{ϵ} = 0. The Jacobi identity is now a relation among the generators of I.

Definition 3.1. [Kim93], [FHST89] Let I be finitely generated by the set y_{α} = {y_{1}, ..., y_{s}}. The set is irreducible if f^{α}y_{α} = 0 implies f^{α} = g^{αβ}y_{β}, where each coefficient g^{αβ} is antisymmetric with respect to its indices. An ideal which has a set of irreducible generators is called irreducible.

Definition 3.2. [Kim93], [FHST89] An ideal I which is finitely generated by the set y_{α_{1}} = {y_{1}, y_{2}, …, y_{s_{1}}} admits a complete set of reducibility functions if there exists Z_{α_{n}}^{α_{n−1}} in the complement of I, such that for n = 2, Z_{α_{2}}^{α_{1}}y_{α_{1}} = 0 and f^{α_{1}}y_{α_{1}} = 0 implies that f^{α_{1}} = g^{α_{2}}Z_{α_{2}}^{α_{1}} + Υ^{α_{1}β_{1}}y_{β_{1}}, where Υ^{α_{1}β_{1}} = −Υ^{β_{1}α_{1}}. For n > 2, Z_{α_{n}}^{α_{n−1}}Z_{α_{n−1}}^{α_{n−2}} ≡ 0 mod I and f^{α_{n−1}}Z_{α_{n−1}}^{α_{n−2}} ≡ 0 mod I implies f^{α_{n−1}} ≡ g^{α_{n}}Z_{α_{n}}^{α_{n−1}} mod I. An ideal which admits a complete set of reducibility functions is reducible.

We consider only ideals which are either irreducible or reducible. Since the Jacobi identity is a relation among the generators of I, when the ideal is irreducible, we have J^{ϵ}y_{ϵ} = 0 implies J^{ϵ} = g^{ϵδ}y_{δ}. When the ideal is completely reducible, we have J^{ϵ}y_{ϵ} = 0 implies J^{ϵ} = g^{α_{2}}Z_{α_{2}}^{ϵ} + Υ^{ϵβ_{1}}y_{β_{1}}.

The quotient I/I^{2} inherits a Lie structure from I and is both an A and an A/Imodule. There is a Lie action of I on A given by z · f = {z, f}, which passes to the quotient so that A/I is a Lie-module over both I and I/I^{2}. Although (A/I, I) is not a Lie-Rinehart pair, the pair (A/I, I/I^{2}) is. The actions μ and ω are given by

\begin{matrix}μ : A/I ⊗ I/I^{2} & → & I/I^{2} \\ \underline{f} ⊗ z̄ & ↦ & \overline{fz} \end{matrix} and \begin{matrix}ω : I/I^{2} ⊗ A/I & → & A/I \\ z̄ ⊗ \underline{f} & ↦ & \underline{\{z, f\}}, \end{matrix}

where and ¯ denote equivalence classes. It is a straightforward exercise to show that the Lie-Rinehart relations are satisfied.

The construction of (K_{A/I}, K_{I/I^{2}}) Before we describe how we construct the homotopy Lie-Rinehart resolution (K_{A/I}, K_{I/I^{2}}) for the pair (A/I, I/I^{2}), we skip ahead to STEP 1, where we review the features of the Koszul-Tate resolution K_{A/I} of A/I (see [Tat57] and [Joz72]), a free differential graded commutative algebra over A which is a projective resolution of A/I.

STEP 1: The Koszul-Tate resolution K_{A/I} The Koszul complex K_{(0)} is a dgca over A and is isomorphic to A⊗⋀{P_{1}, P_{2}, ..., P_{s}}, where the P_{i}’s are assigned degree 1 and are in one-to-one correspondence with the generators y_{i}. (Following the physics literature, we call the P_{i}’s antighosts [HT92].) The differential m_{1} maps each P_{i} to y_{i} and m_{1} is extended as an A-linear graded derivation. The zeroth homology of the Koszul complex is A/I even if higher homologies do not vanish. Using the inductive method Tate introduced in [Tat57], we kill an unwanted nontrivial homology class [z_{q}] of degree q by adjoining a formal variable P_{z_{q}} of degree q +1 to the existing Koszul complex. The variable P_{z_{q}} maps to z_{q} under the differential, killing the unwanted homology. But these new variables may introduce new nontrivial homology on higher levels, which must in turn be killed. More formally, if I is reducible and generated by the set y_{α_{1}} = {y_{1}, ..., y_{s_{1}}}, the zeroth homology of the Koszul complex K_{(0)} = A⊗⋀{P_{α_{1}}} is A/I, but H_{1}(K_{(0)}) may not be zero. The unwanted homology classes form a module generated by the cycles z_{α_{2}} = Z_{α_{2}}^{α_{1}}P_{α_{1}}, so we adjoin degree 2 antighosts P_{α_{2}} and setting m_{1}(P_{α_{2}}) = z_{α_{2}}. The resulting differential graded commutative algebra K_{(1)} = A ⊗ ⋀{P_{α_{1}}, P_{α_{2}}} has homology H_{0}(K_{(1)}) = A/I and H_{1}(K_{(1)}) = 0. Again, H_{2}(K_{(1)}) may contain nontrivial classes, either as a result of introducing the level two antighosts or because they were already present in K_{(0)}. Regardless of origin, we choose a generating set z_{α_{3}} for the nontrivial 2-cycles and adjoin degree 3 antighosts P_{α_{3}} to kill them. The third homology of K_{(2)} may be nonzero, so we continue the process. The complete reducibility of I guarantees that a representative for each nontrivial n-class can be chosen so that each z_{α_{n+1}} = f_{α_{n+1}}^{α_{n}}P_{α_{n}}+ “more”. The limit K_{(∞)} is the Koszul-Tate projective resolution K_{A/I} of A/I and is isomorphic to A ⊗ ⋀P, where P is the graded vector space with basis {P_{α_{1}}, P_{α_{2}}, P_{α_{3}}, … }. The product π_{K} on K_{A/I} is a chain map. It is straightforward to check that H_{m_{1}}(π_{K}) = π_{A/I}, that is to say,

condition (iii) in definition 2.24 is satisfied.

A chain f^{I}P_{I} in K_{A/I}, where P_{I} = P_{α_{n_{1}}} ⋯ P_{α_{ns}}, has homogeneous degree, which we again denote by I. In context, we will not confuse the index I with the degree I. The boundary of an antighost P_{α_{n}} will be denoted as Z_{α_{n}}^{α_{n−1}}P_{α_{n−1}} + ZÎ ; α_{n}PÎ whenever the “linear” term plays an important role. Otherwise, we will set m_{1}(P_{α_{n}}) = Z_{α_{n}}^{I}P_{I}. Whenever possible, an element of K_{A/I} will be denoted simply by X.

When A is a Poisson algebra and the ideal I is a multiplicative ideal, the Koszul-Tate resolution K_{A/I} admits a graded Poisson bracket given by {f^{I}P_{I}, g^{J}P_{J}} = {f^{I}, g^{J}}P_{I}P_{J}.

