Projective modules over polyhedral semirings

I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group.


Introduction
This paper begins the geometric study of module theory over a class of idempotent semirings that are of basic importance in skeletal and tropical geometry: those generated freely by a monoid of monomials. The projective modules over these semirings can be described in terms of a simpler category of partially ordered modules over the underlying monoid.
In the related case of the semiring of convex, piecewise-affine functions on a polyhedron, this latter category can itself be realised in terms of convex geometric data. This last intepretation of the classification suggests explicit descent criteria for modules over such 'polyhedral' semirings.
The only existing work of which I am aware in this direction is [IJK12], which addresses finite projective modules over the real semifield R ∨ = R ⊔ {−∞} (therein denoted T). As far as I know, the classification scheme given in the present paper is new even in that case.

Results
The moral of the paper will be that over semirings generated freely by monomials, projective modules are defined by monomial inequalities.
As a warm-up, we have the following result for the case of the simplest possible semiring, the Boolean semifield B = {−∞, 0}: Theorem (3.9). Let µ be a finite B-module. The following are equivalent: iii) µ is flat; iv) µ is free on a poset; v) µ has unique irredundant primitive decompositions.
A careful extension of the same analysis helps us to understand projectivity of modules over the free or fractional ideal semiring B[A; A + ] associated to a pair of F 1 -algebras (monoids with zero) (A; A + ). We carry out this analysis in §4. In the case that A is a domain, one obtains a complete classification in terms of partially ordered (A; A + )-modules. I do not reproduce the classification here, but refer the reader to corollary 4.13.
The geometric part §5 of the paper concerns modules over the semifield H ∨ = H ⊔ {−∞} associated to a totally ordered group H ⊆ R.
Theorem (5.5). Let H ⊆ R be an additive subgroup. The category of finite projective H ∨modules is anti-equivalent to a category of extended H-integral general linear weight polyhedra and convex, piecewise-affine maps whose linear parts are fundamental weights. 1 By duality for finite projectives these categories are also equivalent -I have just found it more natural to phrase the result in the form of a duality.
Taking H = 0 in this result recovers a more geometric formulation of the (only nontrivial) equivalence i)⇔iv) of theorem 3.9: finite projective B-modules are dual to certain convex cones in the Coxeter complex of GL n .
We cannot immediately extend our classification scheme to the more geometric setting of the semiring of convex, piecewise-affine functions on a polyhedron, since the latter is not actually free on the group of affine functions. Rather, it is the normalisation of the free semiring [Mac15b]. Making this replacement helps us to get a geometric classification: Theorem (5.12). Let ∆ be an H-rational polytope. The category of finite projective CPA(∆, Z)modules is anti-equivalent to a category of convex families of GL weight polyhedra over ∆.
For a more precise description of the latter category and the duality, see §5.2.

Acknowledgement
I'd like to thank Jeff Giansiracusa, a conversation with whom gave me the idea for this paper. 1 This result is closely related to the more specific theorem 1.5 of [IJK12], which states that every projective R ∨ -submodule is isomorphic to a submodule of R n ∨ closed under co-ordinate-wise minimum, as well as maximum. By [JK14,Thm. B], such sets are automatically convex polyhedra, and it is an elementary matter to observe what kinds of supporting half-spaces are allowed. Moreover, [IJK12] even provides criteria to determine when a given submodule of R n ∨ is projective.

Points
It will be convenient to switch between the categories of pointed and unpointed sets. We do this using the strongly monoidal functor − ⊗ F 1 : (Set, ×) → (Set * , ⊗ F 1 = ∧) that adjoins a disjoint base point. This exhibits the category Set * of pointed sets as the universal way to attach a zero object to Set. This functor extends to the pointed versions of all essentially algebraic theories in Set. In particular, a monoid Q can be replaced by a monoid with zero F 1 [Q], and a partially ordered set by a pointed partially ordered set (the point is the minimum). These constructions have the same universal property: POSet * (resp. Mod F 1 [Q] ) is the universal pointed extension of POSet (resp. Mod Q ).
We will adhere to the convention of writing monoids without zero (mainly appearing in §5) and idempotent semirings additively, and pure F 1 -algebras ( §4) multiplicatively. If (Q; Q + ) is a pair of unpointed monoids, we will write F 1 [Q] as a shorthand for the associated F 1 -algebra pair. When H ⊆ R is an additive subgroup, F 1 [H] denotes the F 1 -algebra pair that in [Mac15a] was (more suggestively) labelled F 1 ((t −H )).

Projectives
Let A be a commutative monoid (with or without zero) or semiring. (In fact, the following definition is standard for any commutative algebraic monad in the sense of [Dur07].) ii) any surjection F ։ P has a section;

Definition. An
iii) P is a retract of a free module.
The situation when A is a monoid -with or without zero -is very simple. A free Amodule splits uniquely as a sum of cyclic factors where the coproduct ⊕ is disjoint union in the unpointed case and wedge sum when 0 ∈ A). The factor of a non-zero element x ∈ M is the index of the cyclic submodule to which it belongs. This splitting, and in particular, its index set, is natural in M. In other words, every matrix over A is a product of a permutation and a diagonal matrix. It follows:

Proposition. Let A be a commutative monoid. Every projective A-module splits uniquely as a coproduct of cyclic submodules, each isomorphic to the image of an idempotent in A.
Let us call a monoid (resp. monoid with zero A) a domain if it is cancellative (resp. A \0 is a cancellative submonoid).

