PICARD GROUPS FOR TROPICAL TORIC VARIETIES

arXiv:1709.03130v1 [math.AG] 10 Sep 2017

JAIUNG JUN, KALINA MINCHEVA, AND JEFFREY TOLLIVER

A BSTRACT. From any monoid scheme X (also known as an F1 -scheme) one can pass to a semiring
scheme (a generalization of a tropical scheme) XS by scalar extension to an idempotent semifield S.
In this paper, we investigate the relationship between the Picard groups of X and XS . We prove that
for a given irreducible monoid scheme X (with some mild conditions) and an idempotent semifield S,
the Picard group Pic(X) of X is stable under scalar extension to S. In other words, we show that the
two groups Pic(X) and Pic(XS ) are isomorphic. Moreover, each of these groups can be computed by
considering the correct sheaf cohomology groups. We also construct the group CaCl(XS ) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme XS and prove that CaCl(XS )
is isomorphic to Pic(XS ).

1.

Introduction

In recent years, there has been a growing interest in developing a notion of algebraic geometry
over more general algebraic structures than commutative rings or fields. The search for such a theory
is interesting in its own right, however, the current work relates to two actively growing sub-fields of
that study.
The first one is motivated by the search for “absolute geometry” (commonly known as F1 -geometry
or algebraic geometry in characteristic one) which is first mentioned by J. Tits in [Tit56]; Tits first
hints at the existence of a mysterious field of “characteristic one” by observing a degenerating case
of an incidence geometry Γ(K) associated to a Chevalley group G(K) over a field K; when K = Fq
(a field with q elements), as q → 1, the algebraic structure of K completely degenerates, unlike the
geometric structure of Γ(K) so Tits suggests that limq→1 Γ(K) should be a geometry over the field of
characteristic one.
In [Man95], Y. Manin considers the field of characteristic one from the completely different perspective; for the geometric approach to the Riemann hypothesis. Shortly after, in [Sou04], C. Soulé
first introduced a notion of algebraic geometry over the field F1 with one element. Since then
A. Connes and C. Consani have worked to find a geometric framework which could allow one to
adapt the Weil proof of the Riemann hypothesis for function fields to the Riemann zeta function
(cf. [CC10a], [CC10b], [CC11], [CC14], [CC17a], [CC17b]).
The second field to which this work contributes is a new branch of algebraic geometry called tropical geometry. It studies an algebraic variety X over a valued field k by means of its “combinatorial
shadow”, called the (set-theoretic) tropicalization of X and denoted trop(X ). This is a degeneration of
the original variety to a polyhedral complex obtained from X and a valuation on k. The combinatorial
shadow retains a lot of information about the original variety and encodes some of its invariants. Algebraically, trop(X ) is described by polynomials in an idempotent semiring, which is a more general
object than a ring – a semiring satisfies the same axioms that a ring does except invertibility of addition. However, the tropical variety trop(X ) has no scheme structure. In [GG16], J. Giansiracusa and
N. Giansiracusa combine F1 -geometry and tropical geometry in an elegant way to introduce a notion
of tropical schemes. This is an enrichment to the structure of trop(X ), since a tropical variety is seen
as the set of geometric points of the corresponding tropical scheme. The tropical scheme structure
2010 Mathematics Subject Classification. 14T05(primary), 14C22(primary), 16Y60(secondary), 12K10 (secondary),
06F05 (secondary).
Key words and phrases. tropical geometry, tropical schemes, idempotent semirings, Picard groups.
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is encoded in a “bend congruence” (cf. [GG16]) or equivalently in a “tropical ideal” or a tower of
valuated matroids (cf. [MR14], [MR16]).
A natural question and central motivation for this work, is whether one can find a scheme-theoretic
tropical Riemann-Roch theorem using the notions from J. Giansiracusa and N. Giansiracusa’s theory.
The problem of finding such an analogue in the context of tropical varieties, that is the product of
set-theoretic tropicalization, has already received a lot of attention. In particular it has been solved
in the case of tropical curves. We recall that a tropical curve (the set-theoretical tropicalization of a