The Procedure The construction of (K_{A/I}, K_{I/I^{2}}) relies on four features: the Poisson structure on the Koszul-Tate resolution K_{A/I}, the relations among the generators of I, the fact that both K_{A/I} and K_{I/I^{2}} are projective resolutions of A/I and I/I^{2}, respectively, and an A-module coderivation Ψ from K_{A/I} to K_{I/I^{2}}. A general outline of the process follows:

STEP 1: Choose a Koszul-Tate resolution K_{A/I} for A/I. The differential on K_{A/I} is m_{1}, which must satisfy condition (iii) in definition 2.24. (Already done!) STEP 2: Construct a differential graded K_{A/I}-module K_{I/I^{2}} which is a projective resolution of I/I^{2} (See propositions 3.3 and 3.4). The action μ : K_{A/I} ⊗ K_{I/I^{2}} → K_{I/I^{2}} is free, i.e., K_{I/I^{2}} is isomorphic to K_{A/I} ⊗ Φ, where Φ is a graded vector space over k which is isomorphic to the graded vector space P. (We have already suspended the graded vector space Φ. Therefore, K_{I/I^{2}} is already suspended.) We construct the differential on K_{I/I^{2}} using the coderivation Ψ. The differential l_{1} rests on m_{1}, so the pair (K_{A/I}, K_{I/I^{2}}) has an L(1)-Lie-Rinehart structure. STEP 3: The loop—defining the homotopy Lie-Rinehart structure maps m_{n} and l_{n}. For n ⩾ 2, the loop extends the L(n − 1)-Lie-Rinehart structure to an L(n)-Lie-Rinehart structure.

(a_{n}). Constructing m_{n}. Since the map m_{n} must satisfy the homotopy Lie-Rinehart relation (hLRa_{n}) and be a K_{A/I}-derivation source, the map m_{n} is completely determined once we define it on Φ^{∧n−1} ⊗ A and the basis elements φ_{[1 to n−1]} ⊗ P_{n} of Φ^{∧n−1} ⊗ P, such that φ_{1} ⩽ ⋯ ⩽ φ_{n} with respect to a degree-preserving total ordering of the preghosts: i. Define m_{n} on Φ^{∧n−1} ⊗ A by setting m_{n}(φ_{[1 to n−1]} ⊗ f) =

(−1)^{n}{m_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}.

Here, the inductive assumption (step a_{n−1}iii “above”) guarantees that m_{n} is well-defined on Φ^{∧n−1} ⊗ A. The extended Poisson bracket on K_{A/I} in the definition guarantees that m_{n} will be an A-derivation source. (Does not require proof.)

ii. Verify that ACT_{ID}_{n} = 0 on Φ^{∧n−1} ⊗ A by showing that m_{1}m_{n} = ACT_{OBST}_{n}on that subspace. (Requires proof. See proposition 3.5.) iii. Ensure that m_{n} is well-defined and that ACT_{ID}_{n} = 0 on Φ^{∧n−1} ⊗ P by exploiting the acyclicity of K_{A/I} to define m_{n} on the basis φ_{[1 to n−1]} ⊗ P_{n}of Φ^{∧n−1} ⊗ P with φ_{1} ⩽ ⋯ ⩽ φ_{n}. (Requires proof. See proposition 3.6.) iv. Since ACT_{ID}_{n} is completely determined by its image on the generating set Φ^{∧n−1} ⊗ A and Φ^{∧n−1} ⊗ P (recall Proposition 2.22), it follows that ACT_{ID}_{n} = 0 on all of K_{I/I^{2}}^{∧n−1} ⊗K_{A/I} and therefore ACT_{ID}_{n} = 0 on ⋀ K_{I/I^{2}} ⊗ K_{A/I}. (Does not need proof.)

(b_{n}). Constructing l_{n}. The map l_{n} must rest on m_{n} and is therefore completely determined once we define it on Φ^{∧n}. i. Set l_{n}(φ_{[1 to n]}) = Ψm_{n}(φ_{[1 to n−1]} ⊗P_{n}). Since m_{n} is well-defined on Φ^{∧n−1} ⊗ P, it follows that the map l_{n} is well-defined on Φ^{∧n}. (Does not need proof.)

ii. Verify that J_{ID}_{n} = 0 on Φ^{∧n}. We do so by showing that on the space Φ^{∧n−1} ⊗ P, the sequence of equalities

l_{1}l_{n}(1^{∧n−1} ⊗ Ψ) = Ψm_{1}m_{n} = ΨACT_{OBST}_{n} = J_{OBST}_{n}(1^{∧n−1} ⊗ Ψ)

holds. (Requires proof. See proposition 3.7.) iii. Since J_{ID}_{n} rests on ACT_{ID}_{n} (recall Proposition 2.23) and J_{ID}_{n} = 0 on Φ^{∧n}, it follows that J_{ID}_{n} = 0 on all of K_{I/I^{2}}^{∧n} and therefore J_{ID}_{n} = 0 on ⋀ K_{I/I^{2}}. (Does not need proof.)

STEP 4: Verify that the conditions (i) and (ii) in definition 2.24 are met.

STEP 2: Constructing K_{I/I^{2}} The shLie algebra K_{I/I^{2}} is the free K_{A/I}-module K_{A/I} ⊗ Φ. The graded vector space Φ is spanned by the graded basis {φ_{α_{1}}, φ_{α_{2}}, φ_{α_{3}}, … }, where there is a one-to-one correspondence between the preghosts φ_{α_{i}}, and the antighosts P_{α_{i}}. Each φ_{α_{n}} is assigned degree n. (The vector space Φ is the suspension of the vector space Φ found in [Kje96].) A typical element has the form X^{I}P_{I}^{α_{n}}φ_{α_{n}}, where X^{I}P_{I}^{α_{n}} is an element of K_{A/I}. As before, the degree of X^{I}P_{I}^{α_{n}} in K_{A/I} will be denoted by I. The degree of the element X^{I}P_{I}^{α_{n}}φ_{α_{n}}, then, is the sum (I + n). Whenever possible, an abbreviated form X^{α_{n}}φ_{α_{n}} will be used for a typical element, in which case the degree of X^{α_{n}} will be denoted by α_{n}, again, without confusion.

When I is irreducible, the map l_{1} = m_{1}⊗1 is a differential on K_{I/I^{2}} ≈ K_{A/I} ⊗Φ, whose homology is A/I ⊗ Φ, which is isomorphic to I/I^{2} as A/I-modules. The isomorphism is given by sending \underline{f^{α_{1}}}φ_{α_{1}} ↦ \overline{f^{α_{1}}y_{α_{1}}}. The details are left to the reader. It follows that K_{I/I^{2}} is a resolution of I/I^{2}.

We define a degree 0 A-linear map Ψ : K_{A/I} ⟶ K_{I/I^{2}} by setting Ψ(P_{α_{n}}) = φ_{α_{n}} and extending Ψ as a graded derivation, i.e., Ψ(P_{α_{i}}P_{α_{j}}) = (−1)^{ij}P_{α_{j}}φ_{α_{i}} + P_{α_{i}}φ_{α_{j}}. In the irreducible case, the map Ψ is a chain map.