Flatness and duality over semirings
We now restrict attention to the case that A is a semiring, whence Mod A is semiadditive (i.e. finite coproducts are products).

Definitions. Let
A be a (not necessarily idempotent) semiring, M an A-module. M is said to be: The linear dual of a module is the object a module is said to be reflexive if M→(M ∨ ) ∨ . This is strictly weaker than being dualisable, for while we always have the evaluation map M ⊗ A M ∨ → A, there may be no 'identity matrix' A → M ∨ ⊗ A M.
2.5 Aside. Flatness, at least with this definition, does not make much sense when A is a monoid (more generally, when Mod A is not semiadditive), because the class of flat modules is not closed under finite coproducts. In particular, free modules on more than one generator are never flat. Some authors [Ste71] have studied variants of the notion of flatness adapted to A-modules (or 'A-acts') in the unpointed case.
The relations between these properties are, much as in the case of commutative rings, as follows: • Any coproduct, filtered colimit, or retract of a flat module is flat. In particular, projective modules are flat.
• Any finite coproduct or retract of a dualisable module is dualisable. In particular, finitely generated projective modules are dualisable.
• Dualisable modules are flat, since in this case − ⊗ A M ∨ is left adjoint to − ⊗ A M.
The following fact is no doubt well-known -indeed, the proof for the case of rings [Sta,00HK] carries through with only minor modifications.

Proposition (Equational criterion for flatness).
Let M be a flat module, v : A n → M a homomorphism from a finite free module. Suppose that v satisfies a relation f ∼ g: Then v factors through a finite free module in which f ∼ g.
Proof. Dualising the relation, we obtain an equaliser sequence and hence by flatness of M, an equaliser It follows that v ∈ M n actually lies in K ⊗ M. By writing v as a sum of decomposable elements, one obtains morphisms A m → K ⊆ A n and A m → M that induce the composite on the tensor product. The transpose of A m → A n is the desired factorisation.

B-modules
For the rest of the document, all semirings will be additively idempotent. I continue with the convention of [Mac14] in denoting idempotent semirings and their modules by lowercase Greek letters, and their operations by ∨ ('max') and +. Correspondingly, the closed monoidal structure on the category Mod α of modules over an idempotent semiring α is denoted ⊕ α (i.e. this does not denote the coproduct of modules). As in the introduction, we will denote by B the Boolean semifield, the initial object in the category of idempotent semirings. It will be illuminating to understand B as a monad via its free functor B : Set → Set that takes a set S to the set BS of its finite subsets, with operations 1 Set given by singleton and union, respectively. This monad is the unique algebraic (i.e. commuting with filtered colimits) extension of the monad of power set and union on the category of finite sets. A finite, free B-module is nothing more than the power set of a finite set. The same statements remain valid, mutatis mutandi, with Set replaced by the category Set * of pointed sets. The free B-module functor factorises We will use this in §5 to apply the results of §4, couched in the setting of monoids with zero, in the unpointed regime.
More concretely, a B-module (µ, ∨) is nothing more than a join semilattice, that is, a partially ordered set with finite joins, and a B-linear morphism is a right exact monotone map. In [Mac14], these were called 'spans'.
The following are equivalent for a B-module µ: i) µ is finite as an object of Mod B ; ii) µ is compact as an object of Mod B ; iii) µ is a finite set.

Free module on a poset
Let us denote by POSet the category of partially ordered sets and monotone maps. Since every B-module is in particular a poset, we have a faithful, conservative functor In fact, this functor is monadic; its left adjoint takes a poset (S, ≤) to the poset B(S, ≤) of finitely generated lower subsets. Since any union of lower subsets is lower, union makes this poset a B-submodule of the power set P (S). It is called the free B-module on (S, ≤). By general principles, B : POSet → Mod B is strongly monoidal. It also respects Hom sets in the following way: if S 1 , S 2 are posets, then the assignment defines a monotone map S op 1 × S 2 → Hom(BS 1 , BS 2 ); if S 1 is finite, then this map extends to an isomorphism B(S op 1 × S 2 ) ∼ = Hom B (BS 1 , BS 2 ) so that maps BS 1 → BS 2 are finite monotone correspondences from S 1 to S 2 .
The same logic holds over the category POSet * of pointed posets, that is, posets equipped with a distinguished minimum and monotone maps that preserve this minimum.