curve) is a connected metric graph. One can define divisors, linear equivalence and genus of a graph
in a highly combinatorial way. With these notions M. Baker and S. Norine [BN07] prove a RiemannRoch theorem for finite graphs while G. Mikhalkin and I. Zharkov [MZ08] and later A. Gathmann and
M. Kerber [GK08] solve the problem for metric graphs. Later the problem is revisited by O. Amini
and L. Caporaso [AC13] in the context of wighted graphs. A generalization of the work of M. Baker
and S. Norine to higher dimensions has been carried out by D. Cartwright in [Car15a] and [Car15b].
In [CC17a] A. Connes and C. Consani prove a Riemann-Roch statement for the points of a certain
Grothendieck topos over (the image of) Spec(Z), which can be thought of as tropical elliptic curves.
An important ingredient to the project of A. Connes and C. Consani to attack the Riemann Hypothesis
is to develop an adequate version of the Riemann-Roch theorem for geometry defined over idempotent semifields. Notably, there are several fundamental differences between their statement and the
tropical Riemann-Roch of M. Baker and S. Norine.
Recently, in [FRTU16] T. Foster, D. Ranganathan, M. Talpo and M. Ulirsch investigate the logarithmic Picard group (which is a quotient of the algebraic Picard group by lifts of relations on the tropical
curve) and solve the Riemann-Roch in the context of logarithmic curves (metrized curve complexes).
The solution of a scheme-theoretic tropical Riemann-Roch problem requires several ingredients, such
as a proper framework for “tropical sheaf cohomology” and a notion of divisors on tropical schemes,
in particular, ranks of divisors. In this note, we investigate Picard groups of tropical (toric) schemes as
the first step towards building scheme-theoretic tropical divisor theory and a Riemann-Roch theorem.
To do that, we look at an F1 -model of the tropical scheme, i.e., a monoid scheme.
Monoid schemes are related to tropical (more generally semiring) schemes and usual schemes via
scalar extension. More precisely, if X is a monoid scheme, say X = Spec M for a monoid M, then
XK = Spec K[M] is a scheme if K is a field and a semiring scheme if K is a semifield. Note that XC is
a toric variety if M is integral and finitely generated and XK is a tropical scheme (the scheme theoretic
tropicalization of XC ) if K is the semifield of tropical numbers. When K is a field, the relation between
Pic(X ) and Pic(XK ) has been extensively investigated by J. Flores and C. Weibel in [FW14]. They
show that in this case Pic(X ) is isomorphic to Pic(XK ).
In this paper, our main interest lies in the tropical context, i.e., the case when K is an idempotent
semifield. We investigate the relation between the Picard group Pic(X ) of a monoid scheme X and
Pic(XK ) for an idempotent semifield K. More precisely, when XK is the lift of an F1 -model X (with
an open cover satisfying a mild finiteness condition), we prove that the Picard group is stable under
scalar extension.
Theorem A. Let X be an irreducible monoid scheme and K be an idempotent semifield. Suppose that
X has an open cover satisfying Condition 3.4. Then we have that
Pic(X ) = Pic(XK ) = H1 (X , OX∗ ).
Remark 1.1. One may combine the above result with that of J. Flores and C. Weibel to conclude that
for an irreducible monoid scheme X , the Picard group is stable under scalar extension to a field or an
idempotent semifield.
Recall that a cancellative semiring scheme XK over an idempotent semifield K is a semiring scheme
such that for each open subset U of XK , the semiring OXK (U ) of sections is cancellative. We construct
the group of Cartier divisors and modulo principal Cartier divisors, which we denote by CaCl(XK ).
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These are the naive principal Cartier divisors - the ones defined by a single global section. Then we
can show that
Theorem B. Let XK be a cancellative semiring scheme over an idempotent semifield K. Then
CaCl(XK ) is isomorphic to Pic(XK ).
We remark that the Picard group of a curve is rarely equal to the Picard group of its set-theoretic
tropicalization.
Acknowledgments This work was started at the Oberwolfach workshop ID: 1525 on Non-commutative
Geometry. The authors are grateful for the Institute’s hospitality and would like to thank the organizers for providing the opportunity. K.M would also like to thank Dhruv Ranganathan for many helpful
conversations.
2.

