POLYTOPAL LINEAR ALGEBRA

WINFRIED BRUNS AND JOSEPH GUBELADZE

Abstract. We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous results on the automorphism groups and retractions. These results support a general conjecture proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras. Thereafter a further confirmation of this conjecture is presented by homomorphisms defined on Veronese singularities.

This is a continuation of the project started in [BG1, BG2, BG3]. The higher K-theoretic aspects of polytopal linear objects will be treated in [BG4].

1. Introduction

The present work is a continuation of our study of the similarities between the categories Vect(k) – the category of finitely generated vector spaces over a field k, and its natural extension Pol(k) – the polytopal linear category over k, started in the series of papers [BG1, BG2, BG3]. The category Pol(k) was first introduced explicitly in [BG3] where we studied a special class of morphisms, the retractions.

We recall that the objects of Pol(k) are by definition polytopal k-algebras (discussed in detail in [BGT]), i. e. the standard graded k-algebras k[P] associated to arbitrary finite convex lattice polytopes P ⊂ ℝ^{d} in the following way: the lattice points L_{P} = ℤ^{d} ∩ P form degree one generators of k[P] and they are subject to the binomial relations coming from the affine dependencies inside P.

Alternatively, k[P] is the semigroup ring k[S_{P}] of the semigroup S_{P} ⊂ ℤ^{d} generated by {(x, 1) | x ∈ L_{P} }.

The embedding Vect(k) ⊂ Pol(k), resulting from viewing a vector space V as the degree one component of its symmetric algebra S_{k}(V) = k[X_{1}, …, X_{dim_{k} V}], makes Vect(k) a full subcategory of Pol(k). Obviously, the latter category is far from being additive but it reveals many surprising similarities with Vect(k).

This work provides further results supporting the analogy. In particular, in Section 2 we show that polytopal Picard groups defined as the groups of certain autoequivalences of Pol(k), are trivial (i. e. coincide with Pic(k)). We work exclusively over a field k, which is often assumed to be algebraically closed. But in Remark 2.3 we indicate an approach that enables one to use arbitrary commutative rings. In the general definition of the categories Vect(k) and Pol(k) the hom-sets are replaced by appropriate affine schemes. This definition must already be used for fields in general. The coincidence of Pic^{Pol}(R) with Pic(R) for general (commutative) rings R remains an open question.

The second author was supported by the Max-Planck-Institut für Mathematik at Bonn and TMR grant ERB FMRX CT-97-0107.

In order to present a coherent picture of polytopal linear algebra and to ease references throughout the text, we recall some of the results from [BG1] and [BG2] in Section 3; they concern the automorphism groups and the retractions in Pol(k). In Section 4 we propose a conjecture describing arbitrary homomorphisms in Pol(k). Roughly, it says that the homomorphisms are obtained from the trivial ones by a sequence of standard procedures encoded in the shapes of the underlying polytopes. In particular, the arithmetic of k is irrelevant in the description of Pol(k) because everything is determined on the combinatorial level.

The results obtained so far [BG1, BG2] can be viewed as a confirmation of refined versions of this conjecture for special classes of morphisms, namely automorphisms and retractions.

Thereafter in Section 4 we provide further evidence towards our conjecture by an explicit description of homomorphisms from Veronese subalgebras of polynomial rings. This result, in conjunction with the results from [BG2], provides a complete description of the variety of idempotent endomorphisms of k[P] when k is algebraically closed and P is a lattice polygon (i. e. dim P = 2).

The approach developed in [BG3] suggests a further generalization to the even more general category of polyhedral algebras and their graded homomorphisms. This corresponds to the passage from single polytopes to lattice polyhedral complexes in the sense of [BG3]. However, in this article we do not pursue such level of generality and only remark that even the subclass of simplicial complexes (i. e. the category of Stanley-Reisner rings and their graded homomorphisms) provides interesting possibilities for the generalization of linear algebra.

The arguments in Section 2 below use results presented in Section 3. But we resort to this order of exposition in analogy with the classical hierarchy – Picard groups, retractions, automorphisms. The objects just listed constitute the subject of ‘classical’ algebraic K-theory (Bass [Ba2]). In [BG4] we consider higher K-theoretical aspects of the polytopal generalization of vector spaces.

For the standard terminology in category theory we refer to MacLane [Ma].

In what follows ‘homomorphism’ always means ‘graded homomorphism’.

2. Polytopal Picard groups

For a commutative ring R the group of R-linear covariant autoequivalences of 𝕄(R) – the category of finitely generated R-modules, modulo functor isomorphisms, is isomorphic to Pic(R) – the group of invertible R-modules up to isomorphism [Ba1]. (Recall that a functor F is called R-linear if the induced mappings Hom(M, N) → Hom(F (M), F (N)) are R-homomorphisms for all M, N ∈ |𝕄(R)|.) The isomorphism is given by the assignments F ↦ F (R) and L ↦ L⊗− for F : 𝕄(R) → 𝕄(R) and L ∈ Pic(R) as mentioned.

On the basis on this classical fact we propose the following definition of the polytopal version of Pic(k) for an algebraically closed field k:

Definition 2.1. Pic^{Pol}(k) is the group of those covariant autoequivalences (up to isomorphism) of Pol(k) which satisfy the following condition: the mappings

Hom(A, B) → Hom(F (A), F (B))

are k^{∗}-equivariant and respect the k-variety structures for all A, B ∈ | Pol(k)|.

Some remarks are in order.

First of all, the sets Hom(k[P], k[Q]) carry natural variety structures (see Remark 2.3). Further, any object A ∈ | Pol(k)| is considered to be equipped with the following k^{∗}-action:

(a_{0} ⊕ a_{1} ⊕ a_{2} ⊕ ⋯)^{ξ} = a_{0} ⊕ ξa_{1} ⊕ ξ^{2}a_{2} ⊕ ⋯,

which induces the algebraic action

k^{∗} × Hom(A, B) → Hom(A, B)

as follows:

f^{ξ}(a) = f(a^{ξ}), f ∈ Hom(A, B), ξ ∈ k^{∗}, a ∈ A.

Notice that we obtain the same action on Hom(A, B) by requiring

f^{ξ}(a) = ξf(a) for all f ∈ Hom(A, B), ξ ∈ k^{∗}, a ∈ A_{1}.

Remark 2.2. The key observation justifying this definition is that the same definition, applied to the subcategory Vect(k) ⊂ Pol(k), gives the original Pic(k) (= 0). In fact, any k^{∗}-equivariant algebraic mapping between vector spaces (in this case Hom(A, B), A, B ∈ Vect(k)) is automatically k-linear (due to the infinity of k).

Remark 2.3. The hom-sets Hom(A, B) in Pol(k) are naturally equipped with the structure of an affine k-variety. This remains true for the set of graded homomorphisms between arbitrary finitely generated graded k-algebras (k = k̄) but a special feature of polytopal algebras is that these hom-varieties are defined over ℤ (due to the fact that the schemes Spec(k[P]) themselves are defined over ℤ). This observation makes it possible to define Pic^{Pol}(R) for an arbitrary commutative ring R as follows.

For k an algebraically closed field we write up the equalities defining the Zariski closed subset

Hom(k[P], k[Q]) ⊂ 𝔸_{k}^{mn}, m = # L_{P} , n = # L_{Q},

where an element f ∈ Hom(k[P], k[Q]) is identified with the corresponding matrix

(a_{ij}) ∈ M_{m×n}(k), f(x_{i}) = Σ_{j=1}^{n}a_{ij}y_{j}, x_{i} ∈ L_{P} , y_{j} ∈ L_{Q} .

