Koszul complex for linear monoidal functors

Serge Bouc and Nadia Romero

Abstract

We introduce regular sequences and associated Koszul resolutions for monoids in the category of functors over an essentially small linear symmetric monoidal category. Next we define polynomials over such monoids. We compute the Hochschild cohomology functors and prove a relative analogue of Hilbert’s syzygy theorem for polynomials over tensor idempotent commutative monoids. Keywords: monoidal category, functor category, Koszul resolution, Hochschild cohomology, Hilbert syzygy. AMS MSC (2020): 16Y99, 18D99, 18G10, 18M05, 20J15.

1 Introduction

In recent years, some of the classical tools to study rings and bimodules over them have been generalized to other contexts, in particular to monoidal categories: E.g. in [9], the second author introduces Hochschild cohomology for functors over linear symmetric monoidal categories. It is then a natural question to try to generalize to this framework some other notions. In the classical framework of rings, the construction of Koszul resolutions plays a fundamental role. Such resolutions can be used in particular to compute Hochschild cohomology for polynomial rings over a field, or to give a proof of the Hilbert syzygy theorem (see Application 4.5.6 in [10]).

In this work, we introduce regular sequences (Definition 3.5) and associated Koszul resolutions for monoids in the category F of functors over an essentially small linear symmetric monoidal category. In Section 3, given a monoid A in F, we define its associated Koszul complex and prove first that it is indeed a complex, and then that it is a resolution for a particular A-module, as in the classical case. The definition of the complex as well as the proof of the fact that it is a resolution are completely combinatorial in nature, thus avoiding the use of the tensor product of complexes of functors. More generally, the methods used throughout are elementary, and do not require the use of sophisticated tools.

In Section 4, we define polynomials over the monoids in F. In the classical framework of rings, the results we are interested in are proved for polynomial rings over fields so, in our context, we replace the fields by tensor idempotent commutative monoids (see Definition 4.5). With this, we compute the Hochschild cohomology functors and prove a relative analogue of Hilbert’s syzygy theorem for polynomials over tensor idempotent commutative monoids in F.

An important (and motivating) example where our results apply is the context of Green biset functors. In this case, the Burnside functor is commutative and tensor idempotent, so our results can be applied to polynomials over it.

2 Preliminaries

In what follows R is a commutative ring with identity, denoted simply as 1, and (X , ⋄, 1, α, λ, ρ, s) is an essentially small symmetric monoidal category enriched in R-Mod, in particular, the functor ⋄ : X × X → X is R-bilinear. The category of R-linear functors from X to R-Mod, denoted by F, is also endowed with a tensor product structure ⊗, given by the Day convolution (see [5], for instance). With this, F becomes an abelian, symmetric monoidal, closed category with identity given by I = X (1, ).

Example 2.1. There are many examples of categories satisfying the conditions of X, we mention two of them.

i) The biset category, defined in [2].

ii) The category of correspondences, defined in [3].

In the first case, the monoids in F are called Green biset functors and have been extensively studied in recent years. See, for example, [8]. In the second case, monoids have been considered in Section 7 of [4], under the name of algebra functors. It is shown there in particular that to any finite lattice T, one can associate a monoid F_{T}, and that conversely, any monoid satisfying some additional mild conditions is obtained in this way (Theorem 7.4 of [4]).

Recall that, given a monoid A in F, we denote its product A ⊗ A → A as μ_{A} and its identity morphism I → A as e_{A}. We give an equivalent way of defining a monoid in F, which is a consequence of Remark 3.5 in [9]. We will use both definitions as needed, all along the paper. For more notation and basic facts on monoids in F and modules over them, we invite the reader to take a look at [9].

Given x, y and z, objects in X, consider α_{x, y, z} : x⋄(y⋄z) → (x⋄y)⋄z, λ_{x} : 1⋄x → x and ρ_{x} : x ⋄ 1 → x. Then, A is a monoid in F if for every pair x, y of objects in X, there exist bilinear maps

A(x) × A(y) → A(x ⋄ y),

denoted simply by (a, b) ↦ a × b, and an element ε_{A} ∈ A(1) satisfying the following conditions.

• Associativity. For x, y and z, objects in X, and for any a ∈ A(x), b ∈ A(y) and

c ∈ A(z),

(a × b) × c = A(α_{x, y, z})(a × (b × c)).

• Identity element. For an object x in X and any a ∈ A(x),

a = A(λ_{x})(ε_{A} × a) = A(ρ_{x})(a × ε_{A}).

• Functoriality. Let φ : x → x′ and ψ : y → y′ be arrows in X. Then for any a ∈ A(x) and b ∈ A(y),

A(φ ⋄ ψ)(a × b) = A(φ)(a) × A(ψ)(b).

Definition 2.2. Given a monoid A in F and C a subfunctor of A, we say that C is a submonoid of A if C is a monoid with the monoid structure of A.

The previous definition translates in the following way. Since C is a subfunctor of A, we have a canonical morphism i : C → A in F, so C is a submonoid of A if μ_{A} ∘ (i ⊗ i) has image in C and i ∘ e_{C} = e_{A}.

An important example of submonoid is the following.

Definition 2.3. Let A be a monoid in F. The commutant of A at x ∈ X is

CA(x) = {b ∈ A(x) | a × b = A(s_{x⋄y}^{y⋄x})(b × a), ∀a ∈ A(y), ∀y ∈ X },

where s_{x⋄y}^{y⋄x} : x ⋄ y → y ⋄ x is the symmetry in X. It is a subfunctor of A and, moreover, it is a commutative submonoid of A.