Preliminaries

In this section, we review some basic definitions and properties of monoid and semiring schemes.
We also recall the notion of Picard groups for monoid schemes developed in [FW14] and for semiring
schemes introduced in [Jun17].
2.1 Picard groups for monoid schemes. In what follows, by a monoid we always mean a commutative monoid M with an absorbing element 0M , i.e., 0M · m = 0M for all m ∈ M. Note that if M
is a monoid without an absorbing element, one can always embed M into M0 = M ∪ {0M } by letting
0M · m = 0M for all m ∈ M.
Remark 2.1. We will use the term “monoid schemes” instead of “F1 -schemes” to emphasize that
we are employing the minimalistic definition of F1 -schemes based on monoids following A. Deitmar
[Dei05], instead of any of the more general constructions that exist in the literature (cf. [CC10b]
or [Lor12]).
We recall some important notions which will be used throughout the paper. For the details, we
refer the reader to [Dei05], [Dei08], [CLS12].
Definition 2.2. [Dei05, §1.2 and §2.2] Let M be a monoid.
(1) An ideal I is a nonempty subset of M such that MI ⊆ I. In particular, 0M ∈ I. An ideal I is
said to be prime if M\I is a multiplicative nonempty subset of M.
(2) A maximal ideal of M is a proper ideal which is not contained in any other proper ideal.
(3) The prime spectrum Spec M of a monoid M is the set of all prime ideals of M equipped with
the Zariski topology.
(4) For any f ∈ M, we define the following set:
D( f ) := {p ∈ Spec M | f 6∈ p}.
Then {D( f )} f ∈M forms a basis of Spec M.
One can mimic the construction of the structure sheaf on a scheme to define the structure sheaf
(of monoids) on the prime spectrum Spec M of a monoid M. A prime spectrum Spec M together
with a structure sheaf is called an affine monoid scheme. A monoidal space is a topological space
together with a sheaf of monoids. A monoid scheme is a monoidal space which is locally isomorphic
to an affine monoid scheme. As in the case of schemes, we call a monoid scheme irreducible if the
underlying topological space is irreducible.
Remark 2.3.
(1) Let M be a monoid. Then M has a unique maximal ideal m := M\M × .
(2) Analogously to the classical case, the category of affine monoid schemes is equivalent to the
opposite of the category of monoids. A monoid scheme, in this case, is a functor which is
locally representable by monoids. In other words, one can understand a monoid scheme as a
functor of points. For details, see [PL11].
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Next, we briefly recall the definition of invertible sheaves on a monoid scheme X . We refer the
readers to [FW14] and [CLS12] for details.
Definition 2.4. [FW14, §5] Let M be a monoid and X be a monoid scheme.
(1) By a M-set, we mean a set with an M-action.
(2) An invertible sheaf L on X is a sheaf of OX -sets which is locally isomorphic to OX .
Let M be a monoid and A, B be M-sets. One may define the tensor product A ⊗M B. Furthermore,
if A and B are monoids, A ⊗M B becomes a monoid in a canonical way. See, [CLS12, §2.2, 3.2] for
details.
Remark 2.5. We would like to warn the reader that in the literature the same terminology is used to
denote different things. In [FW14], the authors use the term “smash product” for tensor product of
monoids, whereas in [CLS12], the authors use the term “smash product” only for tensor product of
monoids over F1 , the initial object in the category of monoids. We will use the term tensor product to
stay compatible with the language of schemes and semiring schemes.
Let Pic(X ) be the set of the isomorphism classes of invertible sheaves on X . Suppose that L1 and
L2 are invertible sheaves on X . The tensor product L1 ⊗OX L2 is the sheafification of the presheaf (of
monoids) sending an open subset U of X to L1 (U ) ⊗OX (U) L2 (U ). It is well–known that L1 ⊗OX L2
is an invertible sheaf on X . Also, as in the classical case, the sheafificaiton L1−1 of the presheaf (of
monoids) sending an open subset U of X to HomOX (U) (L1 (U ), OX (U )) becomes an inverse of L1
with respect to the tensor product in the sense that L1 ⊗OX L1−1 ≃ OX . Therefore, Pic(X ) is a group.
One can use the classical argument to prove that Pic(X ) ≃ H1 (X , OX∗ ) (for instance, see [FW14,
Lemma 5.3.]). Recall that for any topological space X and a sheaf F of abelian groups on X , sheaf
cohomology Hi (X , F ) and Čech cohomology Ȟi (X , F ) agree for i = 0, 1. Therefore, we have
Pic(X ) ≃ H1 (X , OX∗ ) ≃ Ȟ1 (X , OX∗ ).