The binomial relations between the x_{i} are preserved by the f(x_{i}) ∈ k[Q]_{1} (the degree 1 component of k[Q]). After passing to the canonical k-linear expansions as linear forms of monomials in the y_{j} and comparing corresponding coefficients (at this point the binomial dependencies between the y_{j} are used) we get a system of homogeneous equalities F_{s} = 0 with F_{s} ∈ ℤ[X_{1}, …, X_{mn}]. Clearly, the polynomials F_{s} only depend on P and Q and the field k does not play a rôle.

We can define the categories Vect(R) and Pol(R) for any ring R as follows: the objects are the polytopal algebras over R and the hom-sets Hom(R[P], R[Q]) are the affine schemes Spec(R^{mn}/(F_{s})) correspondingly. In order to simplify the notation we will consider the underlying rings instead of the affine schemes. Thus Hom(R[P], R[Q]) = R^{mn}/(F_{s}). The equivariant structure on the hom-sets is encoded into the ring homomorphisms

Hom(R[P], R[Q]) → Hom(R[P], R[Q])[X, X^{−1}], a ↦ Xa,

and the composition operation is given by the naturally defined ring homomorphisms

Hom(R[P], R[L]) → Hom(R[P], R[Q]) ⊗ Hom(R[Q], R[L]).

Clearly, if R = k is an algebraically closed field, the two definitions of Pol(R) give equivalent categories.

Now we can define Pic^{Pol}(R) as the group of those covariant autoequivalences of Pol(R) which on the hom-rings induce ring homomorphisms respecting the Spec(ℤ[X, X^{−1}])-equivariant structures. In particular, we have defined Pic^{Pol}(k) for general fields.

The proof we present below for algebraically closed fields yields the equality Pic^{Pol}(k) = 0 for arbitrary fields. We leave this to the interested reader and only remark that the crucial fact is that the main result of [BG1] (Theorem 3.2 below) has been proved for general fields.

The lack of an analogous description of the automorphism groups over a general ring of coefficients is the obstacle in showing that one always has Pic(R) = Pic^{Pol}(R).

Remark 2.4. The last step in the proof of Theorem 2.5 uses the fact from [BGT] that for a polytope P and a field k the polytopal algebra k[cP] is quadratically defined (i. e. by degree 2 equations) whenever c ≥ dim P. This however does not create an additional difficulty in generalizing the result on polytopal Picard groups from fields to arbitrary rings. It is an elementary fact that the condition on a polytopal ring to be quadratically defined depends only on the combinatorial structure of the polytope.

In the remaining part of this section k is an algebraically closed field.

Theorem 2.5. Pic^{Pol}(k) = 0.

Proof. Let [F] ∈ Pic^{Pol}(k). We want to show that there are isomorphisms (σ_{A} : A → F (A))_{| Pol(k)|} such that

F (f) = σ_{B} ∘ f ∘ σ_{A}^{−1}

for every homomorphism f : A → B in Pol(k).

First of all notice that we can work on an arbitrarily fixed skeleton of Pol(k): all the notions we are dealing with are invariant under such a passage. Henceforth Pol(k) is the fixed skeleton. By [Gu] one knows that k[P] determines (up to an affine integral isomorphism) the polytope P (this is so even in the category of all commutative k-algebras). Therefore, we assume that for each object A ∈ | Pol(k)| there is a unique polytope P such that A = k[P] and different polytopal algebras determine non-isomorphic polytopes. By a suitable choice of the skeleton we can also assume that the objects of Vect(k) are of the type k[Δ_{n}] for the unit n-simplices Δ_{n}, n = −1, 0, 1, 2, … (by convention k[Δ_{−1}] = k). Also, we will use the notation k[t] = k[Δ_{0}].

Step 1. We claim that

(i) dim A = dim F (A) (Krull dimension), (ii) dim_{k} A_{1} = dim_{k} F (A)_{1} (k-ranks of the degree 1 components), (iii) F (f) is surjective (the degree 1 component F (f)_{1} is injective) if and only if f

is surjective (f_{1} is injective). In fact, it follows from Theorem 3.2 below that for any polytopal algebra k[P] its

Krull dimension is the dimension of a maximal torus of the linear subgroup

Γ_{k}(P) = Aut_{Pol(k)}(k[P]) ⊂ Aut_{k}k[P]_{1};

it is certainly an invariant of F – hence (i).

The claims (ii) and (iii) follow from the observation that both the injectivity and surjectivity conditions can be reformulated in purely categorical terms by using morphisms originating from k[t].

Step 2. Observe that F restricts to an autoequivalence of Vect(k). This follows from the claims (i) and (ii) in Step 1 and the fact that polynomial algebras are the only polytopal algebras whose Krull dimensions coincide with the k-ranks of the degree 1 components.

Next we correct F on Vect(k) in such a way that F |_{Vect(k)} = 1_{Vect(k)}.

We know that F |_{Vect(k)} ≈ 1_{Vect(k)}. This means that there are elements ρ_{V} ∈ Aut(V) for all V ∈ | Vect(k)| such that

∀ U, V ∈ | Vect(k)| (f : U → V) ↦ (ρ_{V} ∘ f ∘ ρ_{U}^{−1} : U → V).

For each A ∈ | Pol(k)| \ | Vect(k)| put ρ_{A} = 1_{A} and let H : Pol(k) → Pol(k) be the autoequivalence determined as follows: it is the identity on the objects and

H(f) = ρ_{B}^{−1} ∘ f ∘ ρ_{A}

for a morphism f : A → B in Pol(k). Then H represents the neutral element of Pic^{Pol}(k). Now the functor G = H ∘ F is isomorphic to F and it restricts to the identity functor on Vect(k).

Without loss of generality we can therefore assume F |_{Vect_{k}} = 1_{Vect_{k}}.

Step 3. For any positive dimensional object k[P] ∈ | Pol(k)| we fix a bijective mapping Δ_{P} from the set of vertices of Δ_{n−1}, n = # L_{P} to the set of lattice points of P. The resulting surjective homomorphism k[Δ_{n−1}] → k[P] will also be denoted by Δ_{P}.

Every matrix α ∈ GL_{n}(k) gives rise to a graded k-surjective homomorphism

αΔ_{P} : k[Δ_{n−1}] → k[P]

whose degree 1 component is given by

(αΔ_{P})_{1} = (Δ_{P})_{1} ∘ α_{∗},

where α_{∗} is the linear transformation of the k-vector space k[Δ_{n−1}]_{1} determined by α, and (Δ_{P})_{1} is the degree 1 component of Δ_{P}.

By Step 2 F (k[Δ_{n−1}]) = k[Δ_{n−1}] for all n ∈ ℕ. Therefore, for a polytope P, satisfying the condition F (k[P]) = k[P], the claim (iii) in Step 1 implies the following commutative square

Hom(k[Δ_{n−1}], k[P]) \overline{F_{}}Hom(k[Δ_{n−1}], k[P])

[figure: Sur(k[Δ_{n−1}], k[P]); ∪; F; Sur(k[Δ_{n−1}], k[P]); ∪]

where Sur(k[Δ_{n−1}], k[P]) denotes the set of surjective homomorphisms. By the definition of Pic^{Pol}(k) the horizontal mappings are k^{∗}-equivariant automorphisms of k-varieties.