If A and D are monoids in F and f : D → A is a morphism of monoids, we have a restriction functor along f, from A-Mod to D-Mod, which is an additive functor. When considering the arrow e_{A} : I → A, the restriction from A-Mod to F is simply denoted as R_{A}. Now, if C is a submonoid of A, the restriction from A-Mod to C-Mod is denoted by R_{C}^{A}. Since R_{C}^{A} is a forgetful functor, it is faithful and exact. This is the case of the functor R_{A} too.

Let F be a functor in F. Recall that, given a family S = (F_{i})_{i∈I} of subfunctors of F, the intersection H = ⋂_{i∈I} F_{i} of S is the subfunctor of F given by

H(x) = ⋂_{i∈I}(F_{i}(x)), H(φ) = F (φ)|_{H(x)},

for x and y objects and φ : x → y an arrow in X.

Remark 2.4. Let A be a monoid in F and M be an A-module. If S is a family of submodules of M, then the intersection of S is a submodule of M.

Definition 2.5. Let F be a functor in F. Given G, a set of objects in X, and for each x ∈ G, a subset Γ_{x} ⊆ F (x), the subfunctor generated by the data (G, Γ) is the intersection of the family

S = {F′ subfunctor of F | Γ_{x} ⊆ F′(x), ∀x ∈ G}.

Definition 2.6. Let A be a monoid in F and N be an A-module. Consider G and Γ as before and

S′ = {N′ submodule of N | Γ_{x} ⊆ N′(x), ∀x ∈ G}.

We call the intersection of S′ the submodule of N generated by the data (G, Γ).

Remark 2.7. If M is the submodule generated by (G, Γ), then Γ_{x} ⊆ M(x) for all x in G.

Lemma 2.8. Let A be a monoid in F and α_{1}, …, α_{n} be elements in A(1). Let M be the A-submodule of A generated by α_{1}, …, α_{n}. Then, for an object x in X,

M(x) = {A(ρ_{x})(a_{1} × α_{1} + … + a_{n} × α_{n}) | a_{i} ∈ A(x)}.

Proof. Let

F_{x} = {A(ρ_{x})(a_{1} × α_{1} + … + a_{n} × α_{n}) | a_{i} ∈ A(x)}.

Since, by the previous remark, α_{1}, …, α_{n} ∈ M(1) and M is an A-submodule of A, we clearly have F_{x} ⊆ M(x).

Let us show that F (x) = F_{x} defines an A-submodule of A such that α_{1}, …, α_{n} ∈ F (1). With this we will have that M is a submodule of F and, hence, M(x) ⊆ F_{x}. It is easy to see that F (x) is an R-submodule of A(x) and clearly α_{1}, …, α_{n} ∈ F (1). Also, if φ : x → y is an arrow in X, we know that A(φ ∘ ρ_{x}) = A(ρ_{y} ∘ (φ ⋄ 1)), so A(φ) (A(ρ_{x})(a_{1} × α_{1} + … + a_{n} × α_{n})) is equal to

A(ρ_{y}) (A(φ)(a_{1}) × α_{1} + … + A(φ)(a_{n}) × α_{n}) .

Finally, for an object y in X, m ∈ A(y) and any n ∈ A(x × 1), we have

m × A(ρ_{x})(n) = A(y ⋄ ρ_{x})(m × n) = A(ρ_{y⋄x})(m × n).

This shows that F is an A-submodule of A.

Notation 2.9. The functor M of the previous lemma will be denoted by A⟨α_{1}, …, α_{n}⟩, or simply as A⟨α⟩, where α = (α_{1}, …, α_{n}).

3 The Koszul complex

In what follows A is a monoid in F.

Definition 3.1. For an element a ∈ A(1), we define L_{a} : A → A as

L_{a,x} : A(x) → A(x) b ↦ A(λ_{x})(a × b),

for x an object in X. For convenience, we will often denote L_{a} simply as a × .

This defines a natural transformation in A. Furthermore, if a ∈ CA(1), then L_{a} is clearly a morphism of A-modules.

Notation 3.2. Given a natural number n > 0, we denote by A^{n} the coproduct ⊕A of A with itself n times. When considering the binomial coefficient, \binom{n}{p}, for 0 ≤ p ≤ n, we will write K_{p}^{n}(A) for

A^{\binom{n}{p}} = ⨁_{S⊆T |S|=p} A,

where T = {1, …, n}. When necessary, we will write A_{S} for the summand of K_{p}^{n}(A) corresponding to the set S.

To define the Koszul complex, we fix a natural number n > 1 and consider a family of elements α_{1}, …, α_{n} in CA(1). We fix these elements and abbreviate them as α = (α_{1}, …, α_{n}).

We begin by defining, for p ≤ n, the arrow d : K_{p}^{n}(A) → K_{p−1}^{n}(A),

d : ⨁_{S⊆T |S|=p} A ⟶ ⨁_{S⊆T |S|=p−1} A.

Given S = {i_{1}, …, i_{p}} ⊆ T, the arrow d_{S} : A_{S} → K_{p−1}^{n}(A) is given by

d_{S} = ∑_{k=1}^{p}(−1)^{k+1}(α_{i_{k}} × )î_{k},

where the subscript î_{k} indicates that α_{i_{k}} × has as codomain A_{S\{i_{k}}}. In the case p = 1, the arrow d : A^{n} → A is equal to ⊕_{i=1}^{n}d_{{i}} and each d_{{i}} is just L_{α_{i}} : A → A, so we denote d as L_{α}.

Definition 3.3. For A and α as before, we define K_{A}(α) as the sequence in F

\xymatrix{0 \ar[r] & K_{n}^{n}(A) \ar[r]^{d} & … K_{p}^{n}(A) \ar[r]^{d} & K_{p−1}^{n}(A) \ar[r]^{d} & … A^{n} \ar[r]^{L_{α}} & A \ar[r] & 0,}

with A in degree 0. As we will see next, this sequence is a complex of A-modules. We call it the Koszul complex of A for α.