(1)

2.2 Picard groups for semiring schemes. A semiring is a set (with two binary operations - addition and multiplication) that satisfies the same axioms that a ring does, except invertibility of addition.
In this paper by a semiring we mean a commutative semiring with a multiplicative identity. A semifield is a semiring in which every non-zero element has a multiplicative inverse. A semiring A is
idempotent if a + a = a, for all elements a ∈ A. An example of an (idempotent) semifield is the tropical semifield which we denote by T. It is defined on the set R ∪ {−∞}, with operations maximum
and addition. As the name suggests, this semifield is central to tropical geometry.
Definition 2.6. Let A be a semiring.
(1) An ideal I of A is an additive submonoid I of A such that AI ⊆ I.
(2) An ideal I is said to be prime if I is proper and ab ∈ I then a ∈ I or b ∈ I.
(3) A proper ideal I is maximal if the only ideal strictly containing I is A.
Since there are several definitions of semiring schemes and some of them are not equivalent, we
present the following definition which we will use in this paper.
Definition 2.7. Let A be a semiring and X = Spec A be the set of all prime ideals of A. We endow X
with the Zariski topology. The topology on X is generated by the sets of the form D( f ) := {p ∈ X |
f 6∈ p} for all f ∈ A.
Let A be a semiring and S be a multiplicative subset of A. We recall the construction of localization
S−1 A of A at S from [Gol99, §11]. Let M = A × S. We impose an equivalence relation on M in such
a way that (a, s) ∼ (a′ , s′ ) if and only if ∃s′′ ∈ S such that s′′ as′ = s′′ a′ s. The underlying set of S−1 A
is the set of equivalence classes of M under ∼. We let as be the equivalence class of (a, s). Then one
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can define the following binary operations + and · on S−1 A:
a a′
as′ + sa′
+ ′ :=
,
s s
ss′

a a′ aa′
· = ′.
s s′
ss

It is well-known that S−1 A is a semiring with the above operations. Furthermore, we have a canonical
homomorphism S−1 : A → S−1 A sending a to 1a . When S = A − p for some prime ideal p, we denote
the localization S−1 A by Ap .
Let A be a semiring and X = Spec A be the prime spectrum of A. For each Zariski open subset U
of X , we define the following set:
OX (U ) := {s : U →

∏ Ap},
p∈U

where s is a function such that s(p) ∈ Ap and is locally representable by fractions. One can easily see
that OX is a sheaf of semirings on X . An affine semiring scheme is the prime spectrum X = Spec A
equipped with a structure sheaf OX . Next, by directly generalizing the classical notion of locally
ringed spaces, one can define locally semiringed spaces. A semiring scheme is a locally semiringed
space which is locally isomorphic to an affine semiring scheme.
A special case of semiring schemes are the so-called tropical schemes. They are introduced in
[GG16]. These schemes locally are isomorphic to the prime spectrum of a quotient of the polynomial
semiring over the tropical semifield (denoted T[x1 , . . . , xn ]) by a particular equivalence relation, called
a “bend congruence”.
We note that every scheme is a semiring scheme, but never a tropical scheme. The reason is that
the structure sheaf of a tropical scheme is a sheaf of additively idempotent semirings which are never
rings.
One can extend the familiar notions of invertible sheaves and Picard group in the context of
schemes or monoid schemes to the semiring schemes setting. In fact, Čech cohomology for semiring
schemes is introduced in [Jun17] and the following is proved.
Theorem 2.8. [Jun17] Let X be a semiring scheme. Then Pic(X ) is a group and can be computed
via H1 (X , OX∗ ).
We note that OX∗ is a sheaf of abelian groups and thus we can define H1 (X , OX∗ ) in the usual way
and H1 (X , OX∗ ) = Ȟ1 (X , OX∗ ) as in (1).
Example 2.9. In [Jun17], the author also proves that for a projective space PnS over an idempotent
semifield S, one obtains that Pic(PnS ) ≃ Z, as in the classical case. In [FW14], a similar result is
proven for a projective space over a monoid. These two results motivate (among others) the authors
of the current note to study relations among Picard groups of schemes, monoid schemes, and semiring
schemes under “scalar extensions”.
3.