We have the following obvious equalities:

Hom(k[Δ_{n−1}], k[P]) = {γΔ_{P} | γ ∈ M_{n×n}(k)},
Sur(k[Δ_{n−1}], k[P]) = {γΔ_{P} | γ ∈ GL_{n}(k)}.

After the appropriate identifications with M_{n×n}(k) and GL_{n}(k) respectively we arrive at the commutative square of k-varieties

[figure: M_{n×n}(k); F; M_{n×n}(k); GL_{n}(k); ∪; F; GL_{n}(k); ∪]

whose horizontal arrows are algebraic k^{∗}-equivariant automorphisms for the diagonal k^{∗}-actions. Thus the upper horizontal mapping is a linear non-degenerate transformation of the k-vector space M_{n×n}(k) (see Remark 2.2), which leaves the subset GL_{n}(k) ⊂ M_{n×n}(k) invariant, i. e. the matrix degeneracy locus M_{n×n}(k) \ GL_{n}(k) is invariant under F. Then by Corollary 3.3 below there are only two possibilities:

(a) there are α, β ∈ GL_{n}(k) such that F (γΔ_{P}) = (βγα)Δ_{P} for all γ ∈ M_{n×n}(k), or

(b) there are α, β ∈ GL_{n}(k) such that F (γΔ_{P}) = (βγ^{T} α)Δ_{P} for all γ ∈ M_{n×n}(k), where −^{T} is the transposition.

Consider the commutative square

[figure: k[Δ_{n−1}]; γ_{∗}; k[Δ_{n−1}]; k[P]; γΔ_{P}; = = = = = =; k[P],; Δ_{P}],

where γ ∈ M_{n×n}(k) is arbitrary matrix and the degree 1 component of the upper horizontal mapping is the linear transformation given by γ. By applying the functor F to this square and using the equality F |_{Vect(k)} = 1_{Vect(k)} (Step 2) we arrive at the following equalities in the corresponding cases:

(a) (βγα)Δ_{P} = (βα)Δ_{P} ∘ γ_{∗}, (b) (βγ^{T} α)Δ_{P} = (βα)Δ_{P} ∘ γ_{∗}.

Identifying L_{Δ_{n−1}} and L_{P} along Δ_{P} we get the matrix equalities:

(a) βγα = γβα, (b) βγ^{T} α = γβα.

First notice that case (b) is excluded, i. e. there is no matrix β ∈ GL_{n}(k) for which the following holds:

∀ γ ∈ M_{n×n}(k) βγβ^{−1} = γ^{T} .

For example, this follows from Corollary 3.3 which says that the automorphism of the matrix degeneracy locus in M_{n×n}(k) given by γ ↦ γ^{T} in not in the unity component of the linear automorphism group of this locus, provided n ≥ 2.

For case (a) we have

∀ γ ∈ M_{n×n}(k) βγβ^{−1} = γ.

Then β is in the center of M_{n×n}(k). So we can write

F (γΔ_{P}) = (γβα)Δ_{P}

We arrive at the

Claim. For each lattice polytope P with F (k[P]) = k[P], there is α_{P} ∈ GL_{n}(k), n = # L_{P}, such that

∀ γ ∈ M_{n×n}(k) F (γΔ_{P}) = (γα_{P})Δ_{P} .

Step 4. Now we show that for every polytope P the linear automorphism (α_{P})_{∗} of the k-vector space k[P]_{1}, determined by the matrix α_{P} (Step 3), belongs to the closed subgroup

Γ_{k}(P) ⊂ Aut_{k}(k[P]_{1}),

provided F (k[P]) = k[P].

The functor F induces a k^{∗}-equivariant automorphism of the k-variety Hom(k[P], k[t]). On the other hand we have the natural identification

Hom(k[P], k[t]) = maxSpec(k[P]),

where the right hand side denotes the variety of closed points of Spec(k[P]). Therefore, there exists an automorphism ψ of the k-algebra k[P] – not a priori graded, such that the mapping

F : Hom(k[P], k[t]) → Hom(k[P], k[t])

is given by φ ↦ φ ∘ ψ, and, moreover, it is k^{∗}-equivariant. It follows that ψ is a k^{∗}-equivariant automorphism of k[P]. It is easily seen that a k^{∗}-equivariant automorphism is graded (and conversely). Therefore ψ_{1} ∈ Γ_{k}(P). (Here we identify elements of Γ_{k}(P) with their degree one components, which are linear automorphisms of ⨁_{L_{P}} k.)

We let P^{∗} : k[P] → k[t] denote the homomorphism which sends L_{P} to t. For any toric automorphism τ ∈ 𝕋_{k}(P) (i.e. an automorphism for which any element of L_{P} is an eigenvector, 𝕋_{k}(P) is the group of such automorphisms, see Section 3) we have the commutative diagram

[figure: k[Δ_{n−1}]; (Δ_{n−1})^{∗}; k[t]; k[P]; τ^{∗}Δ_{P}; τ^{−1}; k[P]; P^{∗}]

where τ^{∗} refers to the diagonal n×n-matrix corresponding to the degree 1 component of τ.

In view of what has been said above and of Step 3, an application of F to the last commutative diagram yields the equality

(Δ_{n−1})^{∗} = (P^{∗} ∘ τ^{−1} ∘ ψ) ∘ ((τ^{∗}α_{P})Δ_{P} ).

Let ψ^{∗} ∈ M_{n×n}(k) be the matrix of the degree 1 component of ψ, and put ω = α_{P} ψ^{∗}. Then the equality can be reformulated into the condition: (∗) for any τ ∈ 𝕋_{k}(P) the sum of the entries of each row in the matrix τ^{∗}ω(τ^{∗})^{−1}

is 1, that is

((τ^{−1} ∘ ω_{∗} ∘ τ)(x_{i}) = ∑_{j=1}^{n}c_{ij}x_{j}) ⟹ (∑_{j=1}^{n}c_{ij} = 1),

where ω_{∗} is the linear transformation of k[P]_{1} determined by ω, and L_{P} = {x_{1}, …, x_{n}}.

Now we derive from (∗) that ω_{∗} = 1_{k[P]_{1}}, that is (α_{P})_{∗} = ψ^{−1} ∈ Γ_{k}(P). Fix i ∈ [1, n] and τ ∈ 𝕋_{k}(P). We put

ω_{∗}(x_{i}) = ∑_{j=1}^{n} a_{j}x_{j}

and

τ(x_{j}) = b_{j}x_{j}, j ∈ [1, n].

Then

(τ^{−1} ∘ ω_{∗} ∘ τ)(x_{i}) = ∑_{j=1}^{n} \frac{b_{i}a_{j}}{b_{j}} x_{j}.

Consider the Laurent polynomial

l_{i} = ∑_{j=1}^{n} a_{j} \frac{x_{j}}{x_{i}} ∈ k[ℤ^{d}] (= k[z_{1}, z_{1}^{−1}, …, z_{d}, z_{d}^{−1}]).

Without loss of generality we assume that d = dim P and gp(S_{P}) = ℤ^{d+1}.