Even though some of the properties of K_{A}(α) follow in a similar way as in the classical case, we prove them for the sake of completeness.

Lemma 3.4. With A and α as before, K_{A}(α) is a complex of A-modules.

Proof. Since α_{1}, …, α_{n} are in CA(1), the sequence K_{A}(α) is clearly in the category of A-modules.

Next, we prove d ∘ d = 0.

Let us fix S = {i_{1}, …, i_{p}}. If we start by d_{S} = ∑_{k=1}^{p}(−1)^{k+1}(α_{i_{k}} × )î_{k}, then in d ∘ d, for each k we need only to consider the arrow d_{S\{i_{k}}}. Now,

d_{S\{i_{k}}} = ∑_{j=1}^{k−1}(−1)^{j+1}(α_{i_{j}} × )_{î_{k},î_{j}} + ∑_{j=k+1}^{p}(−1)^{j}(α_{i_{j}} × )_{î_{k},î_{j}},

where î_{k}, î_{j} indicates that α_{i_{j}} × has codomain A_{S\{i_{k},i_{j}}}. Hence, the composition d ∘ d_{S} can be written as

∑_{k=1}^{p} (∑_{j=1}^{k−1}(−1)^{k+j}((α_{i_{j}} × α_{i_{k}}) × )_{î_{j},î_{k}} + ∑_{j=k+1}^{p}(−1)^{k+j+1}((α_{i_{j}} × α_{i_{k}}) × )_{î_{j},î_{k}}) ,

since clearly (α_{i_{j}} × )_{î_{j},î_{k}} ∘ (α_{i_{k}} × )î_{k} is equal to ((α_{i_{j}} × α_{i_{k}}) × )_{î_{j},î_{k}}.

Finally, let e and f be in {1, …, p} and, without loss of generality, suppose e < f. Then, in the sum above, for k = e and j = f, we find (−1)^{e+f+1}((α_{i_{f}} × α_{i_{e}}) × )_{î_{f} ,î_{e}}, and for k = f and j = e, we find (−1)^{f+e}((α_{i_{e}} × α_{i_{f}}) × )_{î_{e},î_{f}}. These are the only two times we find the arrow ((α_{i_{e}} × α_{i_{f}}) × )_{î_{e},î_{f}} in the sum above, and all the summands are of this form, hence d ∘ d_{S} = 0 and d ∘ d = 0.

The Koszul resolution

For this section we follow the lines of Matsumura [7].

Definition 3.5. For an A-module M, an element a ∈ CA(1) is called M-regular if, for any object x in X and any m ∈ M(x),

m ≠ 0 ⇒ a × m ≠ 0.

The (ordered) sequence α = (α_{1}, …, α_{n}), with α_{i} ∈ CA(1), is called regular if

1. α_{1} is A-regular, α_{2} is (A/(A⟨α_{1}⟩))-regular,..., α_{n} is (A/(A⟨α_{1}, …, α_{n−1}⟩))-regular.

2. A/(A⟨α⟩) ≠ 0.

Theorem 3.6. If α = (α_{1}, …, α_{n}), with α_{i} ∈ CA(1), is a regular sequence for A, then K_{A}(α) is a resolution for A/I_{α}, with I_{α} = A⟨α⟩. That is, the following sequence is an exact sequence of A-modules,

\xymatrix{0 \ar[r] & K_{n}^{n}(A) \ar[r]^{d} & … \ar[r]^{d} & K_{2}^{n}(A) \ar[r]^{d} & K_{1}^{n}(A) \ar[r]^{L_{α}} & A \ar[r] & A/I_{α} \ar[r] & 0.}

Proof. For this proof we will use the notation α^{i} = (α_{1}, …, α_{i}), for 1 < i < n.

We begin by noticing that, for 1 ≤ p ≤ n, we have the following decomposition of A-modules, given by Pascal’s rule,

K_{p}^{n}(A) = K_{p}^{n−1}(A) ⊕ K_{p−1}^{n−1}(A).  (1)

We denote by ι the embedding K_{p}^{n−1}(A) ↪ K_{p}^{n}(A), and by τ the projection K_{p}^{n}(A) ↠ K_{p−1}^{n−1}(A).

Let us see what happens with the decomposition (1) in the Koszul complex. We write T = T′ ∪ {n} with T′ = {1, …, n − 1}. Then, in K_{p}^{n}(A), we can identify the two summands in the following way

K_{p}^{n−1}(A) = ⨁_{S′⊆T′ |S′|=p}A and K_{p−1}^{n−1}(A) = ⨁_{\begin{matrix}S=S′∪\{n\} \\ S′⊆T′ |S′|=p−1 \end{matrix}}A.

The arrow d : K_{p}^{n}(A) → K_{p−1}^{n}(A) remains the same when restricted to K_{p}^{n−1}(A), that is, d|_{K_{p}^{n−1}(A)} is actually

d : K_{p}^{n−1}(A) → K_{p−1}^{n−1}(A).

Now we suppose p > 1 and restrict d to K_{p−1}^{n−1}(A). In A_{S}, for a set S = S′ ∪ {n} with S′ = {i_{1}, …, i_{p−1}}, the arrow d_{S} is equal to

∑_{k=1}^{p−1}(−1)^{k+1}(α_{i_{k}} × )î_{k} + (−1)^{p+1}(α_{n} × )n̂.