In the reducible case, the graded module K_{I/I^{2}} does not come equipped with a differential; we must build one. The tool we need is the map Ψ. We create l_{1} on K_{I/I^{2}} so that Ψ is a chain map. We set l_{1}(φ_{α_{n}}) = Ψm_{1}(P_{α_{n}}) and extend l_{1} to all of K_{I/I^{2}} so that it rests on m_{1}. The map Ψ is a chain map provided l_{1} is a differential on K_{I/I^{2}}.

Proposition 3.3. l_{1} ∘ l_{1} = 0 on K_{I/I^{2}}.

Proof of Proposition 3.3. We begin by showing that l_{1}(l_{1}(φ_{α_{n}})) = 0. Recall that Ψ is a chain map. It follows that l_{1}(l_{1}(φ_{α_{n}})) = l_{1}(Ψ(m_{1}(P_{α_{n}}))) = Ψ(m_{1}(m_{1}(P_{α_{n}}))) = 0. Since l_{1} rests on m_{1}, it follows that

l_{1}(l_{1}(X^{α_{n}}φ_{α_{n}})) = l_{1}(m_{1}(X^{α_{n}})φ_{α_{n}} + (−1)^{α_{n}}X^{α_{n}}l_{1}(φ_{α_{n}}))
= m_{1}(m_{1}(X^{α_{n}}))φ_{α_{n}} + (−1)^{α_{n}−1}m_{1}(X^{α_{n}})l_{1}(φ_{α_{n}})
+(−1)^{α_{n}}m_{1}(X^{α_{n}})l_{1}(φ_{α_{n}}) + (−1)^{α_{n}}X^{α_{n}}l_{1}(l_{1}(φ_{α_{n}}))
= (−1)^{α_{n}}X^{α_{n}}l_{1}(l_{1}(φ_{α_{n}}))
= 0. □

The spectral sequence argument below proves that K_{I/I^{2}} resolves I/I^{2} as modules.

Proposition 3.4. H_{l_{1}}(K_{I/I^{2}}) = I/I^{2}.

Proof of Proposition 3.4. The module K_{I/I^{2}} is naturally bigraded. The bidegree of the element X^{α_{n}}φ_{α_{n}} is (α_{n}, n). The differential decomposes as l_{1} = (m_{1} ⊗1)+d^{(1)} + d^{(2)} + ⋯ with respect to the filtration of K_{I/I^{2}} by preghost degree, where m_{1} ⊗ 1 has bidegree (−1, 0) and d^{(i)} has bidegree (i − 1, i). The differential on the E_{0} term is m_{1} ⊗ 1. The k^{th} row of the E_{0} term is K_{A/I} ⊗ Φ^{(k)} where Φ^{(k)} is the vector space spanned by the φ_{α_{k}}’s. Each row is exact except in the first slot. Therefore the E_{1} term of the spectral sequence is concentrated in the first column:

E_{1} :  ⋮
↓d̃^{(1)}
A/I ⊗ Φ^{(3)}
↓d̃^{(1)}
A/I ⊗ Φ^{(2)}
↓d̃^{(1)}
A/I ⊗ Φ^{(1)}
↓
0 .

Since the E_{1} term collapses to one column, the homology of K_{I/I^{2}} with respect to l_{1} will be the homology of the E_{1} term.

Let K be the kernel of the map π : A/I ⊗ Φ^{(1)} ⟶ I/I^{2} which sends \underline{f^{α_{1}}} ⊗

φ_{α_{1}} to \overline{f^{α_{1}}y_{α_{1}}}. We need to show that d̃^{(1)}(A/I ⊗ Φ^{(1)}) = K. (⊆): Since l_{1}φ_{α_{n}} =

Z_{α_{n}}^{α_{n−1}}φ_{α_{n−1}} + “more”, it follows that d̃^{(1)}\underline{f^{α_{n}}}φ_{α_{n}} = \underline{f^{α_{n}}Z_{α_{n}}^{α_{n−1}}}φ_{α_{n−1}}. Consider any

element \underline{f^{α_{2}}}φ_{α_{2}} of A/I ⊗ Φ^{(2)}. The image under πd̃^{(1)} is zero because for all α_{2} the sum Z_{α_{2}}^{α_{1}}y_{α_{1}} = 0. (⊇): Suppose π(\underline{f^{α_{1}}}φ_{α_{1}}) = \overline{f^{α_{1}}y_{α_{1}}} = 0. Recalling that the ideal I admits a complete set of reducibility functions, we see that

f^{α_{1}}y_{α_{1}} = h^{α_{1}β_{1}}y_{β_{1}}y_{α_{1}} ⇒ (f^{α_{1}} − h^{α_{1}β_{1}}y_{β_{1}})y_{α_{1}} = 0
⇒ f^{α_{1}} ≡ g^{α_{2}}Z_{α_{2}}^{α_{1}} mod I.

It then follows that d̃^{(1)}(\underline{g^{α_{2}}}φ_{α_{2}}) = \underline{g^{α_{2}}Z_{α_{2}}^{α_{1}}}φ_{α_{1}} = \underline{f^{α_{1}}}φ_{α_{1}}. We conclude that the zeroth cohomology of K_{I/I^{2}} with respect to l_{1} is (A/I ⊗ Φ^{(1)})/K, which is isomorphic to I/I^{2} as modules. For higher cohomologies, suppose \underline{f^{α_{n}}}φ_{α_{n}} is a cocycle. Then

\underline{f^{α_{n}}Z_{α_{n}}^{α_{n−1}}}φ_{α_{n−1}} = 0 ⇒ f^{α_{n}}Z_{α_{n}}^{α_{n−1}} ∈ I
⇒ f^{α_{n}} ≡ g^{α_{n+1}}Z_{α_{n+1}}^{α_{n}} mod I
⇒ \underline{f^{α_{n}}}φ_{α_{n}} = \underline{g^{α_{n+1}}Z_{α_{n+1}}^{α_{n}}}φ_{α_{n}} = d̃^{(1)}\underline{g^{α_{n+1}}}φ_{α_{n+1}}.

Hence, the higher cohomologies are zero. □

The map Ψ is quite useful. We constructed l_{1} so that Ψ was a chain map. In the notation of shLie algebras and shLie modules, we were able to show that J_{ID}_{1} = 0 on Φ because ΨACT_{ID}_{1} = J_{ID}_{1}Ψ and ACT_{ID}_{1} = 0 on Ψ^{−1}Φ. We shall see this pattern of proof again in proposition 3.7.

Let P_{I} = P_{I_{1}} ⋯ P_{I_{s}} ∈ K_{A/I}. Then Ψ(P_{I}) = �(I_{i})P_{I}^{I_{i}}φ_{I_{i}}, where P_{I}^{I_{i}} = P_{I_{1}} ⋯ P̂_{I_{i}} ⋯ P_{I_{s}} and P̂_{I_{i}} indicates that this factor is omitted. The sign �(I_{i}) is the sign produced by moving P_{I_{i}} past P_{I_{1}} ⋯ P_{I_{i−1}}. Recalling that m_{1}P_{α_{n}} = Z_{α_{n}}^{I}P_{I}, we can write l_{1}(φ_{α_{n}}) = �(I_{i})Z_{α_{n}}^{I}P_{I}^{I_{i}}φ_{I_{i}}.