Picard groups of tropical toric schemes

In this section, we prove the main result which states that the Picard group Pic(X ) of an irreducible
monoid scheme X (with some mild conditions) is stable under scalar extension to an idempotent
semifield. Let us first recall the definition of scalar extension of a monoid scheme to a field or an
idempotent semifield.
Let X be a monoid scheme and K a field or an idempotent semifield. Suppose that X is affine, i.e.,
X = Spec M, for some monoid M. Then the scalar extension is defined as:
XK = X ×Spec F1 Spec K := Spec K[M],
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where K[M] is a monoid semiring (when K is an idempotent semifield) or a monoid ring (when K is a
field). This construction can be globalized to define the base change functors from monoid schemes
to semiring schemes (or schemes). See, [GG16, §3.2.].
Remark 3.1. In [Dei08], Deitmar proves that we can obtain every toric variety from integral monoid
scheme X of finite type via extension of scalars. In other words, if K = C is the field of complex
numbers, then XK is a toric variety.
Since the Picard group Pic(X ) of a semiring scheme X can be computed by the cohomology of
the sheaf OX∗ of multiplicative units, we will need to understand K[M]× , the group of multiplicatively
invertible elements of K[M].
Recall that by a cancellative monoid, we mean a monoid M such that if ab = ac, for a ∈ M − {0M }
and b, c ∈ M, then b = c.
Proposition 3.2. Let M be a cancellative monoid and K be an idempotent semifield. Then, we have
K[M]× ∼
= K × × M×.
Proof. For x ∈ K[M], let φ (x) ∈ N be the smallest natural number such that x has a representation of
the form
φ (x)

x=

∑ ak mk ,
k=1

where ak ∈ K and mk ∈ M. Such a representation has a minimal length if and only if each ak 6= 0 and
the elements mk are nonzero and distinct. Since K is an idempotent semifield, two nonzero elements
of K cannot sum to zero and hence we have φ (x) ≤ φ (x + y) for all x, y ∈ K[M].
φ (x)

Let m ∈ M − {0M } and x ∈ K[M]. Write x =

φ (x)

∑ ak mk , so we have that mx =

∑ ak (mmk ).

k=1

k=1

This

is the shortest such expression for mx because each ak 6= 0 and M is cancellative. Hence one has
φ (mx) = φ (x). One also has φ (ax) = φ (x) for a ∈ K × .
φ (y)

Now let y ∈ K[M] be nonzero and x ∈ K[M] be arbitrary. Write y =

∑ ak mk with ak 6= 0 for all
k=1

k = 1, . . . , φ (y). In particular, a1 ∈ K × since K is a semifield. Also, since K is idempotent, we have
that y = a1 m1 + y and hence

φ (xy) = φ (xy + a1 m1 x) ≥ φ (a1 m1 x) = φ (x).
Finally suppose x ∈ K[M]× and let y = x−1 . Then, we have
1 = φ (1) = φ (xy) ≥ φ (x).
Since φ (x) ≤ 1, we have x = am for a ∈ K and m ∈ M. Similarly y = bm′ for b ∈ K and m′ ∈ M. One
easily sees that ab = 1 and mm′ = 1, and hence x ∈ K × × M × .