Flatness and projectivity
The free module on a poset comes equipped with a natural set of generators To understand projectivity, we will need to know when this map admits a section. A natural candidate for a splitting is the (right ind-adjoint) inclusion B(S, ≤) → P (S). As remarked above, it is automatically a B-module homomorphism. This map factors through BS -thus defining an honest adjoint -if and only if (S, ≤) satisfies the condition • any principal lower set S ≤X is finite.
lower finite This has the flavour of a 'finite presentation' condition for posets: it is equivalent that the quotient BS → B(S, ≤) be defined by finitely many relations. It is satisfied, in particular, whenever S is finite. It turns out that this is the only way to split this epimorphism: Proof. Let σ : B(S 2 , ≤) → B(S 1 , ≤) be any section, S 0 ∈ B(S 2 , ≤). Then pσS 0 ⊆ S 0 generates S 0 as a lower set. In particular, for any X ∈ S 2 , X ∈ pσX , i.e. p −1 X ∈ σX . If σ is monotone, then this shows that σp is increasing on B(S 1 , ≤). Thus Y ≤ σX if and only if pY ≤ X , that is, σ is right adjoint to p.
The proof of the lemma 3.1 on monotone sections depends on the fact that a lower set has a unique minimal set of generators. We will use variations of the latter fact, and the lemma, repeatedly in the sequel. 3.4 Example (A flat module with no primitives). The converse to corollary 3.3 is false: while every flat module is a filtered colimit of modules free on a poset, there is no requirement that the transition maps preserve these posets. For example, the set µ of compact open subsets of Z 2 has an expression as a colimit with transition maps given by inverse image along Z/2 k Z ։ Z/2 k−1 Z, and is therefore flat by Lazard's theorem 2.9; however, it has no primitive elements (cf. §3.2 below) and so cannot be free on a poset.

Primitives
To achieve our goal of classifying projective modules, we must still characterise which Bmodules appear through this construction.

Lemma.
Let X be an element of a B-module µ. The following are equivalent: 3.6 Definitions. An element X of a B-module µ satisfying the equivalent conditions of lemma 3.5 is said to be ∨-primitive, or simply primitive if no confusion can arise. (Note that −∞ is never primitive.) The set of primitive elements of µ is denoted Primµ. It is usually not functorial in either direction.
We have tautological maps where Primµ ⊂ µ carries the induced partial order. If these modules surject onto µ, we say that µ has primitive decompositions.
A primitive decomposition of an element X ∈ µ is a lift to B(Primµ). Such a decomposition is said to be irredundant if it is minimal in its fibre of B(Primµ) ։ B(Primµ, ≤). The set of possible primitive decompositions (resp. irredundant decompositions) of elements of µ is precisely B(Primµ) (resp. B(Primµ, ≤)).
In equations, a primitive decomposition of X ∈ µ is an expression X = i∈I X i with X i primitive, and it is irredundant if there are no order relations among different X i .

Aside.
Beware that an irredundant decomposition in µ is not necessarily minimal: one may perfectly well have a relation between primitives in µ, but unless X 3 ≤ X 2 or X 3 ≤ X 1 , both decompositions will be irredundant. Of course, in light of theorem 3.9, this cannot happen for projective modules.
Frees A module is free if and only if B(Primµ) → µ is an isomorphism; that is, if it has unique primitive decompositions.

Theorem.
Let µ be a B-module. The following are equivalent: ii) µ has unique primitive decompositions.
Projectives More generally, a module is free on a poset if and only if it has unique irredundant primitive decompositions.
3.9 Theorem. Let µ be a B-module. The following are equivalent: ii) µ is free on a lower finite poset; iii) µ has unique irredundant primitive decompositions, and the set of primitives is lower finite.
Proof. To complete the proof of this theorem, we must show that projective modules have unique irredundant primitive decompositions.
3.10 Lemma. Any submodule of a B-module having primitive decompositions itself has primitive decompositions.
3.11 Lemma (Sections over primitives). Let S ⊆ µ be a subset, σ any section of If X ∈ µ is primitive, then σX = S ≤X .
Proof. Let σ be a section, and let X ∈ µ. Since Y ∈σX Y = X , σX must certainly be contained in S ≤X . If X is primitive, then in fact X ∈ σX and so σX = S ≤X .
A projective module µ is a submodule of a free module, and hence by lemma 3.10, B(Primµ, ≤) → µ is surjective. By projectivity, it admits a B-linear section. Applying the lemma 3.11 on sections over primitives to S = Primµ shows that it is an isomorphism.

Modules over free semirings
Here we generalise and discuss modules over semirings that are free on a pair (A; A + ) of monoids with A an A + -algebra. Our convention in this section will be that monoids are multiplicative with zero; the category of such pairs is denoted Pair F 1 . Following the remarks of §2.1, the results translate straightforwardly into the unpointed regime.
Usually, A will be a localisation of A + ; correspondingly, we will typically consider idempotent semirings α that are a localisation of their semiring of integers α • := α ≤0 . However, these assumptions are not actually necessary for the conclusions of § §4.1-4.4.
We will later need to assume that A + is integrally closed in A ( §4.5) and that α is normal in the sense of [Mac15b] ( §5.2).
The structure of this section is as follows. The first two subsections and §4.4 are parallel to the structure of §3. In §4.2 we derive formal criteria for the free module on a partially ordered projective A-module to be projective, and in §4.4 we show that conversely, every projective module over the free semiring B[A] on A comes from a partially ordered A-module.
The technical heart of the paper lies in §4.3, where we discuss a method of presenting partial orders via quivers. This will help us to unpack the meanings of the conditions appearing in §4.2. In particular, we define the notion of non-degeneracy of partially ordered free modules, which has crucial finiteness implications.
When A is a domain, we obtain a complete classification in §4.5. The remaining poings are, first, that the partially ordered A-module behind a projective B[A]-module is itself projective, and second, that non-degeneracy is a necessary condition for lower finiteness.