Observe that the assignment

\frac{x_{j}}{x_{i}} ↦ \frac{b_{i}a_{j}}{b_{j}} · \frac{x_{j}}{x_{i}} , j ∈ [1, n],

extends uniquely to a toric automorphism of k[ℤ^{d}] (the torus (k^{∗})^{d} acts tautologically on its coordinate ring k[ℤ^{d}]). Conversely, any toric automorphism of k[ℤ^{d}] can be obtained in this way from some element of 𝕋_{k}(P). Therefore

{ξ(l_{i}) | ξ ∈ (k^{∗})^{d}} = {x_{i}^{−1}((τ^{−1} ∘ ω_{∗} ∘ τ)(x_{i})) | τ ∈ 𝕋_{k}(P)}.

In particular, the sum of the coefficients of the Laurent polynomial ξ(l_{i}) is 1 for any ξ ∈ (k^{∗})^{d}, in other words

l_{i}(ξ_{1}, …, ξ_{d}) = 1

for all ξ_{1}, …, ξ_{d} ∈ k^{∗}. Taking into account the infinity of k we conclude l_{i} = 1, that is ω_{∗} = 1_{k[P]_{1}}.

Step 5. Let Pol(k)′ ⊂ Pol(k) denote the full subcategory whose objects are those polytopal algebras k[P] for which F (k[P]) = k[P]. Now we show that there is G ∈ Pic^{Pol}(k) such that F ≈ G and G|_{Pol(k)′} = 1_{Pol(k)′}.

As in Step 2, this means that we have to show the existence of elements

σ_{P} ∈ Γ_{k}(P), k[P] ∈ | Pol(k)′|,

such that F (f)_{1} = σ_{Q} ∘ f_{1} ∘ σ_{P}^{−1} for any morphism f : k[P] → k[Q] in Pol(k)′. Consider the commutative diagram

[figure: k[Δ_{m−1}]; f̃; k[Δ_{n−1}]; k[P]; Δ_{P}; f; k[Q]; Δ_{Q}]

where f is any morphism in Pol(k)′, m = # L_{P}, n = # L_{Q}, and f̃ is the unique lifting of f. By Step 3 we know that F transforms this square into the square

[figure: k[Δ_{n−1}]; f̃; k[Δ_{m−1}]; k[P]; α_{P} Δ_{P}; F (f); k[Q]; α_{Q}Δ_{Q}]

Therefore, looking at the degree 1 components we conclude

F (f)_{1} = (α_{Q})_{∗} ∘ f_{1} ∘ (α_{P})_{∗}^{−1}.

So by Step (4) the system

{(α_{P})_{∗} | k[P] ∈ | Pol(k)′|}

is the desired one.

Step 6. By the previous step we can assume that F |_{Pol(k)′} = 1_{Pol(k)′}. Now we complete the proof by showing that this assumption implies Pol(k)′ = Pol(k), and therefore F = 1_{Pol(k)}.

Assume P is a lattice polytope. We let denote the unit square and consider all the possible integral affine mappings:

M = {μ : → cP }.

We define the diagram 𝔻(k, P) as follows. It consists of

(1) #(L_{P}) copies of k[t] indexed by the elements of L_{P} = {x_{1}, …, x_{n}},

(2) #(M) copies of k[ ]: {k[ ]_{μ}}_{M}, and the following morphisms between them

k[t]_{x_{i}} → k[ ]_{μ}, t ↦ z ∈ L , if μ(z) = x_{i}.

We claim that

\underrightarrow{lim} 𝔻(k, P) = k[P]

whenever the defining ideal of the toric ring k[P] is generated by quadratic binomials, where the direct limit is considered in the category of all (commutative) k-algebras.

In fact, it is clear that the mentioned limit is always a standard graded k-algebra, whose degree 1 component has k-dimension equal to # L_{P}, and that there is a surjective graded k-homomorphism

𝕃(k, P) = \underrightarrow{lim} 𝔻(k, P) → k[P].

This is so because of the cone over the diagram 𝔻(k, P) with vertex k[P] where t ∈ k[t]_{x_{i}} is mapped to x_{i} and the vertices of from k[ ]_{μ} are mapped accordingly to μ. Therefore, we only need to make sure that t_{i}t_{j} − t_{k}t_{l} = 0 in 𝕃(k, P) whenever x_{i}x_{j} − x_{k}x_{l} = 0 in k[P], where t_{i} is the image of t ∈ k[t]_{x_{i}} in 𝕃(k, P), and similarly for t_{j}, t_{k} and t_{l}. But if x_{i}x_{j} − x_{k}x_{l} = 0 in k[P] then these four points in P belong to Im(μ) for some μ ∈ M and the desired equality is encoded into the diagram 𝔻(k, P).

Having established the equality above for quadratically defined polytopal rings we now show that k[cP] ∈ | Pol(k)′| for every lattice polytope P and all c ≥ dim P.

By Theorem 1.3.3(a) in [BGT] the defining ideal of the toric ring k[cP] is generated by quadratic binomials for c ≥ dim P. Therefore,

𝕃(k, cP) = k[cP]

for such c ∈ ℕ. Now observe that the claims (i) and (ii) in Step 1 imply that either F (k[ ]) = k[ ] or F (k[ ]) = k[T] for a lattice triangle T with 4 lattice points.

Let us show that F (k[ ]) cannot be a ‘triangle ring’. Up to isomorphism there are only two lattice triangles with 4 lattice points

conv ((1, 0), (0, 1), (−1, 0)) and conv ((1, 0), (0, 1), (−1, −1))

By Theorem 3.2 below the triangle T with the property F (k[ ]) = k[T] must have

Figure 1

the same number of column vectors as (see Section 3 for the definition). But this is not the case because the second triangle has no column vectors whereas the first has 5 of them, and the square only 4. (The column vectors are indicated in the figure.) Therefore k[ ] ∈ | Pol(k)′| as well.

By the assumption F |_{Pol(k)′} = 1_{Pol(k)′} we get the natural homomorphism k[cP] → F (k[cP]) (c as above). Arguing similarly for the functor F^{−1}, we get the natural homomorphism k[cP] → F^{−1}(k[cP]), and applying F to the latter homomorphism we arrive at the natural homomorphism F (k[cP]) → k[cP]. But every natural (that is compatible with 𝔻(k, cP)) endomorphism k[cP] → k[cP] must be the identity mapping for reasons of universality. In particular, k[cP] is a k-retract of F (k[cP]). By reasons of dimensions (and claim (i) in Step 1) we finally get the equality F (k[cP]) = k[cP], as required.

Now we are ready to show the equality Pol(k)′ = Pol(k), completing the proof of Theorem 2.4.

Let k[P] ∈ | Pol(k)|. Consider two coprime natural numbers c, c′ ≥ dim P. We have the commutative square, consisting of embeddings in Pol(k),

[figure: k[P]; k[cP]; k[c′P]; k[cc′P]] (∗∗)

where the horizontal (vertical) mappings send lattice points in the corresponding polytopes to their homothetic images (centered at the origin) with factor c and c′ respectively. The key observation is that for the degree one components we get the pull back diagram of k-vector spaces

[figure: k[P]_{1}; k[cP]_{1}; k[c′P]_{1}; k[cc′P]_{1}]

or, equivalently, the following is a pull back diagram of finite sets

[figure: L_{P}; L_{cP}; L_{c′P}; L_{cc′P} .]

Caution. (∗∗) is in general not a pull back diagram of k-algebras. Otherwise, by Theorem 1.3.3 in [BGT], any polytopal algebra would be normal, which is not the case.