We can compute Čech cohomology given the existence of an appropriate open cover. To be compatible with the notation in Proposition 3.2, we use OX× for OX∗ . The following result provides a link
between Pic(XK ) and Čech cohomology of the sheaf K × × OX × (of abelian groups) on X , where
K × is the constant sheaf on X associated to the abelian group K × .
Lemma 3.3. Let X be an irreducible monoid scheme, U be an open subset of X , and K be an idempotent semifield. Then
UK = U ×Spec F1 Spec K
is an open subset of XK .
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Proof. We may assume that X is affine, say X = Spec M, and U = D( f ) for some f ∈ M. Then one
can easily check that
UK = {p ∈ Spec K[M] | 1M · f 6∈ p}.
In other words, UK is nothing but D( f ) when f is considered as an element of K[M].

In what follows, we will assume the following condition for an irreducible monoid scheme.
Condition 3.4. Let X be an irreducible monoid scheme. Suppose that X has an open affine cover
U = {Uα } such that any finite intersection of the sets Uα is isomorphic to the prime spectrum of a
cancellative monoid.
Let X be an irreducible monoid scheme and U = {Ui } be an open cover of X . Let
UK := {Ui ×Spec F1 Spec K}.
It follows from Lemma 3.3 that UK is an open cover of XK for an idempotent semifield K.
Now, under the assumption of Condition 3.4, we have the following.
Theorem 3.5. Let X be an irreducible monoid scheme and U be an open cover of X satisfying
Condition 3.4. Let K be an idempotent semifield, XK = X ×Spec F1 Spec K, and UK = {Uα ×Spec F1
Spec K}. Then, we have
H1 (UK , OX×K ) ∼
= H1 (U , K × ×OX× ).
Proof. Let UαK = Uα ×Spec F1 Spec K. Write Uα1 ...αn = Uα1 ∩ . . . ∩ Uαn , and similarly for UαK1 ...αn . Let
F = K × ×OX× , and G = OX×K . Note that by irreducibility of X , we have F (U ) = K × × (OX (U )× )
for any open subset U ⊆ X .
Fix α1 , . . . , αn . Then we have Uα1 ...αn ∼
= Spec M and UαK1 ...αn ∼
= Spec K[M] for some cancellative
×
monoid M. Then F (Uα1 ...αn ) = K × M × ∼
= K[M]× = G (UαK1 ...αn ), by Proposition 3.2. These isomorphisms induce isomorphisms of Čech cochains Ck (U , F ) ∼
= Ck (UK , G ). The result will follow if these isomorphisms are compatible with the differentials. However, this reduces to checking that the maps F (Uα1 ...αn ) → F (Uα1 ...αn αn+1 ) → G (UαK1 ...αn αn+1 ) and F (Uα1 ...αn ) → G (UαK1 ...αn ) →
G (UαK1 ...αn αn+1 ) agree, which is readily checked.

Our next goal is to show that Pic XK can be computed by using a cover that satisfies the assumption
of Condition 3.4.
Lemma 3.6. Let K be an idempotent semifield. Suppose that a, b ∈ K and a 6= 0 or b 6= 0, then
a + b 6= 0.
Proof. Suppose that a + b = 0. Since K is idempotent we have
a + b = a + a + b = a + (a + b) = a + 0 = a.
Similarly, we obtain that a + b = b. It follows that a = b and hence a + b = a = b as K is idempotent.
This implies that a = b = 0, which contradicts the initial assumption.

Lemma 3.7. Let K be an idempotent semifield and M be a cancellative monoid. Let {Uα } be an open
cover of X = Spec K[M]. Then Uα = X for some α .
Proof. We may assume without loss of generality that each open set in the cover is nonempty. We
first consider the case where Uα = D( fα ), for all α . Then for each p ∈ Spec K[M], we have fα 6∈ p for
some α .
Let I be the ideal generated by { fα }. Then I 6⊆ p for any prime p of Spec K[M]. As in the classical
case, since each proper ideal is contained in a maximal ideal and each maximal ideal is prime, we
have I = K[M]. Hence 1 ∈ I so we can write 1 = ∑α fα gα for some fα ∈ K and gα ∈ M. Since K is
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idempotent and M is cancellative, it follows from Lemma 3.6 that 1 = fα gα for some α and hence
fα ∈ K × . Therefore, we have that X = D( fα ).
For the general case, we may refine the open cover to a cover by sets of the form D( fα ). Then one
of the open sets in the refinement is X and hence this is true for the original cover.