Free semirings and modules
Let (A; A + ) be an F 1 -algebra pair, and define, as in [Mac14], the fractional ideal semiring We also write simply the 'universal valuation' of (A; A + ). This valuation is injective if and only if A + is sharp (i.e. has no invertible elements other than 1). In general, the image of A + in the monoid semiring is its universal sharp quotient. For the purposes of studying B[A]-modules, then, we can and will always assume that A + is sharp.
Note that in contrast to the situation for commutative rings, the fractional ideal functor has a monadic right adjoint forgetful functor which forgets the ∨ operation. In this language, log is the unit of the adjunction.
More generally, if M is an A-module, one can form the disc (or fractional submodule) set with the evident forgetful functor. It follows that B(−; A + ) preserves colimits, and hence the classes of free, projective, and (by the Lazard theorem 2.9) flat modules.

Partially ordered modules
If A ∈ Pair F 1 , then any A-module carries a natural pointed A + -divisibility partial order which is non-degenerate if and only if A + is sharp. An A-linear map is automatically monotone with respect to this order.
The divisibility order on a smash product is the smash product of the orders on the factors. It therefore makes (A, ≤ A + ) into an algebra object in the closed monoidal category (POSet * , ⊗ F 1 ) of pointed posets (cf. §2.1), and (M, ≤ A + ) into an (A, ≤ A + )-module. This defines fully faithful functors and, for each (A; A + ), In the spirit of [GG13], the free B[A]-module on an A-module M can be presented via this partial order which shows that, in particular, B(M) has unique irredundant primitive decompositions. By corollary 3.3, it is moreover flat as a B-module.

Definition.
A partially ordered module, or po-module, over a F 1 -algebra pair (A; A + ) is an (A, ≤ A + )-module object in (POSet * , ⊗ F 1 ). That is, it is an A-module together with a partial order such that

The category of partially ordered A-modules is abbreviated POMod
It is closed monoidal and compactly generated.
By analogy with the case of B-modules, we can define a 'free' B[A]-module on any partially ordered A-module: here a lower submodule is 'finitely generated' if it is the lower hull of a finite A + -submodule; and hence a monadic adjunction Proof. Indeed, we have already observed that B(−; A + ) preserves projectivity, and so assuming M projective, by the monotone section lemma 3.1 we must check that the canonical generator has a right adjoint. Condition i) is enough to show that the right adjoint exists as a map of posets, and ii) is the condition that it be A-linear.
We will expand upon the meanings of the other two conditions in §4.3.

Partial orders from quivers
In the case of projective A-modules -or more generally, direct sums of cyclic modules -we can give a fairly explicit method for defining partial orders. We will use this method as an auxiliary tool to obtain a good classification theorem 4.13. Let M ∼ = i∈I A i be such a module, and let Q be a (possibly infinite) quiver with vertex set I. Suppose that we are given the structure of a representation of Q on M -that is, for each edge in Q from i to j, a map A i → A j . If we choose generators x i ∈ A i for the cyclic factors of M -since A + is sharp, this is the same as choosing a cyclic A + -structure for each factor -then such can be represented by attaching an element of A ։ Hom A (A i , A j ) to each edge of Q.
With a choice of generators, we can exchange a quiver representation for a presentation of a partial order on M. In particular, this presentation is finite if and only if Q has finitely many edges. Conversely, any presentation R ⊂ M 2 of this form -with the right-hand term always the chosen generator of its cyclic factor -can be obtained from a quiver representation, for which R is the set of edges via the natural projection M \ 0 → I. More invariantly, the action of the path category Path(Q) of Q defines an A-invariant pre-order on M. It is the transitive and A-invariant closure of the relation defined by the set in (1). The union of this pre-order with the A + -divisibility order is an A-module pre-order where Path A + denotes the A + -linear extension of the path category. The latter is degenerate if there is a cycle γ : i → i in Path(Q) such that 1 ≤ A + γ ∈ End(A i ); otherwise, it defines the structure of a po-A-module on M. We call it the partial order presented by Q M.
Lower saturation Partial orders on M can be equivalently described by producing, for each x ∈ M, the lower hull M ≤x . For lower saturated partial orders (cf. proposition 4.2, ii) it is enough to define the lower hull for x i the generators of the cyclic factors of M, since by definition in that case, From such a module we can produce a quiver whose edges i → j are a generating set for M ≤x i ∩ A j as an A + -module. This quiver presents the partial order via (1).
Conversely, the partial order presented by a quiver action Q M has lower sets whose definition manifestly commutes with the action of A. In other words, quiver partial orders are lower saturated. ii) ≤ can be presented by a quiver.

Proposition
Lower finiteness When (M, ≤) is lower finite, in particular each M ≤x i is finitely generated, and so by sharpness of A + has a unique set of primitive generators. Let us call the quiver Q with these generators as its set of edges the canonical quiver. (If A + ≤i ∩ A + i = A + i , then this algorithm yields a trivial loop at i, which we may exclude.) It has finitely edges departing from each vertex. In particular: Non-degeneracy Unfortunately, being presented by a quiver with finitely many edges departing each vertex is not sufficient to guarantee lower finiteness; by the formula 4, it is the action of the path algebra at i that we need to worry about, and the latter is infinite whenever there is a cycle at i. We need a way to disregard such cycles. (Non-degeneracy). Let M be a direct sum of cyclic A-modules. A partial ordering on M is said to be non-degenerate if the induced partial order on each cyclic factor is the A + -divisibility order ≤ A + .