This can be written as d_{S′} + (−1)^{p+1}L_{α_{n}}, taking into account that in d_{S′} the subindex î_{k} means that the codomain is A_{(S′\{i_{k}})∪{n}} and L_{α_{n}} : A_{S} → A_{S′}. That is, d|_{K_{p−1}^{n−1}(A)} is given by d + (−1)^{p−1}L_{α_{n}}, with

d : K_{p−1}^{n−1}(A) → K_{p−2}^{n−1}(A) and with L_{α_{n}} : K_{p−1}^{n−1}(A) → K_{p−1}^{n−1}(A)

being the product by α_{n} in each summand. With this, the (split) short exact sequence of A-modules

\xymatrix{0 \ar[r] & K_{p}^{n−1}(A) \ar[r]^{ι} & K_{p}^{n}(A) \ar[r]^{τ} & K_{p−1}^{n−1}(A) \ar[r] & 0,}

induces a short exact sequence of complexes

\xymatrix{0 \ar[r] & K_{A}(α^{n−1}) \ar[r] & K_{A}(α) \ar[r] & K_{A}^{[1]}(α^{n−1}) \ar[r] & 0,}

where K_{A}^{[1]}(α^{n−1}) is the complex obtained by shifting the degrees of K_{A}(α^{n−1}) up by 1 and the differentials are the same. Indeed, for 1 ≤ p ≤ n let us see the commutativity of the following diagram

\xymatrix{0 \ar[r] & K_{p}^{n−1}(A) \ar[d]^{d} \ar[r]^{ι} & K_{p}^{n}(A) \ar[d]^{d} \ar[r]^{τ} & K_{p−1}^{n−1}(A) \ar[d]^{d} \ar[r] & 0 \\ 0 \ar[r] & K_{p−1}^{n−1}(A) \ar[r]^{ι} & K_{p−1}^{n}(A) \ar[r]^{τ} & K_{p−2}^{n−1}(A) \ar[r] & 0.}

In the case p = 1, on the bottom line, ι is the identity arrow and, instead of K_{p−2}^{n−1}(A), we have 0. By the description of d|_{K_{p}^{n−1}(A)}, above, the square on the left commutes. For the square on the right, suppose p > 1. Since L_{α_{n}} has image in K_{p−1}^{n−1}(A), then clearly τd = dτ.

Since F is an abelian category, we have the following long exact sequence of homology

[figure: …; H_{p+1}(K_{A}^{[1]}(α^{n−1})); δ; H_{p}(K_{A}(α^{n−1})); H_{p}(K_{A}(α))]

\xymatrix{H_{p}(K_{A}^{[1]}(α^{n−1})) \ar[r]^{δ} & …}

Notice that H_{p}(K_{A}^{[1]}(α^{n−1})) = H_{p−1}(K_{A}(α^{n−1})), so

δ : H_{p−1}(K_{A}(α^{n−1})) → H_{p−1}(K_{A}(α^{n−1})).

Now, H_{p−1}(K_{A}(α^{n−1})) is a subquotient of K_{p−1}^{n−1}(A) and (since in F all limits and colimits are defined pointwise) the connecting morphism δ is given, in an object x, through the composition (ι_{x})^{−1}d_{x}(τ_{x})^{−1} (a short for taking inverse image by τ_{x}, then image by d_{x}, and then taking inverse image by ι_{x}). So, by the description of d|_{K_{p−1}^{n−1}(A)}, given before, we have δ = (−1)^{p−1}L_{α_{n}}.

To finish the proof, we observe first that, by Lemma 2.8, ImL_{α} = I_{α}. Next we proceed, as in the classical case, by induction on n.

For n = 1, the sequence of A-modules

\xymatrix{0 \ar[r] & A \ar[r]^{L_{α_{1}}} & A \ar[r] & A/I_{α_{1}} \ar[r] & 0}

is exact because α_{1} is A-regular.

Now suppose the theorem to be true for n − 1. Then, by induction hypothesis, for p > 1, in the previous long exact sequence of homology we have

\xymatrix{0 = H_{p}(K_{A}(α^{n−1})) \ar[r] & H_{p}(K_{A}(α)) \ar[r] & H_{p−1}(K_{A}(α^{n−1})) = 0.}

So H_{p}(K_{A}(α)) = 0 for p > 1. Finally, for p = 1 we have

\xymatrix{0 \ar[r] & H_{1}(K_{A}(α)) \ar[r] & H_{0}(K_{A}(α^{n−1})) \ar[r]^{δ} & H_{0}(K_{A}(α^{n−1})) \ar[r] & …}

But H_{0}(K_{A}(α^{n−1})) = A/A⟨α^{n−1}⟩ and δ = L_{α_{n}}. Since α_{n} is (A/A⟨α^{n−1}⟩)-regular, we must have H_{1}(K_{A}(α)) = 0.

4 Functors of polynomials

General properties

We begin by recalling the following standard definition and notation.

Definition 4.1. Let t_{1}, …, t_{n} be variables. Given a monomial t_{1}^{i_{1}} … t_{n}^{i_{n}}, where the exponents i_{ν} are nonnegative integers, the vector i = (i_{1}, …, i_{n}) of exponents is called a multi-index of size n. We write the monomial symbolically as

t^{i} = t_{1}^{i_{1}} … t_{n}^{i_{n}}.

For the multi-index (0, …, 0), we fix t^{0} = t_{1}^{0} … t_{n}^{0} = 1.

Given an R-module U, we can consider the R-module U[t_{1}, …, t_{n}]. With the previous definition, a polynomial f(t) = f(t_{1}, …, t_{n}) in U[t_{1}, …, t_{n}], can be written in exactly one way in the form

f(t) = ∑_{i} a_{i}t^{i},

where i runs through all the multi-indices (i_{1}, …, i_{n}), the coefficients a_{i} are in U and only finitely many of them are different from zero. Finally, U[t_{1}, …, t_{n}] contains U as the submodule of constant polynomials, that is, polynomials with all coefficients equal to zero except, possibly, the one corresponding to the multi-index (0, …, 0).