Remark 3.8. A special case of Lemma 3.7, when M is a free monoid generated by n elements, is
proved in [Jun17, Lemma 4.20] and referred to as a “tropical partition of unity”.
Theorem 3.9. Let K be an idempotent semifield, M be a cancellative monoid, and X = Spec K[M].
Then Pic(X ) = 0, and more generally, we have
Hk (X , OX× ) = 0,

for k ≥ 1.

Proof. We in fact show that the sheaf cohomology of any sheaf of abelian groups on X vanishes. It
suffices to show that, for any sheaves F and G of abelian groups, if f : F → G is surjective then
the corresponding map F (X ) → G (X ) is surjective. From the surjectivity of f : F → G , it follows
that for every open set U ⊆ X , and for every s ∈ G (U ), there is a covering {Ui } of U , and elements
ti ∈ F (Ui ) such that f (ti ) = s|Ui for all i. From Lemma 3.7 the result follows.

Corollary 3.10. Let X be an irreducible monoid scheme and U = {Uα } be an affine open cover
satisfying Condition 3.4. Let K be an idempotent semifield and let XK = X ×Spec F1 Spec K. Let
UK = {Uα ×Spec F1 Spec K} be a cover of XK coming from U . Then, we have
H1 (XK , OX×K ) = H1 (UK , OX×K ).
Proof. Combine the previous result with the Serre’s version ( [Ser55, Théorème 1 of n◦ 29]) of Leray’s
theorem.

We now consider similar results for the cohomology of K
analogue of Lemma 3.7 for cancellative monoids.

× ×O × .
X

The following lemma is an

Lemma 3.11. Let M be a cancellative monoid. Let {Uα } be an open cover of X = Spec M. Then
Uα = X for some α .
Proof. As in Lemma 3.7, we may assume that each Uα is nonempty and satisfies Uα = D( fα ) for
some fα ∈ M. Let I be the ideal of M generated by { fα }. Then I is not contained in any prime
ideal. Since the maximal ideal of any monoid is prime then I is not contained in the maximal ideal
m = (M − M × ). Hence 1 ∈ I. Then 1 = fα g for some α and some g ∈ M. Then fα is a unit as
desired.

Corollary 3.12. Let M be a cancellative monoid and X = Spec M. Then, we have
Hk (X , K

×

×OX× ) = 0,

for k ≥ 1.

Proof. A similar argument to the one of Theorem 3.9 yields the desired result.



Corollary 3.13. Let X be an irreducible monoid scheme and U = {Uα } be an affine open cover of
X satisfying Condition 3.4. Then, we have
H1 (X , K

×

×OX× ) = H1 (U , K

×

×OX× ).

Proof. Combine the previous result with the Serre’s version ( [Ser55, Théorème 1 of n◦ 29]) of Leray’s
theorem.

We are now able to express Pic XK in terms of X .
Proposition 3.14. Let X be an irreducible monoid scheme and U = {Uα } is an affine open cover
satisfying Condition 3.4. Let K be an idempotent semifield and let XK = X ×Spec F1 Spec K. Then, we
have
Pic(XK ) ∼
= H1 (X , K × ×OX× ).
8

Proof. Let UK = {Uα ×Spec F1 Spec K}. Then, by Theorem 3.5 and Corollaries 3.10 and 3.13, we have
Pic(XK ) ∼
= H1 (UK , OX×K ) ∼
= H1 (U , K

×

×OX× ) ∼
= H1 (X , K

×

×OX× ).


Proposition 3.15. Let X be an irreducible monoid scheme. Then, we have
Pic(X ) = H1 (X , K

×

×OX× ).

Proof. Since the constant sheaf on an irreducible space is flasque, we have that H1 (X , K × ) = 0.
Then, it follows that
H1 (X , K

×

×OX× ) = H1 (X , K × ) × H1 (X , OX× ) = 0 × Pic(X ) = Pic(X ).