STEP 3: The loop We prove the three propositions needed to complete the inductive step of the loop. Then we will return to n = 2 and define m_{2} at the one location where both K_{A/I} and K_{I/I^{2}} are not exact.

Proposition 3.5. (3a_{n}ii) If we set m_{n}(φ_{[1 to n−1]} ⊗ f)=(−1)^{n}{m_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}, for φ_{1} ⩽ ⋯ ⩽ φ_{n−1}, then

m_{1}(m_{n}(φ_{[1 to n−1]} ⊗ f)) = ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ f).

Proof of Proposition 3.5. We omit sign arguments and specific signs, opting instead for ±. Using the definition, m_{1}m_{n}(φ_{[1 to n−1]} ⊗ f)

= ±m_{1}{m_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}
= ±{m_{1}(m_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1})), f} ± Z_{[1 to n−1]}^{A}{m_{1}(P_{A}), f},

where m_{n−1}(φ_{[1 to n−2]} ⊗P_{n−1}) = Z_{[1 to n−1]}^{A}P_{A}. Now, ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗f) contains one term ±m_{2}(l_{n−1}(φ_{[1 to n−1]}) ⊗ f) which uses the (n − 1, 0)-unshuffle. But since

l_{n−1}(φ_{[1 to n−1]_{)}} = Ψm_{n−1}(φ_{[1 to n−1]_{⊗}}P_{n−1}) = Z_{[1 to n−1]}^{A}P_{A}^{β}φ_{β}

and m_{2} is K_{A/I}-linear,

m_{2}(Z_{[1 to n−1]}^{A}P_{A}^{β}φ_{β} ⊗ f) = ±Z_{[1 to n−1]}^{A}P_{A}^{β}{m_{1}P_{β}, f} = ±Z_{[1 to n−1]}^{A}{m_{1}(P_{A}), f}.

Since m_{1}m_{n−1} is equal to ACT_{OBST}_{n−1}, we may rewrite ±{m_{1}m_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f} as ±{ACT_{OBST}_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}, which must equal the remaining terms in the map ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ f).

We organize the remaining terms by unshuffle. Let σ be an (i, j−2)-unshuffle. Applying σ to φ_{[1 to n−1]} produces two hands φ_{σ[1 to i]} and φ_{σ[i+1 to n−1]}. In ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ f), there are two terms in which the φ_{i}’s appear in the σ order. They are

(𝕄_{j}𝕃_{i}σ) = ±m_{j}(l_{i}(φ_{σ[1 to i]}) ∧ φ_{σ[i+1 to n−1]} ⊗ f)

and

(𝕄_{i+1}𝕄_{j−1}σ) = ±m_{i+1}(φ_{σ[1 to i]} ⊗ m_{j−1}(φ_{σ[i+1 to n−1]} ⊗ f)).

There is a unique (j−2, i)-unshuffle ρ which switches the σ hands, i.e., φ_{ρ[j−1 to n−1]} = φ_{σ[1 to i]} and φ_{ρ[1 to j−2]} = φ_{σ[i+1 to n−1]}. The two terms in which the φ_{i}’s appear in the ρ order, written in terms of σ are

(𝕄̃_{i+2}𝕃̃_{j−2}σ) = ±m_{i+2}(l_{j−2}(φ_{σ[i+1 to n−1]}) ∧ φ_{σ[1 to i]} ⊗ f)

and

(𝕄̃_{j−1}𝕄̃_{i+1}σ) = ±m_{j−1}(φ_{σ[i+1 to n−1]} ⊗ m_{i+1}(φ_{σ[1 to i]} ⊗ f)).

Organized in this way, the terms remaining in ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ f) can be rewritten as

\frac{1}{2} ∑_{2<j<n^{i+j=n+1}} ∑_{(i,j−2)^{σ}} 𝕄_{j}𝕃_{i}σ + 𝕄_{i+1}𝕄_{j−1}σ + 𝕄̃_{i+2}𝕃̃_{j−2}σ + 𝕄̃_{j−1}𝕄̃_{i+1}σ.

Without loss of generality, we now compute the sum

𝕄_{j}𝕃_{i}e + 𝕄_{i+1}𝕄_{j−1}e + 𝕄̃_{i+2}𝕃̃_{j−2}e + 𝕄̃_{j−1}𝕄̃_{i+1}e

for the identity (i, j − 2)-unshuffle e and show that the sum equals the two terms in

±{ACT_{OBST}_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}

for which φ_{[1 to n−2]} ⊗P_{n−1} is split into the hands φ_{[1 to i]} and φ_{[i+1 to n−2]} ⊗P_{n−1}.

The terms of the sum 𝕄_{j}𝕃_{i}e + 𝕄_{i+1}𝕄_{j−1}e + 𝕄̃_{i+2}𝕃̃_{j−2}e + 𝕄̃_{j−1}𝕄̃_{i+1}e expand to become

(𝕄_{j}𝕃_{i}e) = ±{m_{j−1}(l_{i}(φ_{[1 to i]}) ∧ φ_{[i+1 to n−2]} ⊗ P_{n−1}, f}  (A)
±{Z_{[1 to i]}^{A}, f}m_{j−1}(φ_{[i+1 to n−1]} ⊗ P_{A}),  (B)
(𝕄̃_{j−1}𝕄̃_{i+1}e) = ±{Z_{[i+1 to n−1]}^{B}, {Z_{[1 to i]}^{A}, f}}P_{B}P_{A}  (C)
±{Z_{[1 to i]}^{A}, f}m_{j−1}(φ_{[i+1 to n−1]} ⊗ P_{A}),  (D)
(𝕄_{i+1}𝕄_{j−1}e) = ±{Z_{[1 to i]}^{A}, {Z_{[i+1 to n−1]}^{B}, f}}P_{A}P_{B}  (E)
±{Z_{[i+1 to n−1]}^{B}, f}m_{i+1}(φ_{[1 to i]} ⊗ P_{B}),  (F)
(𝕄̃_{i+2}𝕃̃_{j−2}e) = ±{Z_{[i+1 to n−1]}^{B}m_{i+1}(φ_{[1 to i]} ⊗ P_{B}), f}  (G)
±{Z_{[i+1 to n−1]}^{B}, f}m_{i+1}(φ_{[1 to i]} ⊗ P_{B}),  (H)

Terms which are identical up to sign cancel, i.e., (B) cancels with (D), (F) cancels with (H). After exchanging P_{B} and P_{A} in (C), we can combine (C) with (E) and use the Jacobi identity to produce ±{{Z_{[1 to i]}^{A}, Z_{[i+1 to n−1]}^{B}}P_{A}P_{B}, f}, where we have brought P_{A}P_{B} inside the bracket. Since m_{i+1}(φ_{[1 to i]} ⊗ Z_{[i+1 to n−1]}^{B}) = ±{Z_{[1 to i]}^{A}, Z_{[i+1 to n−1]}^{B}}P_{A}, the term above becomes ±{m_{i+1}(φ_{[1 to i]}⊗ Z_{[i+1 to n−1]}^{B})P_{B}, f}. To this we add (G), yielding one of the desired terms: ±{m_{i+1}(φ_{[1 to i]} ⊗ m_{j−1}(φ_{[i+1 to n−2]} ⊗ P_{n−1})), f}. This term and (A) are the two terms in ±{ACT_{OBST}_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f} for which φ_{[1 to n−2]}⊗ P_{n−1} is split into the hands φ_{[1 to i]} and φ_{[i+1 to n−2]} ⊗ P_{n−1}. Therefore, for each (i, j − 2)-unshuffle σ, we produce a term of the form ±{m_{j−1}(l_{i}(first σ hand) ⊗ (second σ hand)), f} and a term of the form ±{m_{i+1}((first σ hand)⊗ m_{j−1}(second σ hand)), f} with the appropriate signs. As j runs from 2 to n, j − 1 runs from 1 to n − 1, so as we run through all (i, j − 2)-unshuffles and divide by two, we produce ±{ACT_{OBST}_{n−1}(φ_{[1 to n−2]} ⊗ P_{n−1}), f}. □