We keep the variable t whether we have one or several variables. That is, if we have only one variable, we continue to write t^{i}, instead of t_{1}^{i_{1}}.

Given an object F in F, we define the functor of polynomials in F, denoted by F [t_{1}, …, t_{n}], or simply by F_{n}, as follows. In an object x in X it is defined as the R-module F_{n}(x) := F (x)[t_{1}, …, t_{n}] and, in an arrow φ : x → y in X, as the R-linear map

F_{n}(φ) : F_{n}(x) → F_{n}(y), p(t) = ∑_{i} a_{i}t^{i} ↦ F (φ)(p(t)) := ∑_{i} F (φ)(a_{i})t^{i},

for p(t) ∈ F_{n}(x).

Proposition 4.2. Let t_{1}, …, t_{n} be variables and F be an object in F. Then F_{n} is an object in F. Also, if A is a monoid in F, then A_{n} inherits a monoid structure from A,

A_{n}(x) × A_{n}(y) → A_{n}(x ⋄ y),

by sending p(t) = ∑_{i} a_{i}t^{i} ∈ A_{n}(x) and q(t) = ∑_{j} b_{j}t^{j} ∈ A_{n}(y) to

(p × q)(t) := ∑_{i, j}(a_{i} × b_{j})t^{i+j} ∈ A_{n}(x ⋄ y).

The identity element is given by the identity element of A, ε(t) := ε, seen as a constant polynomial in A_{n}(1).

Proof. The first assertion is clear, since F_{n}(φ) extends F (φ) and F is a functor. The rest of the proposition also follows easily, since the product defined in A_{n} extends the one in A. We just notice, as usual, that (p × q)(t) = ∑_{k} c_{k}t^{k}, where

c_{k} = ∑_{i+j=k} a_{i} × b_{j}

and k is a multi-index.

Remark 4.3. Let t_{1}, …, t_{n} be variables, F an object in F and A a monoid in F. From the previous proposition, we have the following observations:

1. Sending F to F_{n} defines an endofunctor of the category F.

2. The functor F_{n} is isomorphic to a coproduct, running over all the multi-indices, of copies of F. For the monoid A, this is an isomorphism of (A, A)-bimodules.

3. Let u and v be variables. Then F is a subfunctor of F [u]. More generally, F [u] is a subfunctor of F [u, v] and, in fact, F [u, v] = (F [u])[v]. On the other hand, A is a submonoid of A[u] and A[u] is a submonoid of A[u, v]. In particular, any A[u]-module is an A-module. This observation extends in an obvious way to n variables.

4. If A is a commutative monoid, then A_{n} is commutative too.

5. Each variable t_{i} can be found in A_{n}(1) as εt_{i}. Moreover εt_{i} ∈ CA_{n}(1).

Lemma 4.4. Let C and D be monoids in F.

1. Let t be a variable. Then there are isomorphisms of monoids

C[t] ⊗ D ≅ (C ⊗ D)[t] ≅ C ⊗ D[t].

2. Let n, m ∈ ℕ, and u_{1}, …, u_{n}, v_{1}, … v_{m} be variables. Then there is an isomorphism of monoids

C[u_{1}, …, u_{n}] ⊗ D[v_{1}, …, v_{m}] ≅ (C ⊗ D)[u_{1}, …, u_{n}, v_{1}, … v_{m}].

Proof. 1. Since the tensor product of monoids is commutative up to isomorphism, it suffices to prove the first isomorphism C[t] ⊗ D ≅ (C ⊗ D)[t]. Now C[t] = ⨁_{n∈ℕ} Ct^{n}, and

Ct^{n} ≅ C in F. It follows that we have isomorphisms in F

C[t] ⊗ D = ( ⨁_{n∈ℕ} Ct^{n}) ⊗ D = ⨁_{n∈ℕ}(Ct^{n} ⊗ D) ≅ ⨁_{n∈ℕ}(C ⊗ D) ≅ (C ⊗ D)[t],

where the composed isomorphism Φ : C[t]⊗D → (C⊗D)[t] is given by the identification Φ_{n} of Ct^{n} ⊗ D with (C ⊗ D)t^{n}, for all n ∈ ℕ. In particular, the restriction Φ_{0} of Φ to C ⊗ D = Ct^{0} ⊗ D is the identity morphism C ⊗ D = Ct^{0} ⊗ D → (C ⊗ D)t^{0} = C ⊗ D, so Φ maps the identity element of C[t] ⊗ D to the identity element of (C ⊗ D)[t].

Hence all we have to check is that Φ is compatible with the product of monoids, which follows from the following commutative diagram, for all n, m ∈ ℕ,

[figure: C[t] ⊗ D; ⊗; C[t] ⊗ D; Φ⊗Φ; (C ⊗ D)[t]; ⊗; (C ⊗ D)[t]; Ct^{n} ⊗ D; ⊗; Ct^{m} ⊗ D; Φ_{n}⊗Φ_{m}; (C ⊗ D)t^{n}; ⊗; (C ⊗ D)t^{m}; Ct^{n} ⊗ Ct^{m}; μ_{C[t]}; ⊗; D ⊗ D; μ_{D}; (C ⊗ D)t^{n}; ⊗; μ_{(C⊗D)[t]}; (C ⊗ D)t^{m}; Ct^{n+m}; ⊗; D; Φ_{n+m}; (C ⊗ D)t^{n+m}.]

2. This follows from Assertion 1, applying n times the left hand side isomorphism and m times the right hand side isomorphism.

The following notion already appears (without a name) in Proposition 5.8 of [9]:

Definition 4.5. Let A be a monoid in F. We say that A is tensor idempotent if μ_{A} : A ⊗ A → A is an isomorphism in F, or equivalently, an isomorphism of (A, A)- bimodules.