Definition
Non-degeneracy of a partial order entails that the element of A attached to any path in a presenting quiver Q with the same start and end point i actually lies in A + ⊆ End(A i ), and so for any x i ∈ A i . Such cycles can therefore be disregarded in the presentation (3), and we may restrict attention to the set Q ⊆ Path(Q) of acyclic paths. The latter is finite as soon as Q is.

Lemma. Let (M, ≤) be lower saturated and finitely presented. If ≤ is non-degenerate, then it is lower finite.
All that remains to obtain an equivalence lower finite ⇔ finitely presented + non-degenerate is to show that lower finite orders are non-degenerate. We conclude this for domains in §4.5.

Primitives
Free modules on a po-module have the following distinguishing features: • they have unique irredundant primitive decompositions (cf. §3.2); • the set of primitive elements, together with −∞, is an A-submodule.
If, in general, these criteria are satisfied, we can by passing to primitive elements reverse the procedure and obtain a po-A-module from a B[A]-module. This is possible for finite projective B[A]-modules, by a partial converse to proposition 4.2:

Lemma. Every projective B[A]-module is free on a partially ordered A-module.
Proof. We have seen that free B[A]-modules are, in particular, free B-modules on a poset. By this and lemma 3.10, any projective B[A]-module has primitive decompositions. Now let µ be projective, APrimµ ⊆ µ the A-submodule generated by the primitive elements. Then APrimµ generates µ as a B-module, that is, B(APrimµ, ≤) ։ µ is surjective. By projectivity of µ, it has a B[A]-linear section. By applying the lemma 3.11 on sections over primitives to S = APrimµ, the section surjects onto the set of elements of B(APrimµ, ≤) of the form (APrimµ) ≤X with X ∈ Primµ. Since these generate it as a B[A]module, it is an isomorphism. (In particular, APrimµ = Primµ is an A-submodule of µ.)

Domains
It remains to determine whether it is necessary that M be projective, that is, that only projective A-modules can yield projective B[A]-modules. We have a complete result in the case that A is an F 1 -domain.

Definition (Domain). A monoid pair (A;
A is a localisation of A + , and A + is integrally closed (i.e. saturated) in A.
I remind the reader that A + is always assumed to be sharp. Since A + is integrally closed in A, the multiplicative torsion of A must belong to A + and therefore be trivial. ii) Let p : i A i ։ M be a surjection from a free module, where each A i ≃ A is free cyclic.

Proposition. Let M be a partially ordered A-module. i) If B(M; ≤) is finitely generated over B[A], then M is finitely generated over A. ii) Suppose A is a domain. If B(M; ≤) is projective over B[A], then M is free over
By eliminating indices i such that p A i is strictly contained in another p A j , and identifying indices with the same image, we may assume that p is an irredundant generator, i.e. that no factor can be removed without destroying surjectivity of p. Using the natural identification i B(A i ) ∼ = B( i A i ), we will study σ via its components

Lemma. If x ∈ A i , then x ∈ σ i pv.
Proof. By A-linearity, it will be enough to show this for x a generator of A i . Irredundancy implies that px is not in the image of A j for any j = i. Thus σ i px contains a lift f x of px, and f fixes px.
Applying σ i to the relation f px = px shows that f acts by an automorphism of σ i px. Since A + is sharp, this automorphism must permute the finite set of primitive generators. In particular, the orbit of f x is finite. Thus f is torsion, therefore 1.

Lemma. M is a direct sum of the cyclic modules p A i .
Proof. Since A i and A j are free cyclic, there are identifications A i ≃ A ≃ A j . We will show that if v i , v j , are the images of 1 in M under p with respect to these two identifications, then any relation of the form By lemma 4.11, applying σ j (resp. σ i ) to the relation shows that there exist e i ∈ σ j v i (resp. e j ∈ σ i v j ) such that in A, and so in particular e i e j f i = f i . By cancellativity of A \ 0, e i e j = 1. Applying p to the relation σ j v i ≤ σv i gives e i v j ≤ v i , and vice versa. We deduce It remains to show that the cyclic factors of M are free. This follows from a 'high-brow' argument based on the fact that B is strongly monoidal and commutes with equalisers.
Since B(M) is by hypothesis projective, the tensor sum B(M) ⊕ B[A] − is left exact. By taking primitives, this implies that M ⊗ A − commutes with equalisers. It follows from the argument for the equational criterion for flatness 2.6 that any relation in a cyclic summand of M must be trivial.

Corollary (Classification). Let A be an F 1 -domain. A B[A]-module µ is finite projective if and only if: i) it has unique irredundant primitive decompositions;
ii) Primµ is a free A-submodule of µ, whose induced order is iii) non-degenerate, and iv) can be presented by a finite quiver.