Proposition 3.6. (3a_{n}iii) For n ⩾ 3, one can define m_{n}(φ_{[1 to n−1]} ⊗P_{n}) for each element of the totally ordered basis for Φ^{∧n−1} ⊗ P so that m_{n} is well-defined and so that

m_{1}(m_{n}(φ_{[1 to n−1]} ⊗ P_{n})) = ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ P_{n}).

Proof of Proposition 3.6. We ensure that m_{n} is well-defined on Φ^{∧n−1} ⊗ P by defining m_{n} on the basis elements φ_{[1 to n−1]} ⊗ P_{n} with φ_{1} ⩽ ⋯ ⩽ φ_{n} with respect to the total ordering of the preghosts. The image of φ_{[1 to n−1]_{⊗}}P_{n} under ACT_{OBST}_{n} is in K_{A/I}. The map ACT_{OBST}_{n} has degree −2, so for n ⩾ 3, the degree of ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗P_{n}) is at least one, at which level K_{A/I} is exact. So if we can show that

m_{1}ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ P_{n}) = 0,

then a pre-image exists which we can set equal to m_{n}(φ_{[1 to n−1]} ⊗ P_{n}). Since ACT_{OBST}_{n} = ACT_{n} − m_{n}m_{1} and that ACT_{n} is a symmetric chain map, we find that

m_{1}ACT_{OBST}_{n} = m_{1}(ACT_{n} − m_{n}m_{1})
= m_{1}ACT_{n} − m_{1}m_{n}m_{1}
= ACT_{n}m_{1} − m_{1}m_{n}m_{1}
= (ACT_{n} − m_{1}m_{n})m_{1}.

Let us begin by examining m_{n} on an ordered basis element φ_{[1 to n−1]} ⊗ P_{n} with |φ_{i}| = 1 for all i. For such a basis element we have

m_{1}ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ P_{n}) = (ACT_{n} − m_{1}m_{n})m_{1}(φ_{[1 to n−1]} ⊗ P_{n})
= (ACT_{n} − m_{1}m_{n})(φ_{[1 to n−1]} ⊗ y_{n}).

Since we already know that ACT_{ID}_{n}(φ_{[1 to n−1]} ⊗ y_{n}) = 0, we can replace ACT_{n} − m_{1}m_{n} with (−1)^{n−1}m_{n}m_{1}, yielding m_{n}m_{1}(φ_{[1 to n−1]} ⊗ y_{n}), which equals 0.

The proof is completed by strong induction. Let φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}} be the k^{th} basis element in the ordered list and suppose ACT_{ID}_{n}(φ_{[1 to n−1]_{i}} ⊗ P_{n_{i}}) = 0 for all basis elements before the k^{th} one, where once again,

m_{1}ACT_{OBST}_{n}(φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}}) = (ACT_{n} − m_{1}m_{n})m_{1}(φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}}).

The element m_{1}(φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}}) can only have pieces in K_{I/I^{2}}^{∧p} ⊗ K_{A/I} for 0 ⩽ p ⩽ n − 1 and, moreover, the pieces in K_{I/I^{2}}^{∧n−1} ⊗ K_{A/I} contain only basis elements of order less than that of φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}}. So for m_{1}(φ_{[1 to n−1]_{k}} ⊗ P_{n_{k}}), we can replace ACT_{n} − m_{1}m_{n} with m_{n}m_{1}. It follows that

(ACT_{n} −m_{1}m_{n})m_{1}(φ_{[1 to n−1]_{k}} ⊗P_{n_{k}}) = m_{n}(m_{1}(m_{1}(φ_{[1 to n−1]_{k}} ⊗P_{n_{k}}))) = 0. □

Proposition 3.7. (3b_{n}ii) If we set l_{n}(φ_{[1 to n]}) = Ψm_{n}(φ_{[1 to n−1]} ⊗ P_{n}), then J_{ID}_{n} = 0 on Φ^{∧n}.

Proof of Proposition 3.7. As outlined above, we will prove that J_{ID}_{n} = 0 on Φ^{∧n} by showing that on Φ^{∧n−1} ⊗ P, the following equalities hold:

l_{1}l_{n}(1^{∧n−1} ⊗ Ψ) = Ψm_{1}m_{n} = ΨACT_{OBST}_{n} = J_{OBST}_{n}(1^{∧n−1} ⊗ Ψ).

The first equality holds because the map Ψ is a chain map. Since ACT_{ID}_{n} = 0, it follows that m_{1}m_{n} = ACT_{OBST}_{n}. The second equality holds because Ψm_{1}m_{n} = ΨACT_{OBST}_{n}. Showing that ΨACT_{OBST}_{n} = J_{OBST}_{n}(1^{∧n−1} ⊗ Ψ) is more difficult because Ψ does not commute with m_{i} for i > 1. We will need to organize ACT_{OBST}_{n} and J_{OBST}_{n} by unshuffle. Let σ be any (i, j − 2)-unshuffle. The two terms in the sum ACT_{OBST}_{n} for which φ_{[1 to n−1]} ⊗ P_{n} is split into the hands φ_{σ[1 to i]} and φ_{σ[i+1 to n−1]} ⊗ P_{n} are

±m_{j} (l_{i}(φ_{σ[1 to i]}) ∧ φ_{σ[i+1 to n−1]} ⊗ P_{n})
±m_{i+1} (φ_{σ[1 to i]} ⊗ m_{j−1}(φ_{σ[i+1 to n−1]} ⊗ P_{n})) .

We will show that when Ψ is applied to each of the terms above, their sum becomes

±l_{j} (l_{i}(φ_{σ[1 to i]}) ∧ φ_{σ[i+1 to n−1]} ∧ φ_{n})
±l_{i+1} (φ_{σ[1 to i]} ∧ l_{j−1}(φ_{σ[i+1 to n−1]} ∧ φ_{n})) .

This fact completes the proof because we can expand J_{OBST}_{n}(φ_{[1 to n]}) so that φ_{n} is never moved. For any (i, j − 1)-unshuffle which moves φ_{n}, we will use the graded symmetry of the shLie structure maps to move φ_{n} back to the last position. With this approach,

J_{OBST}_{n}(φ_{[1 to n]}) = ∑_{i>0,j>1^{i+j=n+1}} ∑_{(i,j−2)^{σ}} ( l_{j} (l_{i}(φ_{σ[1 to i]}) ∧ φ_{σ[i+1 to n]})

±l_{i+1} (φ_{σ[1 to i]} ∧ l_{j−1}(φ_{σ[i+1 to n]}))) .