Example 4.6. Clearly, the identity functor I is a monoid which is tensor idempotent. Since any subfunctor of it is an ideal, then any quotient of I in F is a monoid. So, by Proposition 5.8 in [9], it is also tensor idempotent. In particular, if F is the category of biset functors (see Example 2.1), then the Burnside functor RB is tensor idempotent, and any quotient biset functor of RB is a tensor idempotent monoid. For example, the simple biset functor S_{1,k}, where k is any field, and the functor of rational representations kR_{ℚ}, where k is a field of characteristic 0 (see [2]), are tensor idempotent.

Proposition 4.7. Suppose that A is a tensor idempotent commutative monoid in F. Then:

1. A_{n} ⊗ A_{n} ≅ A_{2n} as monoids in F.

2. Also, via this isomorphism, if we let A_{2n} = A[u_{1}, …, u_{n}, v_{1}, …, v_{n}], then the kernel of μ_{A_{n}} : A_{n} ⊗ A_{n} → A_{n} is equal to the ideal A_{2n}⟨u_{1} − v_{1}, …, u_{n} − v_{n}⟩.

Proof. 1. Taking C = D = A and m = n in Assertion 2 of Lemma 4.4, we get that A_{n} ⊗ A_{n} ≅ (A ⊗ A)_{2n} as monoids in F. Now since A is commutative, the morphism μ_{n} : A_{n} ⊗ A_{n} → A_{n} is a morphism of monoids. Then the assumption implies that A ⊗ A ≅ A as monoids, so (A ⊗ A)_{2n} ≅ A_{2n} as monoids, and A_{n} ⊗ A_{n} ≅ A_{2n} as claimed.

2. Let u = (u_{1}, …, u_{n}), v = (v_{1}, …, v_{n}) and t = (t_{1}, …, t_{n}). Then

A[u] = ⨁_{i} Au^{i}, A[v] = ⨁_{j} Av^{j}, and A[t] = ⨁_{k} At^{k},

where i, j, k run through multi-indices of size n. The morphism

μ_{A_{n}} : A[u] ⊗ A[v] → A[t]

is induced by the morphisms μ_{i,j} : Au^{i} ⊗ Av^{j} ≅ A ⊗ A →^{μ_{A}} A ≅ At^{i+j}. We have a commutative diagram

[figure: A_{n}; A_{n}; A_{n} ⊗ A_{n}; A[u]; ⊗; A[v]; ≅; (A ⊗ A)[u, v]; (( ⨁_{i} Au^{i}); ⊗; ⨁_{i,j} μ_{i,j}; ( ⨁_{j} Av^{j})); ≅; ⨁_{i,j}(A ⊗ A)u^{i}v^{j}; ⨁_{i,j} μ_{A}; ≅; ⨁_{k} At^{k}; ⨁_{i,j} Au^{i}v^{j}; π; A_{n}; A_{2n}]

where the bottom right vertical isomorphism is induced by μ_{A} : A ⊗ A → A, and the bottom horizontal morphism π : A_{2n} → A_{n} is induced by the morphisms Au^{i}v^{j} → At^{i+j} which are the identity on A, and map u^{i}v^{j} to t^{i+j}. It follows that for all l ∈ {1, …, n}, the elements u_{l} and v_{l} of A_{2n}(1) are both mapped to t_{l} by π. Hence Ker π contains the ideal J = A_{2n}⟨u_{1} − v_{1}, …, u_{n} − v_{n}⟩ of A_{2n} generated by u_{1} − v_{1}, …, u_{n} − v_{n}.

Now A[u] is a submonoid of A[u, v], and we clearly have A[u, v] = A[u] + J. Thus J ≤ Ker π ≤ A[u] + J, so Ker π = J + (Ker π ∩ A[u]). But the restriction of π to A[u] is an isomorphism, as it is induced by the morphisms Au^{i} → At^{i} which are the identity on A and map u^{i} to t^{i}, for all i ∈ ℕ. Then Ker π ∩ A[u] = 0, and Ker π = J, as was to be shown.

Hochschild cohomology and Hilbert syzygy theorem

In what follows, we refer to Section 4 of [1] for the notion of projectivity relative to a functor R : C → D between categories C and D. Suppose that C and D are abelian categories and that R is a faithful exact additive functor admitting a left adjoint. We will say that an object M of C is D-projective if it is projective relative to R in the sense of [1]. Similarly, we will say that a complex L in C is D-split if R(L) is a split complex in D.

Now for an object M of C, we say that a complex

… → L_{i} → L_{i−1} → … L_{0} → M → 0  (L)

in C is a D-split resolution of M if it is exact in C and if R(L) is a split complex in D. Since R is exact, this is equivalent to saying that L is exact and R(L) is split exact in D. With this terminology, Lemma 4.6 of [1] says that every object M of C admits a D-split resolution L, where the objects L_{i}, for i ≥ 0, are D-projective, and that such a resolution is unique up to homotopy.

In our context, we consider a monoid A in F, and the functor R will be the restriction functor R_{A} : A-Mod → F. Then, every A-module M admits an F-split resolution by F-projective objects, and such a resolution is unique up to homotopy.

From now on, we suppose that A is a tensor idempotent commutative monoid in F, as in Proposition 4.7. In particular, A_{n}-bimodules coincide with A_{2n}-modules.