That the latter square of finite sets is in fact a pull back diagram becomes transparent after thinking of it as the (isomorphic) diagram consisting of the inclusions:

[figure: L_{P (ℤ^{d})}; ⊂; L_{P ( \frac{ℤ^{d}}{c})}; L_{P ( \frac{ℤ^{d}}{c′})}; ∩; ⊂; L_{P ( \frac{ℤ^{d}}{cc′})}; ∩]

where P (H) refers to the lattice polytope P ⊂ ℝ^{d} with respect to the intermediate lattice ℤ^{d} ⊂ H ⊂ ℝ^{d}. Here again we assume that d = dim P and gp(S_{P}) = ℤ^{d+1}.

Since k[cP], k[c′P], k[cc′P] ∈ Pol(k)′ and F is the identity functor on Pol(k)′, an application of F to the square (∗∗) yields the square of graded homomorphisms

[figure: F (k[P]); k[cP]; k[c′P]; k[cc′P],]

with the same arrows into k[cc′P] as (∗∗). The same arguments as in the proof of (iii), Step 1 show that the degree 1 component of this square is a pull-back diagram, in other words F (k[P])_{1} = k[P]_{1}. Therefore, F (k[P])_{1} and k[P]_{1} generate the same algebras, i. e. k[P] = F (k[P]).

3. Automorphisms and retractions

In this section we survey those results of [BG1] and [BG2] that are related to polytopal linear algebra.

As far as automorphisms are concerned, k will be assumed to be a general, not necessarily algebraically closed field.

As remarked in Section 2, for a lattice polytope P ⊂ ℝ^{d} the group Γ_{k}(P) = gr. aut(k[P]) is a linear k-group in a natural way. It coincides with GL_{n}(k) in the case of the unit (n − 1)-simplex P = Δ_{n−1}. The groups Γ_{k}(P) have been named polytopal linear groups in [BG1].

An element v ∈ ℤ^{d}, v ≠ 0, is a column vector (for P) if there is a facet F ⊂ P such that x + v ∈ P for every lattice point x ∈ P \ F. The facet F is called the base facet of v. The set of column vectors of P is denoted by Col(P). A pair (P, v),

v

Figure 2. A column structure

v ∈ Col(P), is called a column structure. Let (P, v) be a column structure and P_{v} ⊂ P be the base facet for v ∈ Col(P). Then for each element x ∈ S_{P} we set ht_{v}(x) = m where m is the largest non-negative integer for which x + mv ∈ S_{P}. Thus ht_{v}(x) is the ‘height’ of x above the facet of the cone C(S_{P}) corresponding to P_{v} in direction −v. It is an easy observation that x + ht_{v}(x) · v ∈ S_{P_{v}} ⊂ S_{P} for any x ∈ S_{P}.

Column vectors are just the dual objects to the roots of the normal fan N (P) in the sense of Demazure (see [Oda, Section 3.4]). (For the relationship of Theorem 3.2 with the results of Demazure [De] and Cox [Cox] on the automorphism groups of complete toric varieties see [BG1, Section 5].)

Let (P, v) be a column structure and λ ∈ k. We identify the vector v, representing the difference of two lattice points in P, with the degree 0 element (v, 0) ∈ gp(S_{P}) ⊂ k[gp(S_{P})]. Then the assignment

x ↦ (1 + λv)^{ht_{v}(x)}x.

gives rise to a graded k-algebra automorphism e_{v}^{λ} of k[P]. Observe that e_{v}^{λ} becomes an elementary matrix in the special case when P = Δ_{n−1}, after the identifications k[Δ_{n−1}] = k[X_{1}, …, X_{n}] and Γ_{k}(P) = GL_{n}(k). Accordingly e_{v}^{λ} is called an elementary automorphism.

The following alternative description of elementary automorphisms is essential in showing that all automorphisms in Pol(k) are tame (see Remark 4.2(c) below). We may assume dim P = d and gp(S_{P}) = ℤ^{d+1}. By a suitable integral unimodular change of coordinates we may further assume that v = (0, −1, 0, …, 0) and that P_{v} lies in the subspace ℝ^{d−1} (thus P is in the upper halfspace). Consider the standard unimodular simplex Δ_{d−1} (i. e. the one with vertices at the origin and the standard coordinate unit vectors). Clearly, P is contained in a parallel integral shift of cΔ_{d−1} for a sufficiently large natural number c. Then we have a graded k-embedding k[P] → k[cΔ_{d−1}], the latter ring being just the cth Veronese subring of the polynomial ring k[X_{1}, …, X_{d}]. Moreover, v = X_{1}/X_{2}. Now the automorphism of k[X_{1}, …, X_{d}] mapping X_{2} to X_{2} + λX_{1} and leaving all the other variables fixed induces an automorphism of k[cΔ_{d−1}] and the latter restricts to an automorphism of k[P], which is nothing but the elementary automorphism e_{v}^{λ} above.

As usual, 𝔸_{k}^{s} denotes the additive group of the s-dimensional affine space.

Proposition 3.1. Let v_{1}, …, v_{s} be pairwise different column vectors for P with the same base facet F = P_{v_{i}}, i = 1, …, s. The mapping

φ : 𝔸_{k}^{s} → Γ_{k}(P), (λ_{1}, …, λ_{s}) ↦ e_{v_{1}}^{λ_{1}} ∘ ⋯ ∘ e_{v_{s}}^{λ_{s}},

is an embedding of algebraic groups. In particular, e_{v_{i}}^{λ_{i}} and e_{v_{j}}^{λ_{j}} commute for all i, j ∈ {1, …, s} and (e_{v_{1}}^{λ_{1}} ∘ ⋯ ∘ e_{v_{s}}^{λ_{s}})^{−1} = e_{v_{1}}^{−λ_{1}} ∘ ⋯ ∘ e_{v_{s}}^{−λ_{s}}.

The image of the embedding φ given by Lemma 6.1 is denoted by 𝔸(F). Of course, 𝔸(F) may consist only of the identity map of k[P], namely if there is no column vector with base facet F.

Put n = dim(P)+1. The n-torus 𝕋_{n} = (k^{∗})^{n} acts naturally on k[P] by restriction of its action on k[gp(S_{P})] that is given by

(ξ_{1}, …, ξ_{n})(e_{i}) = ξ_{i}e_{i}, i ∈ [1, n].

Here e_{i} is the i-th element of a fixed basis of gp(S_{P}) = ℤ^{n}. This gives rise to an algebraic embedding 𝕋_{n} ⊂ Γ_{k}(P), whose image we denote by 𝕋_{k}(P). It consists precisely of those automorphisms of k[P] which multiply each monomial by a scalar from k^{∗}.

The (finite) automorphism group Σ(P) of the semigroup S_{P} is also a subgroup of Γ_{k}(P). It is exactly the group of automorphisms of P as a lattice polytope.

Next we recapitulate the main result of [BG1]. It should be viewed a polytopal generalization of the standard linear algebra fact that any invertible matrix over a field can be reduced to a diagonal matrix (generalized to toric automorphisms in the polytopal setting) using elementary transformations on columns (rows). Moreover, we have normal forms for such reductions reflecting the fact that the elementary transformations can be carried out in an increasing order of the column indices.