Without loss of generality, we shall show that the desired equality holds for the identity (i, j − 2)-unshuffle e. The e-terms of ACT_{OBST}_{n}(φ_{[1 to n−1]} ⊗ P_{n}) are

m_{j} (l_{i}(φ_{[1 to i]}) ∧ φ_{[i+1 to n−1]} ⊗ P_{n})
= ±Z_{[1 to i]}^{A}P_{A}^{A_{a}}m_{j}(φ_{A_{a}} ∧ φ_{[i+1 to n−1]} ⊗ P_{n})  (𝕄_{j}𝕃_{i})

and

m_{i+1} (φ_{[1 to i]} ⊗ m_{j−1}(φ_{[i+1 to n−1]} ⊗ P_{n}))
= ±{Z_{[1 to i]}^{A}, Z_{[i+1 to n]}^{B}}P_{A}P_{B}  ({}_{(i,j−2)})
±Z_{[i+1 to n]}^{B}P_{B}^{B_{b}}m_{i+1}(φ_{[1 to i]} ⊗ P_{B_{b}}).  (𝕄_{i+1}𝕄_{j−1})

Applying Ψ to 𝕄_{j}𝕃_{i} produces

±Z_{[1 to i]}^{A}P_{A}^{A_{a}}l_{j}(φ_{A_{a}} ∧ φ_{[i+1 to n]}) ± Z_{[1 to i]}^{A}m_{j}(φ_{[i+1 to n]} ⊗ P_{A}^{A_{a}})φ_{A_{a}}.

The map Ψ acts on P_{A}P_{B} in the {}_{(i,j−2)} term to produce two terms ±P_{B}P_{A}^{A_{a}}φ_{A_{a}}± P_{A}P_{B}^{B_{b}}φ_{B_{b}}. Using the derivational property of the bracket and the definition of m_{j}, the first term of {}_{(i,j−2)} equals ±m_{j}(φ_{[i+1 to n]} ⊗ Z_{[1 to i]}^{A})P_{A}^{A_{a}}φ_{A_{a}}. Adding this term to Ψ(𝕄_{j}𝕃_{i}), we have

±Z_{[1 to i]}^{A}P_{A}^{A_{a}}l_{j}(φ_{A_{a}} ∧ φ_{[i+1 to n]}) ± m_{j}(φ_{[i+1 to n]} ⊗ Z_{[1 to i]}^{A}P_{A}^{A_{a}})φ_{A_{a}},

which equals ±l_{j} (l_{i}(φ_{[1 to i]}) ∧ φ_{[i+1 to n]}) . Similarly, the sum of Ψ(𝕄_{i+1}𝕄_{j−1}) and the second term of Ψ({}_{(i,j−2)}) is the second desired term ±l_{i+1} (φ_{[1 to i]} ∧ l_{j−1}(φ_{[i+1 to n]})) . We produce all the terms in J_{OBST}_{n}(φ_{[1 to n]}) as we run through all (i, j − 2)-unshuffles with 2 < j < n. □

Return to n = 2: We can define m_{2}(φ_{α_{n}} ⊗ P_{β_{m}}) inductively on φ_{α_{n}} ⊗ P_{β_{m}} with φ_{α_{n}} ⩽ φ_{β_{m}}, just as in proposition 3.6, except for φ_{α_{1}} ⊗ P_{β_{1}} with φ_{α_{1}} ⩽ φ_{β_{1}}. Since m_{2} must be a chain map, m_{1}(m_{2}(φ_{α_{1}} ⊗P_{β_{m}}) should equal m_{2}(φ_{α_{1}} ⊗m_{1}(P_{β_{1}})). Having chosen structure functions C_{α_{1}β_{1}}^{γ_{1}} in A, we find that m_{2}(φ_{α_{1}} ⊗ m_{1}(P_{β_{1}})) = {y_{α_{1}}, y_{β_{1}}} = C_{α_{1}β_{1}}^{γ_{1}}y_{γ_{1}}, which is in I. Therefore a pre-image exists under m_{1}; we set m_{2}(φ_{α_{1}} ⊗ P_{β_{1}}) = C_{α_{1}β_{1}}^{γ_{1}}P_{γ_{1}}.

We have completed the definition of m_{2} and in doing so, have finished the definition of l_{2} on K_{I/I^{2}} as well.

STEP 4: To complete the construction of the homotopy Lie-Rinehart resolution (K_{A/I}, K_{I/I^{2}}), we must verify that H_{m_{1}}(m_{2}) = m̂_{2} on I/I^{2} ⊗ A/I and that H_{l_{1}}(l_{2}) = l̂_{2} on I/I^{2} ∧ I/I^{2}, i.e., conditions (ii) and (i) in definition 2.24 are satisfied.

Let λ_{1} be a splitting from A/I into K_{A/I} and let λ_{2} be a splitting of I/I^{2} into K_{I/I^{2}}. Then for \overline{z} ⊗ \underline{f} ∈ I/I^{2} ⊗ A/I, the map m_{2} maps λ_{2}(I/I^{2}) ⊗ λ_{1}(A/I) into A. It follows that H_{m_{1}}(\overline{z} ⊗\underline{f}) = \underline{m_{2}(λ_{2}(\overline{z}) ⊗ λ_{1}(\underline{f}))}. Suppose λ_{2}(\overline{z}) = gα_{1} ; \overline{z}φ_{α_{1}}, which

implies that \overline{z} = \overline{gα_{1} ; \overline{z}y_{α_{1}}}. Then

\underline{m_{2}(gα_{1} ; \overline{z}φ_{α_{1}} ⊗ λ_{1}(\underline{f}))} = \underline{gα_{1} ; \overline{z}\{y_{α_{1}}, λ_{1}(\underline{f})\}}
= \underline{\{gα_{1} ; \overline{z}y_{α_{1}}, λ_{1}(\underline{f})\} − \{gα_{1} ; \overline{z}, λ_{1}(\underline{f})\}y_{α_{1}}}
= \underline{\{gα_{1} ; \overline{z}y_{α_{1}}, λ_{1}(\underline{f})\}}
= m̂_{2}(\overline{z} ⊗ \underline{f}).

Similarly, for any \overline{z} ∧ \overline{w}, there exists an element gα_{1} ; \overline{z}φ_{α_{1}} ∧ hβ_{1} ; \overline{w}φ_{β_{1}} in K_{I/I^{2}} ∧ K_{I/I^{2}} whose class is \overline{z} ∧ \overline{w}. Then

H_{l_{1}}(l_{2})(\overline{z} ∧ \overline{w}) = [l_{2}(gα_{1} ; \overline{z}φ_{α_{1}} ∧ hβ_{1} ; \overline{w}φ_{β_{1}})]
= [gα_{1} ; \overline{z}hβ_{1} ; \overline{w}C_{α_{1}β_{1}}^{γ_{1}}φ_{γ_{1}} + gα_{1} ; \overline{z}m_{2}(φ_{α_{1}} ⊗ hβ_{1} ; \overline{w})φ_{β_{1}} − hβ_{1} ; \overline{w}m_{2}(φ_{β_{1}} ⊗ gα_{1} ; \overline{z})φ_{α_{1}}]
= \overline{gα_{1} ; \overline{z}hβ_{1} ; \overline{w}C_{α_{1}β_{1}}^{γ_{1}}y_{γ_{1}} + gα_{1} ; \overline{z}\{y_{α_{1}}, hβ_{1} ; \overline{w}\}y_{β_{1}} − hβ_{1} ; \overline{w}\{y_{β_{1}}, gα_{1} ; \overline{z}\}y_{α_{1}}}
= \overline{\{gα_{1} ; \overline{z}y_{α_{1}}, hβ_{1} ; \overline{w}y_{β_{1}}\}}
= l̂_{2}(\overline{z} ∧ \overline{w}).