Theorem 4.8. Let A be a tensor idempotent commutative monoid in F. Let C = A_{2n} = A[u_{1}, …, u_{n}, v_{1}, …, v_{n}] and α_{i} = u_{i} − v_{i} for i = 1, …, n. Also, let M be an A_{n}-bimodule and, for a positive integer p, consider HH^{p}(A_{n}, M), the Hochschild cohomology of A_{n} with coefficients in M (see [9]). Then:

1. The sequence α = (α_{1}, …, α_{n}) is a regular sequence for C and the Koszul resolution of C/(C⟨α⟩) ≅ A_{n},

\xymatrix{0 \ar[r] & K_{n}^{n}(C) \ar[r]^{d} & … \ar[r]^{d} & K_{2}^{n}(C) \ar[r]^{d} & K_{1}^{n}(C) \ar[r]^{L_{α}} & C \ar[r]^{μ_{An}} & A_{n} \ar[r] & 0} (K_{C}(α)_{μ})

is an F-split resolution of A_{n} by F-projective A_{2n}-modules.

2. HH^{p}(A_{n}, M) = 0 for p > n and HH^{p}(A_{n}, A_{n}) ≅ K_{p}^{n}(A_{n}) for p = 0, …, n.

Proof. 1. Let C′ = A[u_{1}, …, u_{n}]. We notice that, by Remark 4.3, C = C′[v_{1}, …, v_{n}] and also C = C′[α_{1}, …, α_{n}]. The last equality implies that α is a regular sequence for C and hence, by Theorem 3.6, we know that K_{C}(α)_{μ} is an exact sequence of Cmodules. We must show now that C and the K_{i}^{n}(C), with 1 ≤ i ≤ n are F-projective and that R_{C}(K_{C}(α)_{μ}) is a split complex. Since C and the K_{i}^{n}(C), with 1 ≤ i ≤ n, are projective C-modules (each is a coproduct of copies of C), then, by Remark 4.5 in [1], they are projective with respect to R_{C}, i.e. F-projective.

To prove that

\xymatrix{0 \ar[r] & R_{C}(K_{n}^{n}(C)) \ar[r]|{d} & … \ar[r]^{d} & R_{C}(K_{1}^{n}(C)) \ar[r]^{L_{α}} & R_{C}(C) \ar[r]^{μ_{An}} & R_{C}(A_{n}) \ar[r] & 0}

is split, we consider first the restriction to C′. That is, each of the objects R_{C′}^{C}(A_{n}), R_{C′}^{C}(C) and R_{C′}^{C}(K_{i}^{n}(C)) with 1 ≤ i ≤ n is isomorphic to a coproduct of copies of C′, by Remark 4.3. Hence, they are all projective C′-modules and R_{C′}^{C}(K_{C}(α)_{μ}) is a split complex of C′-modules. Hence, R_{C}(K_{C}(α)_{μ}) = R_{C′}(R_{C′}^{C}(K_{C}(α)_{μ})) is a split complex.

2. By Lemma 5.1 and Remark 5.2 in [9], the bar resolution and the Koszul resolution of A_{n} are homotopy equivalent. Hence, we can calculate the Hochschild cohomology of A_{n} using K_{C}(α)_{μ}. Then, clearly HH^{p}(A_{n}, M) = 0 for p > n.

Again using the Koszul resolution, in the Hochschild cochain complex of A_{n}, for 1 ≤ p ≤ n, we have the morphism

Φ : H_{C}(K_{p}^{n}(C), A_{n}) ⟶ H_{C}(K_{p+1}^{n}(C), A_{n}),

which, in an object x of X, sends an arrow t ∈ Hom_{C-Mod}(K_{p}^{n}(C), (A_{n})_{x}) to t ∘ d ∈ Hom_{C-Mod}(K_{p+1}^{n}(C), (A_{n})_{x}). Now, from Section 3, we know that d is the sum of some L_{α_{i_{k}}} : C → C, with L_{α_{i_{k}}} = α_{i_{k}} × . Hence t ∘ d will be the sum of some t(α_{i_{k}} × ). Since t is an arrow of C-modules, we have t(α_{i_{k}} × ) = α_{i_{k}} × t( ). But, by the previous proposition and Lemma 2.8, this is equal to 0. Hence t ∘ d = 0 and, since this holds for any x, we have Φ = 0. This means that, for 1 ≤ p ≤ n, we have HH^{p}(A_{n}, A_{n}) = H_{C}(K_{p}^{n}(C), A_{n}).

Now, K_{p}^{n}(C) = ⨁_{\begin{matrix}S⊆T \\ |S|=p \end{matrix}} C, with T = \{1, …, n\}, so, by Section 3 of [9], we have

H_{C}( ⨁_{\begin{matrix}S⊆T \\ |S|=p \end{matrix}} C, A_{n}) ≅ ⨁_{\begin{matrix}S⊆T \\ |S|=p \end{matrix}} H_{C}(C, A_{n}) ≅ ⨁_{\begin{matrix}S⊆T \\ |S|=p \end{matrix}} A_{n} = K_{p}^{n}(A_{n}).

Finally, by Section 6 of [9], we know that HH^{0}(A_{n}, A_{n}) = A_{n} = K_{0}^{n}(A_{n}).

In the next proposition we make use of some results concerning the tensor product over a monoid in F, which can be found in the Appendix.

Proposition 4.9. (Relative Hilbert syzygy Theorem) Let A be a tensor idempotent commutative monoid in F and t_{1}, …, t_{n} be variables. Then, any A[t_{1}, …, t_{n}]-module M admits a finite F-split resolution

0 → L_{n} → L_{n−1} → … → L_{1} → M → 0

by F-projective A[t_{1}, …, t_{n}]-modules.

Proof. We continue to denote A_{2n} by C and apply the functor −⊗_{A_{n}}M to the resolution K_{∗} := K_{C}(α)_{μ} of A_{n} of Theorem 4.8. Since A_{n} ⊗_{A_{n}} M ≅ M as A_{n}-modules, we get a complex

0 → K_{n}^{n}(C) ⊗_{A_{n}} M → … → K_{1}^{n}(C) ⊗_{A_{n}} M → C ⊗_{A_{n}} M → M → 0 (K_{∗} ⊗_{A_{n}} M)

of A_{n}-modules.