Theorem 3.2. Let P be a convex lattice n-polytope and k a field. Every element γ ∈ Γ_{k}(P) has a (not uniquely determined) presentation

γ = α_{1} ∘ α_{2} ∘ ⋯ ∘ α_{r} ∘ τ ∘ σ,

where σ ∈ Σ(P), τ ∈ 𝕋_{k}(P), and α_{i} ∈ 𝔸(F_{i}) such that the facets F_{i} are pairwise different and # L_{F_{i}} ≤ # L_{F_{i+1}}, i ∈ [1, r − 1].

We have dim Γ_{k}(P) = # Col(P) + n + 1 (the left hand side is the Krull dimension of the group scheme Γ_{k}(P)), and 𝕋_{k}(P) is a maximal torus in Γ_{k}(P), provided k is infinite.

A corollary of this result is the complete description of the graded automorphisms of all determinantal rings:

Corollary 3.3. Let k be any field, X an m × n matrix of indeterminates, and R = k[X]/I_{r+1}(X) the residue class ring of the polynomial ring k[X] in the entries of X modulo the ideal generated by the (r + 1)-minors of X, 1 ≤ r < min(m, n). Let G^{0} denote the image of GL_{m}(k) × GL_{n}(k) in GL_{mn}(k) under the mapping

GL_{m}(k) × GL_{n}(k) → Aut_{k}(R_{1})

given by

∀M ∈ R_{1} = ⨁_{i,j} kX_{ij} = M_{m×n}(k) (γ_{1}, γ_{2})(M) = γ_{1}Mγ_{2}^{−1}.

Then m ≠ n implies G^{0} = G. In case m = n we have G/G^{0} = ℤ_{2} where the other class is represented by the matrix transposition. Moreover, scheme theoretically G^{0} is the k-rational locus of the unity component of G.

This is Corollary 3.4 in [BG1]. Here we just sketch how it follows from Theorem 3.2. In the case r = 1 the determinantal ring is just the polytopal ring corresponding to the polytope Δ_{m−1} × Δ_{n−1}, that is the coordinate ring of the Segre embedding ℙ^{m−1} ×ℙ^{n−1} → ℙ^{mn−1} and Theorem 3.2 applies. For higher r we look at the singular locus of R which is exactly the coordinate ring of the locus of matrices with rank at most r − 1. Since the singular locus is invariant under the automorphisms, the induction process goes through.

Now we describe those results from [BG2] that are relevant in Section 4 below. At this point we need to require that the field k is algebraically closed. The case of arbitrary fields remains open.

Let P ⊂ ℝ^{n} be a lattice polytope of dimension n and F ⊂ P a face. Then there is a uniquely determined retraction

π_{F} : k[P] → k[F], π_{F} (x) = 0 for x ∈ L_{P} \F.

Retractions of this type will be called face retractions, and facet retractions if F is a facet. In the latter case we write codim(π_{F}) = 1.

Now suppose there are an affine subspace H ⊂ ℝ^{n} and a vector subspace W ⊂ ℝ^{n} with dim W + dim H = n, such that

L_{P} ⊂ ⋃_{x∈L_{P} ∩H}(x + W).

(Observe that dim(H ∩P) = dim H.) The triple (P, H, W) is called a lattice fibration of codimension c = dim W, whose base polytope is P ∩ H; its fibers are the maximal lattice subpolytopes of (x + W) ∩ P, x ∈ L_{P} ∩H (the fibers may have smaller dimension than W). P itself serves as a total polytope of the fibration. If W = ℝw is a line, then we call the fibration segmental and write (P, H, w) for it. Note that

w  H

Figure 3. A lattice segmental fibration

the column structures give rise to lattice segmental fibrations in a natural way.

For a lattice fibration (P, H, W) let L ⊂ ℤ^{n} denote the subgroup spanned by L_{P}, and let H_{0} be the translate of H through the origin. Then one has the direct sum decomposition

L = (L ∩ W) ⊕ (L ∩ H_{0}).

Equivalently,

gp(S_{P}) = L ⊕ ℤ = (gp(S_{P}) ∩ W_{1}) ⊕ gp(S_{P ∩H_{1}}).

where W_{1} is the image of W under the embedding ℝ^{n} → ℝ^{n+1}, w ↦ (w, 0), and H_{1} is the vector subspace of ℝ^{n+1} generated by all the vectors (h, 1), h ∈ H.

For a fibration (P, H, W) one has the naturally associated retraction

ρ_{(P,H,W)} : k[P] → k[P ∩ H];

it maps L_{P} to L_{P ∩H} so that fibers are contracted to their intersection points with the base polytope P ∩ H.

The following is a composition of Theorems 2.2 and 8.1 in [BG2]:

Theorem 3.4. (a) If A is a retract of a polytopal algebra k[P] and dim A ≤ 2 then

A is polytopal itself, i. e. A is of type either k or k[t] or k[cΔ_{1}] for some c ∈ ℕ.

(b) If dim P = 2 and f : k[P] → k[P] is any idempotent endomorphism of codimension 1 (f^{2} = f and dim Im(f) = 2) then either

(i) P admits a lattice segmental fibration (P, H, w) and γ ∘f ∘γ^{−1} = ι∘ρ_{(P,H,w)}

for some γ ∈ Γ_{k}(P) and an embedding ι : k[P ∩ H] → k[P], or

(ii) γ ∘ f ∘ γ^{−1} = ι ∘ π_{E} for some γ ∈ Γ_{k}(P), an edge E ⊂ P and an embedding

ι : k[E] → k[P].

Remark 3.5. The conjectures (A) and (B) in [BG2] say that both statements (a) and (b) of Theorem 3.4 generalize to arbitrary dimensions. They should be thought of as analogues of the fact that idempotent matrices are conjugate to ‘subunit’ matrices (having diagonal entries 0 and 1 and entries 0 everywhere else). That one has to restrict oneself to codimension 1 idempotent endomorphisms in the higher analogue of (b) is explained by explicit examples in [BG2, Section 5].

The splitting embeddings ι will be described in the next section.

4. Tame homomorphisms

Assume we are given two lattice polytopes P, Q ⊂ ℝ^{d} and a homomorphism f : k[P] → k[Q] in Pol(k). Under certain conditions there are several standard ways to derive new homomorphisms from it.

First assume we are given a subpolytope P′ ⊂ P and a polytope Q′ ⊂ ℝ^{n}, d ≤ n, such that f(k[P′]) ⊂ k[Q′]. Then f gives rise to a homomorphism f′ : k[P′] → k[Q′] in a natural way. (Notice that we may have Q ⊂ Q′.) Also if P ≈ P̃ and Q ≈ Q̃ are lattice polytope isomorphisms, then f induces a homomorphism f̃ : k[P̃] → k[Q̃]. We call these types of formation of new homomorphisms polytope changes.

Now consider the situation when Ker(f) ∩ S_{P} = ∅. Then f extends uniquely to a homomorphism f̄ : k[S̄_{P}] → k[S̄_{Q}] of the normalizations. Here S̄_{P} = {x ∈ gp(S_{P}) | x^{m} ∈ S_{P} for some m ∈ ℕ} and similarly for S̄_{Q}. (It is well known that k[S̄] = \overline{k[S]} for any affine semigroup S ⊂ ℤ^{d} where on the right hand side we mean the normalization of the domain k[S], [BH, Ch. 6].) This extension is given by

f̄(x) = \frac{f(y)}{f(z)}, x = \frac{y}{z} , x ∈ S̄_{P} , y ∈ S_{P} , z ∈ S_{Q}.