We conclude that the homotopy Rinehart algebra R for the homotopy Lie-Rinehart resolution (K_{A/I},K_{I/I^{2}}) is a model for the Rinehart algebra R for the Lie-Rinehart pair (A/I,I/I^{2}).

4. Comparing the homotopy Rinehart algebra R with the BRST algebra A

Following [FV75], [BF83] and [BV85], the classical BRST algebra A for the Poisson reduction of the Poisson algebra A by a finitely presented coisotropic ideal I is a differential graded Poisson algebra, built from a specific choice of Koszul-Tate resolution K_{A/I} of A/I. The BRST algebra A is formed by tensoring the graded commutative algebra ⋀ N with K_{A/I}, where N and P are isomorphic as vector spaces over k. The basis elements of N are denoted by {η^{α_{1}}, η^{α_{2}}, η^{α_{3}}, ...}, and are called ghosts. The result is

A ≈ ⋀ N ⊗ A ⊗ ⋀ P

(see [FHST89], [HT88], [Sta92], [HT92], [Sta96] etc.). Each η^{α_{n}} has the same degree as the corresponding P_{α_{n}}, namely n. If we let N^{I} denote a string of ghosts η^{α_{n_{1}}} ∧ ⋯ ∧ η^{α_{n_{i}}} = η^{α_{n_{[1 to i]}}} , the degree of N^{I} is the sum ∑_{1}^{i} n_{p}, which we shall denote by I without confusion. There are two significant gradings on the BRST algebra A. Let N^{I}f_{I}^{J}P_{J} be an element of A. The first grading is the internal degree, which is the difference between the ghost degree gh(N^{I}f_{I}^{J}P_{J}) = I and the antighost degree antigh(N^{I}f_{I}^{J}P_{J}) = J. The second grading is by ghost number. The ghost number gh#(N^{I}f_{I}^{J}P_{J}) = i if N^{I} = η^{α_{n_{[1 to i]}}} . The multiplication on A is given by the multiplication in each of its three pieces and obeys the Koszul sign convention, that is to say,

(N^{I}f_{I}^{J}P_{J})(N^{K}f_{K}^{L}P_{L}) = (−1)^{JK}N^{I}N^{K}f_{I}^{J}f_{K}^{L}P_{J}P_{L}.

The Poisson bracket on K_{A/I} is extended to all of A by setting {N, N} = {N, A} = 0 and {η^{α_{n}}, P_{β_{m}}} = δ_{β_{m}}^{α_{n}} (where δ_{β_{m}}^{α_{n}} is the Kronecker delta). If we view N as the dual of P, this bracket formula is the usual symplectic structure on N ⊕ P [Sta92].

When I is irreducible, Batalin, Fradkin and Vilkovisky ([BF83] and [BV85]) were the first to define a differential D on A which is a Poisson derivation and whose zeroth cohomology H^{0}(A, D) ≈ (A/I)^{I} [BF83]. Later it was shown that the BRST algebra (A, D) is a model for the entire Rinehart complex in the irreducible case (see [HT92]).

A K_{A/I}-linear map F_{k} : ⋀^{k} K_{I/I^{2}} → K_{A/I} is completely determined by where it sends each element of the ordered basis φ_{α_{n_{[1 to k]}}} . Once we set η^{α_{n}} equal the k-dual of φ_{α_{n}}, the map F_{k} can be represented by

∑_{α_{n_{1}} ⩽···⩽α_{n_{k}}} η^{α_{n_{[1 to k]}}} F_{k}(φ_{α_{n_{[1 to k]}}} ).

It follows that the homotopy Rinehart algebra R ≈ Hom_{k}(⋀(Φ), K_{A/I}). This fact allows us to compare the homotopy Rinehart algebra with the BRST algebra A.

Theorem 4.1. Given a Lie-Rinehart pair (A/I, I/I^{2}) and a specific Koszul-Tate resolution K_{A/I} of A/I, the BRST algebra A = ⋀ N ⊗ K_{A/I} is isomorphic to the homotopy Rinehart algebra R = Hom_{K_{A/I}}(⋀(K_{I/I^{2}}), ⋀(K_{I/I^{2}}) ⊗ K_{A/I}) as K_{A/I}-modules and as algebras (but not necessarily as differential graded algebras).

Proof of Theorem 4.1. It is straightforward to show that the map sending

F_{k} ↦ ∑_{α_{n_{1}} ⩽···⩽α_{n_{k}}} η^{α_{n_{(1 to k)}}} F_{k}(φ_{α_{n_{[1 to k]}}} )

is bijective and respects the K_{A/I}-module structures on both R and A. We need to show that it is a map of algebras. Let F_{i} : ⋀^{i}(Φ) → K_{A/I} and G_{j} : ⋀^{j}(Φ) → K_{A/I}. Recall that the product F_{i} ⌣ G_{j} : ⋀^{i+j}(Φ) → K_{A/I}, evaluated on φ_{α_{p_{[1 to i+j]}}} is

∑_{(i,j)^{σ}} �(σ) �(G_{j}; φ_{α_{p_{σ[1 to i]}}} )F_{i}(φ_{α_{p_{σ[1 to i]}}} )G_{j}(φ_{α_{p_{σ[i+1 to i+j]}}} ).

This product can be represented in A by

∑_{α_{p_{1}} ⩽···⩽α_{p_{i+j}}} η^{α_{p_{[1 to i+j]}}} (F_{i} ⌣ G_{j})(φ_{α_{p_{[1 to i+j]}}} ).

Similarly, F_{i} = ∑_{α_{r_{1}} ⩽···⩽α_{r_{i}}} η^{α_{r_{(1 to i)}}} F_{i}(φ_{α_{r_{[1 to i]}}} ) and G_{j} = ∑_{α_{q_{1}} ⩽···⩽α_{q_{j}}} η^{α_{q_{(1 to j)}}} G_{j}(φ_{α_{q_{[1 to j]}}} ).