Consider the restriction from (A_{n}, A_{n})-bimodules to right A_{n}-modules. We continue to denote this functor by R_{A_{n}}^{C}, but taking into account that the A_{n}-modules are viewed as right modules. By the proof of the previous theorem, we know that R_{A_{n}}^{C}(K_{∗}) is a split exact complex. Hence it is contractible and, since ⊗_{A_{n}} M is an additive functor, the complex (R_{A_{n}}^{C}(K_{∗})) ⊗_{A_{n}} M is contractible too. Now, by Lemma 5.2 of the Appendix, this complex is isomorphic to R_{I}^{A_{n}}(K_{∗} ⊗_{A_{n}} M) = R_{A_{n}}(K_{∗} ⊗_{A_{n}} M). Hence, R_{A_{n}}(K_{∗} ⊗_{A_{n}} M) is a split exact complex in F. Finally, since R_{A_{n}} is faithful and exact, it follows that the complex (K_{∗} ⊗_{A_{n}} M) is exact in A_{n}-Mod. Hence it is an F-split resolution of M.

Now, each K_{i}^{n}(C), for i > 0, is a direct sum of copies of the (C, C)-bimodule C. Also, C ≅ A_{n} ⊗ A_{n} = A_{n} ⊗_{I} A_{n} (we notice in the Appendix that the tensor product over I is the usual tensor product of F), so

C ⊗_{A_{n}} M ≅ (A_{n} ⊗_{I} A_{n}) ⊗_{A_{n}} M ≅ A_{n} ⊗_{I} (A_{n} ⊗_{A_{n}} M) ≅ A_{n} ⊗_{I} M.

This shows in particular that C ⊗_{A_{n}} M is F-projective, by Lemma 4.3 of [1], since A_{n} ⊗_{I} M is in the image of the left adjoint to the restriction functor from A_{n}-modules to F. Moreover, K_{i}^{n}(C) ⊗_{A_{n}} M ≅ K_{i}^{n}(A_{n}) ⊗_{I} M, so all the terms K_{i}^{n}(C) ⊗_{A_{n}} M, for i > 0, are also F-projective. Hence, the complex K_{∗} ⊗_{A_{n}} M, isomorphic to

0 → K_{n}^{n}(A_{n}) ⊗_{I} M → … → K_{1}^{n}(A_{n}) ⊗_{I} M → A_{n} ⊗_{I} M → M → 0,

is an F-split resolution of M by F-projective A_{n}-modules. This completes the proof.

5 Appendix

Let A be a monoid in F, M a right A-module with action σ_{r} : M ⊗ A → M and N a left A-module with action σ_{l} : A ⊗ N → N. The tensor product over A of M and N, denoted by M ⊗_{A} N, is the coequalizer of the arrows given by the actions (see Section VII of [6]),

\xymatrix{M ⊗ A ⊗ N \ar[r]^{σ_{r}⊗N} \ar[r]_{M⊗σ_{l}} & M ⊗ N \ar[r] & M ⊗_{A} N,}

which exists since F is bicomplete. This construction defines an additive functor

Mod-A × A-Mod → F, (M, N) ↦ M ⊗_{A} N.

Notice that, for F and T objects in F, the tensor product over I, F ⊗_{I} T, is the usual tensor product of F.

Remark 5.1. Let A, C and D be monoids in F, let M be a (C, A)-bimodule and N be an (A, D)-bimodule. The following facts are easily deduced from the definition.

1. If Q is a D-module, then

(M ⊗_{A} N) ⊗_{D} Q ≅ M ⊗_{A} (N ⊗_{D} Q)

and the isomorphism is natural in the three variables.

2. M ⊗_{A} N is a (C, D)-bimodule.

3. We have M ⊗_{A}A ≅ M and A⊗_{A}N ≅ N in F and these are natural isomorphisms of C-modules and right D-modules, respectively.

Lemma 5.2. Let A, D and E be monoids in F, M be a (D, A)-bimodule and N be an (A, E)-bimodule. Let R_{A}^{D, A} denote the restriction functor R_{I⊗A^{op}}^{D⊗A^{op}} from (D, A)- bimodules to right A-modules and R_{E}^{D, E} be defined in a similar way. Then, there is an isomorphism

R_{E}^{D, E}(M ⊗_{A} N) ≅ (R_{A}^{D, A}M) ⊗_{A} N

of right E-modules, which is functorial in M and N.

Proof. Indeed, by construction, we have an exact sequence

M ⊗ A ⊗ N →^{δ} M ⊗ N → M ⊗_{A} N → 0

of (D, E)-bimodules, where δ = σ_{r} ⊗ N − M ⊗ σ_{l}. Applying the functor R_{E}^{D, E} to this sequence gives the exact sequence

R_{E}^{D, E}(M ⊗ A ⊗ N) → R_{E}^{D, E}(M ⊗ N) → R_{E}^{D, E}(M ⊗_{A} N) → 0.

Moreover, there are canonical isomorphisms of right E-modules

R_{E}^{D, E}(M ⊗ A ⊗ N) ≅ (R_{A}^{D, A}M) ⊗ A ⊗ N,
R_{E}^{D, E}(M ⊗ N) ≅ (R_{A}^{D, A}M) ⊗ N,

so we get the exact sequence

(R_{A}^{D, A}M) ⊗ A ⊗ N → (R_{A}^{D, A}M) ⊗ N → R_{E}^{D, E}(M ⊗_{A} N) → 0.

Thus R_{E}^{D, E}(M ⊗_{A} N) ≅ (R_{A}^{D, A}M) ⊗_{A} N, as claimed.

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Serge Bouc, CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens, France.

serge.bouc@u-picardie.fr

Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico.

nadia.romero@ugto.mx