For any natural number c the subalgebra of k[S̄_{P}] generated by the homogeneous component of degree c is naturally isomorphic to the polytopal algebra k[cP], and similarly for k[S̄_{Q}]. Therefore, the restriction of f̄ gives rise to a homomorphism f^{(c)} : k[cP] → k[cQ]. We call the homomorphisms f^{(c)} homothetic blow-ups of f. (Note that k[cP] is often a proper overring of the cth Veronese subalgebra of k[P].)

One more process of deriving new homomorphisms is as follows. Assume that homomorphisms f, g : k[P] → k[Q] are given such that

∀x ∈ L_{P} N(f(x)) + N(g(x)) ⊂ Q,

where N(−) denotes the Newton polytope and + is the Minkowski sum in ℝ^{d}. Then we have z^{−1}f(x)g(x) ∈ k[Q] where z = (0, …, 0, 1) ∈ S_{Q}. Clearly, the assignment

∀x ∈ L_{P} x ↦ z^{−1}f(x)g(x)

extends to a Pol(k)-homomorphism k[P] → k[Q], which we denote by f ⋆ g. We call this process Minkowski sum of homomorphisms.

All the three mentioned recipes have a common feature: the new homomorphisms are defined on polytopal algebras of dimension at most the dimension of the sources of the old homomorphisms. As a result we are not able to really create a nontrivial class of homomorphisms using only these three procedures. This possibility is provided by the fourth (and last in our list) process.

Suppose P is a pyramid with vertex v and basis P_{0} such that L_{P} = {v} ∪ L_{P_{0}}, that is P = join(v, P_{0}) in the terminology of [BG2, BG3]. Then k[P] is a polynomial extension k[P_{0}][v]. In particular, if f_{0} : k[P_{0}] → k[Q] is an arbitrary homomorphism and q ∈ k[Q] is any element, then f_{0} extends to a homomorphism f : k[P] → k[Q] with f(v) = q. We call f a free extension of f_{0}.

Conjecture 4.1. Any homomorphism in Pol(k) is obtained by a sequence of taking free extensions, Minkowski sums, homothetic blow-ups, polytope changes and compositions, starting from the identity mapping k → k. Moreover, there are normal forms of such sequences for idempotent endomorphisms.

Observe that for general homomorphisms we do not mean that the constructions mentioned in the conjecture are to be applied in certain order so that we get normal forms: we may have to repeat a procedure of the same type at different steps. However, the results mentioned in Section 3 and Theorem 4.3 below show that for special classes of homomorphisms such normal forms are possible.

We could call the homomorphisms obtained in the way described by Conjecture 4.1 just tame. Then we have the tame subcategory Pol(k)_{tame} (with the same objects), and the conjecture asserts that actually Pol(k)_{tame} = Pol(k).

Remark 4.2. (a) The correctness of Conjecture 4.1 may depend on whether or not k is algebraically closed. For instance, some of the arguments in [BG2] only go through for algebraically closed fields.

(b) The current notion of tameness is weaker then the one for retractions and surjections in [BG2]. This follows from Example 5.2 and 5.3 of [BG2] in conjunction with the observation that all the explicitly constructed retractions in [BG2] are tame in the new sense.

(c) Theorems 3.2 and 3.4 can be viewed as substantial refinements of the conjecture above for the corresponding classes of homomorphisms. Observe that the tameness of elementary automorphisms follows from their alternative description in Section 3. We also need the tameness of the following classes of homomorphisms: automorphisms that map monomials to monomials, retractions of the type ρ_{(P,H,w)} and π_{F} and the splitting embeddings ι as in Theorem 3.4. This follows from Theorem 4.3 and Corollary 4.4 below.

The next result shows that certain basic classes of morphisms in Pol(k) are tame.

Theorem 4.3. Let k be a field (not necessarily algebraically closed). Then

(a) any homomorphism from k[cΔ_{n}], c, n ∈ ℕ, is tame,

(d) if ι : k[cΔ_{n}] → k[P] (c ∈ ℕ) splits either ρ_{(P,H,W)} for some lattice fibration (P, H, W) or π_{E} for some face E ⊂ P then there is a normal form for representing ι in terms of certain basic tame homomorphisms.

Corollary 4.4. For every field k the homomorphisms respecting monomial structures are tame, and we have the inclusion Vect_{ℕ}(k) ⊂ Pol_{tame}(k), where Vect_{ℕ}(k) is the full subcategory of Pol(k) spanned by the objects of the type k[cΔ_{n}], c, n ∈ ℕ.

Proof. We only need to consider homomorphisms respecting monomial structures.

Assume f : k[P] → k[Q] is a homomorphism respecting the monomial structures and such that Ker(f) ∩ S_{P} = ∅. By a polytope change we can assume P ⊂ cΔ_{n} for a sufficiently big natural number c, where n = dim P and Δ_{n} is taken in the lattice ℤL_{P}. In this situation there is a bigger lattice polytope Q′ ⊃ Q and a unique homomorphism g : k[cΔ_{n}] → k[Q′] for which g|_{L_{P}} = f|_{L_{P}}. By Theorem 4.3 f is tame.

Consider the situation when the ideal I = (Ker(f) ∩ S_{P})k[P] is a nonzero prime monomial ideal and there is a face P_{0} ⊂ P such that Ker(f) ∩ L_{P_{0}} = ∅ and f factors through the face projection π : k[P] → k[P_{0}], that is π(x) = x for x ∈ L_{P_{0}} and π(x) = 0 for x ∈ L_{P} \ L_{P_{0}}. In view of the previous case we are done once the tameness of face projections has been established.

Any face projection is a composite of facet projections. Therefore we can assume that P_{0} is a facet of P. Let (ℝP)_{+} ⊂ ℝP denote the halfspace that is bounded by the affine hull of P_{0} and contains P. There exists a unimodular (with respect to ℤ L_{P}) lattice simplex Δ ⊂ (ℝP)_{+} such that dim Δ = dim P, the affine hull of P_{0} intersects Δ in one of its facets and P ⊂ cΔ for some c ∈ ℕ. But then π is a restriction of the corresponding facet projection of k[cΔ], the latter being a homothetic blow-up of the corresponding facet projection of the polynomial ring k[Δ_{n}] – obviously a tame homomorphism.

Proof of Theorem 4.3. We will use the notation {x_{0}, …, x_{n}} = L_{Δ_{n}}. Any lattice point x ∈ cΔ_{n} has a unique representation x = a_{0}x_{0} + ⋯ + a_{n}x_{n} where the a_{i} are nonnegative integer numbers satisfying the condition a_{0} +⋯+a_{n} = c. The numbers a_{i} are the barycentric coordinates of x in the x_{i}.

Let f : k[cΔ_{n}] → k[P] be any homomorphism.

First consider the case when one of the points from L_{cΔ_{n}} is mapped to 0. In this situation f is a composite of facet projections and a homomorphism from k[cΔ_{m}] with m < n. As observed in the proof of Corollary 4.4 facet projections are tame. Therefore we can assume that none of the x_{i} is mapped to 0. By a polytope change we can also assume L_{P} ⊂ {X_{1}^{a_{1}} ⋯ X_{r}^{a_{r}}Y^{b}Z | a_{i}, b ≥ 0}, r = dim P − 1.