Alternatively, the last condition iv) can be replaced by the two conditions iva) M is a lower submodule of M ⊗ A K A , and ivb) (M, ≤) is finitely presented, that is, compact as an object of POMod A .
Proof. By lemma 4.8, a projective module µ is free on (Primµ, ≤), and by proposition 4.10 Primµ is itself finite projective. By proposition 4.2, the order on Primµ is lower finite and lower saturated; loc. cit. also handles the converse.
The remaining meat of the theorem is therefore that lower finiteness and lower saturation equate to conditions iii) and iv), and that iv) is equivalent to iva)+ivb). More precisely, lower saturation is equivalent to iva), and lower finiteness is equivalent to non-degeneracy together with finite presentation.
(ls ⇔ iva)) This follows from unravelling explicitly the definition of lower saturation: for all x, y ∈ M and f ∈ A such that x ≤ f y, there exists a symbol [ When M ∼ = i∈I A i is a direct sum of cyclic modules, it follows that every relation can be deduced from one of the formx ≤ y with y a generator of its cyclic factor.
When A is a domain and M is free, this works for all f = 0, hence the result.
(lf ⇔ ndg + fp) After lemmas 4.4 and 4.7, it remains to show that lower finiteness implies non-degeneracy. Any degenerate relation f x ≤ x in M entails an infinite sequence of relations f n x ≤ x. Since M is free, we may localise our study to a cyclic factor and assume M = A, and by lower saturation, that x = 1.
The condition that the set { f n } n∈N be contained in a finite A + -submodule is precisely that f be integral over A + . Since we have assumed that A + is integrally closed in A, nondegeneracy is a necessary condition for lower finiteness.
4.14 Aside (Duality over F 1 ). There is in general no good theory of duality for modules over F 1 -algebras -since A ⊕n ≃ A n = Hom(A ⊕n , A) as soon as n > 1, free modules are typically not reflexive. However, in light of corollary 4.13, at least for certain po-modules over a domain A we can 'borrow' the theory of duality for finite projective B[A]-modules, and define a restricted dual M * := Prim (Hom(BM, BA)).
The result implies that if M is a lower finite and lower saturated partially ordered, finite, free A-module, then so is M * , and that this operation is an involution (though still not actually a duality with respect to ⊗ A ) on such modules.
We end this section with some counterexamples.

Example (Projective B[A]
-module with non-projective A-module of primitives). A more involved argument in lemma 4.12 shows that the result holds for M with the divisibility order whenever A is idempotent-free. The same line of thought yields a counterexample with idempotents to part ii) of proposition 4.10. Let The idempotent ξ = e 1 e 2 fixes all elements except 0 and 1, and multiplication by f i is injective on its image -so induces an isomorphism between the fixed set of ξ and the principal ideal ( f 1 ) = ( f 2 ). Now define a module M over A with two generators and the relations ( f 1 , 0) = (0, f 2 ), (e 2 , 0) = (0, ξ), (ξ, 0) = (0, e 1 ).
Both of the generating cyclic submodules are free, but M itself is not, so by proposition 2.2 it cannot be projective. Note also: • the fixed set of ξ on M is isomorphic to the fixed set of ξ of A itself; in particular, it is projective cyclic; • the base change of A to Z splits as a product Z × Z[e ±1 1 ], whereupon M becomes the union of a trivial line bundle over the G m part and a rank two free module over the point -in particular, it is projective. When working with monoids one somewhat inevitably encounters pathological examples like this one, as these kinds of relations can arise after a simple sequence of relative tensor products between finite free monoids ≃ N m . For example, take Z 2 ∨ /(e 1 + λ ∨ e 2 = e 1 ), where λ ≤ 0. The only primitives in this module are the translates of e 2 ; in particular, it is not generated by primitives. It follows that it cannot be embedded in a free module, and is in particular non-reflexive.
Proof. We will prove directly the equational criterion (prop. 2.6). Let v : Γ n ∨ → Γ ′ ∨ be a Γlinear homomorphism, Γ ∨ ⇒ Γ n ∨ a relation. These data are determined by a set of (up to) n with ( f i ), (g i ) ∈ Γ n . Let i f , resp. i g , be the index for which the maximum on the left (resp. right) is realised. If i f = i g , then f i f = g i g and the projection Γ n → Γ on the i f th factor factors v. Otherwise, we may identify the i f th and i g th factor via the primitive of the given relation, and project out the other co-ordinates.
On the other hand, if the rank of Γ ′ is strictly greater than that of Γ, then submodules generated by n > 1 elements are in general infinitely presented, and so cannot be projective. For instance, this applies to the Z ∨ -submodule of Z 2 lex generated by (0, 0) and (0, 1).

Polyhedra
In this section, we will work with additively notated monoids without zero. The associated fractional submodule functor B factors through adjoining zero. Cf. §2.1.