The product of F_{i} and G_{j} in A is

∑_{{αr_{1} ⩽···⩽αr_{i} \atop αq_{1} ⩽···⩽αq_{j}}} �(G_{j}; φ_{α_{p_{σ[1 to i]}}} )η^{α_{r_{[1 to i]}}} η^{α_{q_{[1 to j]}}} F_{i}(φ_{α_{r_{[1 to i]}}} )G_{j}(φ_{α_{q_{[1 to j]}}} ),

but the terms η^{α_{r_{[1 to i]}}} η^{α_{q_{[1 to j]}}} are no longer in the proper order. Returning the η ’s in each term to the proper order produces the sign �(σ) corresponding to the (i, j)-unshuffle σ which places n_{[1 to i+j]} into the hands r_{[1 to i]} and q_{[1 to j]}. The sum above becomes

∑_{α_{p_{1}} ⩽α_{p_{i+j}}} η^{α_{p_{[1 to i+j]}}} (∑_{(i,j)^{σ}} �(σ) �(G_{j}; φ_{α_{p_{σ[1 to i]}}} )F_{i}(φ_{α_{p_{σ[1 to i]}}} )G_{j}(φ_{α_{p_{σ[i+1 to i+j]}}} )) ,

which equals ∑_{α_{p_{1}} ⩽···⩽α_{p_{i+j}}} η^{α_{p_{[1 to i+j]}}} (F_{i} ⌣ G_{j})(φ_{α_{p_{[1 to i+j]}}} ). □

The bidegree (ghost number, internal degree) on A agrees with the bidegree (external degree, (suspended internal degree − external degree)) on R.

The differentials {Q, } and ⟨D_{M}, ⟩ when I is irreducible The differential D is an inner-derivation on A, i.e. D = {Q, }, where the element Q ∈ A has total degree +1 and {Q, Q} = 0. The element Q (called the BRST charge) is a sum ∑_{n=0}^{∞} Q_{n}, where Q_{n} has ghost number n + 1. The Jacobi identity for the Poisson bracket guarantees that D has square zero. We construct the BRST charge Q using methods from homological perturbation theory. The BRST algebra A is filtered by ghost degree and Q_{n} is defined by induction on n once we select a set of generators {y_{α}} for I and structure functions C_{αβ}^{γ} [Sta92]. We set Q_{0} = η^{α}y_{α} and find that Q_{1} must equal −\frac{1}{2}η^{α}η^{β}C_{αβ}^{γ}P_{γ} in order to kill the nonzero piece of {Q_{0}, Q_{0},} with ghost degree 2. This process continues: for every i > 1, the terms of {∑_{n=0}^{i−1} Q_{n}, ∑_{n=0}^{i−1} Q_{n}} have ghost degree at least i + 1. An element Q_{i} with ghost degree i + 1 is selected to kill the terms with ghost degree i + 1.

With respect to the filtration of A by ghost number, {Q, } decomposes as 1 ⊗ m_{1} + δ̂_{R} + δ_{2} + .... In terms of the BRST differential, 1 ⊗ m_{1} is {Q_{0}, }_{|_{⋀ P}} and

δ̂_{R} = {Q_{1}, }_{|_{⋀ N}} + {Q_{0}, }_{|_{A}} + {Q_{1}, }_{|_{⋀ P}}

is, loosely speaking, a “lifting” of the Rinehart differential δ_{R} on R. The E_{1} term of the associated spectral sequence is isomorphic to ⋀ N ⊗ A/I and the differential d_{1} for the E_{1} term which δ̂_{R} induces on H_{1⊗m_{1}}(A) is the map δ_{R}.

The filtration of R by external degree is the same as the filtration of A by ghost number. The differential ⟨D_{M}, ⟩ breaks up into the sum ∑_{i=1}^{∞} ⟨m_{i}, ⟩, where ⟨m_{i}, ⟩ increases external degree by i − 1. For any map F_{k} ∈ R, the map ⟨m_{1}, F_{k}⟩ is m_{1}F_{k} because l_{1} = 0 on ⋀(Φ). Suppressing the sum over the ordered basis (it is henceforth understood), we write F_{k} as η^{α_{n_{[1 to k]}}} F_{k}(φ_{α_{n_{[1 to k]}}} ) and the map m_{1}F_{k} is represented in A by η^{α_{n_{[1 to k]}}} m_{1}F_{k}(φ_{α_{n_{[1 to k]}}} ), which equals (1 ⊗ m_{1})(η^{α_{n_{[1 to k]}}} F_{k}(φ_{α_{n_{[1 to k]}}} )). So ⟨m_{1}, ⟩, when realized on A, is 1 ⊗ m_{1}. We conclude that in the irreducible case, the E_{1} terms of the spectral sequences associated with {Q, −} and ⟨D_{M}, ⟩ are isomorphic. The differentials for the E_{1} terms are induced by δ̂_{R} and ⟨m_{2}, ⟩, respectively; we claim that δ̂_{R} = ⟨m_{2}, ⟩ . Because both δ̂_{R} and ⟨m_{2}, ⟩ are derivations, a quick check on the ghosts, antighosts and the elements of A suffices. For η^{α}, we see that ⟨m_{2}, η^{α}⟩ (φ_{β} ∧ φ_{γ}) = η^{α}l_{2}(φ_{β} ∧ φ_{γ}) = η^{α}(C_{βγ}^{δ}φ_{δ}) = C_{βγ}^{α} and

\{−\frac{1}{2}η^{a}η^{b}C_{ab}^{δ}P_{δ}, η^{α}\} (φ_{β} ∧ φ_{γ}) = \frac{1}{2}η^{a}η^{b}C_{ab}^{α}(φ_{β} ∧ φ_{γ}) = \frac{1}{2}C_{βγ}^{α} − \frac{1}{2}C_{γβ}^{α} = C_{βγ}^{α}.

For P_{α}, we find that ⟨m_{2}, P_{α}⟩ (φ_{β}) = m_{2}P_{α}(φ_{β}) = −m_{2}(φ_{β} ⊗ P_{α}) = C_{αβ}^{δ}P_{δ} and

\{−\frac{1}{2}η^{a}η^{b}C_{ab}^{δ}P_{δ}, P_{α}\} (φ_{β}) =

\frac{1}{2}η^{a}C_{aα}^{δ}P_{δ}(φ_{β}) − \frac{1}{2}η^{b}C_{αb}^{δ}P_{δ}(φ_{β}) = −\frac{1}{2}C_{βα}^{δ}P_{δ} + \frac{1}{2}C_{αβ}^{δ}P_{δ} = C_{αβ}^{δ}P_{δ}.

And for f ∈ A, it is straightforward to compute that ⟨m_{2}, f⟩ (φ_{β}) = m_{2}f(φ_{β}) = m_{2}(φ_{β} ⊗ f) = {y_{β}, f} and {η^{α}y_{α}, f}(φ_{β}) = η^{α}{y_{α}, f}(φ_{β}) = {y_{β}, f}. We conclude that (A, {Q, }) and (R, ⟨D_{M}, ⟩) are equivalent as models for the Rinehart cohomology of the Lie-Rinehart pair (A/I, I/I^{2}) when the ideal I is irreducible.

When the ideal is reducible, we know that (R, ⟨D_{M}, ⟩) is also a model for the Rinehart cohomology, but we do not know, except in cases arising from particularly nice symplectic settings (see [FHST89]), whether (A, {Q, }) is a model for the Rinehart cohomology. This problem will be addressed in a future paper.

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Lars Kjeseth lkjeseth@elcamino.cc.ca.us

Mathematical Sciences Division El Camino College U.S.A. Torrance 16007 Crenshaw Blvd, Torrance, CA 90506