Consider the polynomials φ_{x} = Z^{−1}f(x) ∈ k[X_{1}, …, X_{r}, Y], x ∈ L_{cΔ_{n}}. Then the φ_{x} are subject to the same binomial relations as the x. One the other hand the multiplicative semigroup k[X_{1}, …, X_{r}, Y] \ {0}/k^{∗} is a free commutative semigroup and, as such, is an inductive limit of free commutative semigroups of finite rank. Therefore, by Lemma 4.5 below there exist polynomials ψ, η_{i} ∈ k[X_{1}, …, X_{r}, Y], i ∈ [0, n], and scalars t_{x} ∈ k^{∗}, x ∈ L_{cΔ_{n}}, such that φ_{x} = t_{x}ψη_{0}^{a_{0}} ⋯ η_{n}^{a_{n}} where the a_{i} are the barycentric coordinates of x. Clearly, t_{x} are subject to the same binomial relations as the x ∈ L_{cΔ_{n}}. Therefore, after the normalizations ψ ↦ t_{x_{0}} and η_{i} ↦ t_{x_{0}}^{−1}t_{x_{i}}η_{i} (i ∈ [0, n]) we get φ_{x} = ψη_{0}^{a_{0}} ⋯ η_{n}^{a_{n}}. But the latter equality can be read as follows: f is obtained by a polytope change applied to Ψ ⋆ Θ^{(c)}, where

(i) Ψ : k[cΔ_{n}] → k[Q], Ψ(x) = ψZ, x ∈ L_{cΔ_{n}}, (ii) Θ : k[Δ_{n}] → k[Q], Θ(x_{i}) = η_{i}Z, i ∈ [0, n], and Q is a sufficiently large lattice polytope so that it contains all the relevant lattice polytopes. Now Ψ is tame because it can be represented as the composite map

k[cΔ_{n}] ⟶^{L_{cΔn}→t} k[t] ⟶^{t↦ψZ} k[Q]

(the first map is the cth homothetic blow-up of k[Δ_{n}] → k[t], x_{i} ↦ t for all i ∈ [0, c]) and Θ is just a free extension of the identity embedding k → k[Q].

(b) First consider the case of lattice segmental fibrations.

Consider the rectangular prism Π = (cΔ_{n}) × (mΔ_{1}). By a polytope change (assuming m is sufficiently large) we can assume that P ⊂ Π so that H is parallel to cΔ_{n}: The lattice point (x, b) ∈ Π will be identified with the monomial

mΔ_{1}  cΔ_{n} Π  H

Figure 4

X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}}Y^{b}Z whenever we view it as a monomial in k[Π], where the a_{i} are the corresponding barycentric coordinates of x (see the proof of (a) above). (In other words, the monomial X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}} is identified with the point x = (c − a_{1} − ⋯ − a_{n})x_{0} + a_{1}x_{1} + ⋯ + a_{n}x_{n} ∈ cΔ_{n}.)

Assume A : k[cΔ_{n}] → k[m′Δ_{1}] is a homomorphism of the type A(x_{i}) = aZ, i ∈ [0, c] for some a ∈ k[Y] satisfying the condition a(1) = 1. Consider any homomorphism B : k[Δ_{n}] → k[Π′], Π′ = Δ_{n} × (m′Δ_{1}) that splits the projection ρ′ : k[Π′] → k[Δ_{n}], ρ′(ZY^{b}) = Z and ρ′(X_{i}Y^{b}Z) = X_{i}Z for i ∈ [1, n], b ∈ [0, m′]. The description of such homomorphisms is clear – they are exactly the homomorphisms B for which

B(x_{0}) = B_{0} ∈ Z + (Y − 1)(Zk[Y] + X_{1}Zk[Y] + ⋯ + X_{n}Zk[Y]),

B(x_{i}) = B_{i} ∈ X_{i}Z + (Y − 1)(Zk[Y] + X_{1}Zk[Y] + ⋯ + X_{n}Zk[Y]), i ∈ [1, n],

deg_{Y} B_{i} ≤ m′ for all i ∈ [0, n].

Clearly, all such B are tame.

In case m ≥ max{m′ + c deg_{Y} B_{i}}_{i=0}^{n} we have the homomorphism A ⋆ B^{(c)} : k[cΔ_{n}] → k[Π] which obviously splits the projection ρ : k[Π] → k[cΔ_{n}] defined by

ρ(X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}}Y^{b}Z) = X_{1}^{a_{1}} ⋯ X_{n}^{a_{n}}Z.

Assume ι splits ρ_{(P,H,w)}. The standard facts on Newton polytopes imply the following: the polynomials ψ and η_{i}, mentioned in the proof of (a), that correspond to ι, satisfy the conditions: ψ ∈ k[Y] and η_{i} ∈ k[Y] + X_{1}k[Y] + ⋯ + X_{n}k[Y]. Is is also clear that upon evaluation at Y = 1 we get ψ(1), η_{i}(X_{1}, …, X_{n}, 1) ∈ k^{∗}, i ∈ [0, n]. Therefore, after the normalizations ψ ↦ ψ^{−1}(1)ψ, η_{i} ↦ η_{i}(X_{1}, … X_{n}, 1)^{−1}η_{i} we conclude that ι is obtained by a polytope change applied to A ⋆ B^{(c)} as above (with respect to a = ψ, B_{0} = η_{i}Z, i ∈ [0, n].

For a lattice fibration (P, H, W) of higher codimension similar arguments show that ι is obtained by a polytope change applied to A ⋆ B^{(c)}, where

B is a splitting of a projection of the type ρ_{(P′,H′,W′)} such that the base polytope P′ ∩ H′ is a unit simplex and A is a homomorphism defined by a single polynomial whose Newton polytope is parallel to W′.

We skip the details for splittings of face projections and only remark that similar arguments based on Newton polytopes imply the following. All such splittings are obtained by polytope changes applied to A ⋆ B^{(c)} where B is a splitting of a face projection onto a polynomial ring and A is again defined by a single polynomial.

Lemma 4.5. Assume we are given an integral affine mapping α : cΔ_{n} → ℝ_{+}^{d} for some natural numbers c, n and d. Then there exists an element v ∈ ℤ_{+}^{d} and a integral affine mapping β : Δ_{n} → ℝ_{+}^{d} such that α = v + cβ.

Proof. Assume α(cx_{i}) = (a_{i1}, …, a_{id}), i ∈ [0, n] (the x_{i} as above). Consider the vector

v = ( min{a_{i1}}_{i=0}^{n}, …, min{a_{id}}_{i=0}^{n}).

It suffices to show that all the vectors α(cx_{i})−v are cth multiples of integral vectors. But for any index l ∈ [1, d] the lth component of either α(cx_{i}) − v or α(cx_{j}) − v for some j ≠ i is zero. In the first case there is nothing to prove and in the second case the desired divisibility follows from the fact that α(cx_{i}) − α(cx_{j}) = (α(cx_{i}) − v) − (α(cx_{j}) − v) is a cth multiple of an integral vector (because α is integral affine).

Remark 4.6. Theorems 3.2, 3.4 and 4.3 provide a possibility for computing the Γ_{k}(P)-variety of idempotent endomorphisms Idemp(k[P]) for a polygon P and an algebraically closed field k (Γ_{k}(P) acts by conjugation): the orbits of codimension 1 idempotent endomorphisms are naturally associated to the segmental fibration structures and edges of P, and all codimension 2 idempotent endomorphisms pass through k[t].

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Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

  E-mail address: Winfried.Bruns@mathematik.uni-osnabrueck.de

A. Razmadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia

  E-mail address: gubel@rmi.acnet.ge