Weight polyhedra
Let H be a totally ordered group -for example, an additive subgroup of R. We will study finite modules over the semifield via partially ordered H-modules (or 'H-acts'). This setting enjoys two substantial simplifications over the general case: • since H is a group, lower saturation is automatic (cf. part iva of corollary 4.13); • since H is totally ordered, non-degeneracy is automatic; and so all we have to worry about is finite presentation. cut out by relations defining the partial order of M. When (M, ≤) is finitely presented, it is a polyhedral subset defined by inequalities between co-ordinates. We formalise the sets arising in this way as follows:

Example.
If ∆ 1 is one-dimensional (i.e. a GL 1 weight polyhedron), then ∆ 1 ≃ H as an H-act and the differential of any map ∆ 1 → ∆ 2 is fixed to be the map Z I 2 → Z sending all basis vectors to 1. To specify such a map, it is therefore enough to say where a single point 0 ∈ ∆ 1 goes. In particular, Aff GL H (H, ∆ 2 ) = ∆ 2 . Conversely, a map ∆ 2 → ∆ 1 is necessarily a co-ordinate projection. In particular, the set Aff GL H (∆ 2 , H) =: Aff GL H (∆ 2 ) of such is a finite H-module.
Co-ordinate functions on a weight polyhedron Let N be a torsor under Hom Z (V Z , H). Then in particular N is a weight polyhedron, and Proof. Let ∆ ⊆ N be a weight polyhedron. A monotone homomorphism Aff GL H (∆) → H is a point p of N such that for every pair of affine functions F, G on V , F| ∆ ≤ G| ∆ implies F(p) ≤ G(p). Since ∆ is cut out from N by such inequalities, p ∈ ∆.
The more subtle part of the proof is to show that when (M, ≤) is finitely presented, conversely, the elements of V(M) 'separate points' of M. More precisely: (Constructing monotone functionals). Let M ∈ POMod free H be finitely presented.

Lemma
in particular,N is a free H ∨ -module.

Bundles of weight polyhedra
Let H ⊆ R be an additive subgroup, and let ∆ ⊆ N be an H-rational, strongly convex polyhedron in an H-affine space N. More precisely, ∆ ⊆ N(QH) is a subset defined over the divisible hull QH of H cut out by finitely many affine equations over H and having at least one vertex. By multiplying out denominators, the condition of H-rationality is the same as rationality over QH. 3 To ∆ we will associate the monoid pair of H-affine functions Aff(∆, H) on ∆ with integral slopes and values in H, with integers the (saturated, sharp) submonoid Aff + (∆, H) of functions bounded above by zero. We do not add strata to ∆ at infinity -but see §5.3. Since Aff(∆, H) = Aff (N, H) is a group, we retain the benifit that we enjoyed in §5.1 that po-modules are automatically lower saturated. However, non-degeneracy is no longer automatic. We will be interested in the 'polyhedral' semiring CPA(∆, H ∨ ) of convex piecewise-affine functions on ∆ with integer slopes and values in H ∨ . This is generated as a B-module by the pair Aff(∆, H), and is closely related to the free semiring.

Definition.
Let ∆ ⊂ N be a strongly convex H-rational polyhedron. A convex family of weight polyhedra E over ∆ is a convex polyhedron ∆ E inside an H-affine space N E /N such that for each point p ∈ ∆(H ′ ) defined over an ordered extension H ′ of H, π −1 (p) is an H ′ -rational GL weight polyhedron. It is enough to check for finite extensions of H ′ . The dimension of the fibre at any interior point is called the rank of the family.
A morphism of convex families of weight polyhedra is an H-affine map over N that restricts to a morphism in Poly GL H ′ on the fibre over any H ′ -point.
Note that π is necessarily surjective, and the rank does not depend on the point.

Dual to a partially ordered
to be the subset of pairs (p, φ) for which φ is monotone. 3 Beware that this category of polytopes differs from the category Poly GL H discussed in §5.1 in several key ways. First, by H-invariance GL weight polyhedra are never strongly convex. Second, the co-ordinates of the supporting half-spaces for a weight polyhedron are required to lie in H, rather than merely in QH. Finally, while it is always natural to add strata at (minus) infinity to a weight polyhedron, in this section for simplicity we will consider our base polyhedra to be 'punctured' at infinity.

Co-ordinate functions on families of weight polyhedra
Let ∆ × H denote the trivial affine line bundle over ∆ -more precisely, the affine bundle whose H ′ -points are ∆(H ′ ) × H ′ .
If E is a convex family of weight polyhedra, then the set Aff GL H (E) of affine functions E → ∆ × H whose relative differentials are fundamental weights has a natural structure of a partially ordered, finite, free module over Aff(∆, H). This can be viewed as a problem on the B[A]-module side and can be solved by considering instead its 'normalisation' as in [Mac15b]. In geometric terms, this is achieved by replacing POMod A with the associated stack for the rigid analytic topology Spec(A; A + ). gives a section of B(Primµ, ≤) → µ. By the lemma 3.11 on sections over primitives, this necessarily commutes with the two embeddings of Primµ. It follows that σ is an isomorphism. This part of the proof does not require A to be a domain.

Normal projectives
The remainder of the argument depends crucially on the fact that a finitely generated A + -module has a unique minimal set of generators. With a little work, the same holds true in the integrally closed regime as well: 5.9 Lemma (Unicity of generators). Let A be an F 1 -field, M a free A-module. Let K be the integral closure of a finitely generated A + -submodule of M. There is a unique minimal set of generators of K as an integrally closed fractional submodule.
Proof. We write in additive notation. By intersecting K with the cyclic factors of M and choosing a trivialisation, we may reduce to the case M = A. In this case, integral closure of K coincides with convexity [Mac15b, lemma 5.4].
Let (X i ) i∈I and (Y j ) j∈J be two minimal sets of generators of K as a convex A + -submodule. Fixing an index 0 ∈ I, this give us equations so either X 0 can be eliminated from its generating set by cancelling it from the right, or Since all m ℓ are non-zero, there's exactly one index j ∈ J for which n j and m j0 are both non-vanishing, and m ji = 0 for all i = 0. Thus m j = m j0 = 1 and so Y j = X 0 .
The proof of the monotone section lemma 3.1 applies, in light of lemma 5.9, and shows that the section is automatically a right adjoint.
Primµ is free. We need clarify only two points: • to prove that the submodule spanned by each generator of M was free, we used the fact that B is strongly monoidal and commutes with equalisers. These properties also hold for ν B = ν B[A]⊕ B[A] (−); the first is clear, and the second follows from the fact that the normalisation of a semiring is always flat.
• The crucial lemma 4.11 that M is a direct sum of cyclics uses the uniqueness of primitive generators for fractional submodules of a free cyclic module, and so passes with another application of lemma 5.9.
That said, the proof of proposition 4.10 runs verbatim.

Corollary. Every projective ν B[A]-module is the normalisation of a projective B[A]module.
Not every partially order on a free A-module M is induced from its inclusion into the normalisation of B(M, ≤), as example 5.7 shows. However, since Hom(α, H ∨ )→Hom( ν α, H ∨ ) for any semiring α and totally ordered group H [Mac15b, Prop. 1.2.ii)], the geometric dual V factors through this replacement.
The missing element of the proof is to show that conversely, M → Aff GL ∆ (V(M, ≤)) is an order isomorphism for M the module of primitives in a normal projective B[A]-module. This follows from a corrected version of lemma 5.4:

Lemma (Constructing monotone functionals). Let M ∈ POMod free
A be finitely presented, and suppose the partial order on M is that induced by its inclusion into B(M, ≤). For any F ≤ G ∈ M there exists a φ F,G : M → H such that φ F,G G = 0 and φ F,G F > 0.
Proof. By the same argument as in lemma 5.4, it will be enough to find an identification A dF→ A and algebra homomorphism A → H satisfying the inequalities 5 and φ F,G > 0. By non-degeneracy, we may assume dF = dG.
We have F ≤ γ:dG→dF γG in ν B(A dF ). Since ν B(A dF ) admits a representation as semiring of convex piecewise-affine functions ([Mac15b] ex. 5.6), there is some point q ∈ ∆(QH) at which γ γG(q) < F(q). If γ 0 is the path achieving the maximum in this formula, then identifying A dF with A by setting γ 0 G = 0 gives us the desired functional. between the category of finite projective CPA(∆)-modules and the category of convex families of extended GL weight polyhedra over ∆ with convex piecewise-affine maps whose vertical linear parts are fundamental weights.

Extensions
The results of §5.2 were couched in the setting of F 1 -fields, but using corollary 4.13 it can easily be extended to the case of domains: we simply need to identify the lower free Astructures on partially ordered K A -modules. We obtain a supply of F 1 -domains by partially compactifying our polyhedra at infinity. This means that we replace Aff(∆, H) with the submonoid Aff(∆, H) of functions bounded above on ∆. We may also define various partial compactifications of ∆ dual to intermediate saturated submonoids Aff(∆) ⊆ A ⊆ Aff(∆).
A free A-structure on a module M over K A is the same data as a reduction of the structure group of V(M) from Aff(∆, H) to the group of bounded affine functions. For instance, one may fix a trivialisation V(M) ≃ ∆ × H n (which even determines an A + -structure). The A-structure M A is then the set of functions on V(M) bounded above by some co-ordinate function of the trivialisation.
Such a reduction defines a lower A-structure on M if and only if every function bounded above on V(M, ≤) is bounded above on all of V(M). This is the case if and only if there exists a constant subset K × ∆ ⊆ V(M, ≤) ⊆ V(M) whose fibre K ⊆ H n has non-empty interior. More generally, if∆ is not compact, such subsets must exist on rays approaching the boundary.

5.13
Example. Let∆ = [−∞, 0] with co-ordinate X , and let E be the rank two family of polyhedra cut out by the equation Y 1 ≤ Y 2 + X . Its restriction to X = −∞ is free of rank one. The dual module Aff GL ∆ (E) is not lower finite because of the function Y 1 − X , and so this does not correspond to a projective CPA Z (∆, H)-module.
However, its base change to ∆ := (−∞, 0] is projective; it is the module presented by the quiver It is trivialised by the change of basis Y 2 ↔ Y 2 + X ; in fact, it is the pullback of a projective B-module. This provides us with a trivial, and in particular projective, extension over −∞. 5.14 Aside (Other geometric classes of modules). If we drop the non-emptiness requirement in the definition of weight polyhedra -and hence that E → ∆ be surjective -we obtain a class of modules that can be obtained by 'pushing forward' projective modules from subpolyhedra. More interestingly, by pushing forward projective modules from boundary strata one can construct modules in polyhedra whose rank jumps up at infinity. The construction of example 5.13 gives also families whose rank decreases at the boundary. Finally, it will be important in the future to understand modules that correspond to families of 'valuated matroids' -but that is another story for another day.