Combining the two previous propositions gives the following theorem.
Theorem 3.16. Let X be an irreducible monoid scheme and U = {Uα } is an affine open cover
satisfying Condition 3.4. Let K be an idempotent semifield and let XK = X ×Spec F1 Spec K. Then,
∼ Pic(X ).
Pic(XK ) =
4.

Cartier divisors on cancellative semiring schemes

In this section, we construct a Cartier divisor on a cancellative semiring scheme X , following
the idea of Flores and Weibel [FW14]. We show that the Picard group Pic(X ) is isomorphic to the
group Cart(X ) of Cartier divisors modulo principal Cartier divisors. In what follows, by an integral
semiring, we mean a semiring without zero divisors.
Let A be an integral semiring and p ∈ Spec A. We will call an element f ∈ A× (multiplicatively)
cancellable, if the following condition holds:
a f = b f implies a = b,

∀a, b ∈ A.

(2)

By a cancellative semiring, we mean a semiring A such that any nonzero element a ∈ A is cancellable.
Definition 4.1. By an integral semiring scheme, we mean a semiring scheme X such that for any
open subset U of X , OX (U ) is an integral semiring. Moreover, if OX (U ) is cancellative for any open
subset U of X , we call X a cancellative semiring scheme.
Remark 4.2. Classically, a ring without zero divisors is multiplicatively cancellative and vice versa.
However, a semiring without zero divisors is not necessarily cancellative - such as the polynomial
semiring over the tropical semifield T[x1 , . . . , xn ]. Nonetheless, any cancellative semiring is integral
and hence any cancellative semiring scheme is an integral semiring scheme.
Lemma 4.3. Let X be an integral semiring scheme. Then X has a unique generic point.
Proof. As in the classical case, if X is an integral semiring scheme, then X is irreducible and any
irreducible topological space has a unique generic point.

Lemma 4.4. Let X be an integral semiring scheme with a generic point η . Then, for any affine open
subset U = Spec A ⊆ X , we have
OX,η ≃ A(0) ,
where A(0) := Frac(A). In particular, OX,η is a semifield.
Proof. Since X is integral, A is an integral semiring. In particular, p = (0) ∈ Spec A is the generic
point of Spec A. Again, since X is integral, it follows that p is also the generic point of X . Therefore

we have OX,η = A(0) .
9

Definition 4.5. Let X be an integral semiring scheme and U = Spec A be any affine open subset. We
define the function field K(X ) of X as follows:
K(X ) := OX,η .
Example 4.6. Let X = Spec T[x], an affine line. Let A = T[x]. One can easily see that A is an integral
semiring and hence the generic point is η = (0). Therefore, we have
g
K(X ) = OX,η := { | g ∈ A, f ∈ A\{0}} = T(x).
f
Proposition 4.7. Let X be a cancellative semiring scheme with the function field K. Let K be the
constant sheaf associated to K on X . Then OX× is a subsheaf (of abelian groups) of K × .
Proof. First, suppose that X is affine, i.e., X = Spec A for some cancellative semiring A. Notice that,
since A is cancellative, for any f ∈ A, we have a canonical injection i f : A f → Frac(A). Therefore, as
in the classical case, for each open subset U of X , we have
\

OX (U ) =

A f ⊆ Frac(A) = K (U ).

(3)

D( f )⊆U

It follows that OX× (U ) ⊆ K × (U ). In general, one can cover X with affine open subsets and the
argument reduces to the case when X is affine; this is essentially due to Lemma 4.4.

Thanks to Proposition 4.7, we can define a Cartier divisor on a cancellative semiring scheme X as
follows:
Definition 4.8. Let X be a cancellative semiring scheme with the function field K. Let K be the
constant sheaf associated to K on X . A Cartier divisor on X is a global section of the sheaf of abelian
groups K × /OX× .
We recall the notion of Cartier divisors on a cancellative monoid scheme. Let X be a cancellative
and irreducible monoid scheme with a generic point η . Denote by K := OX,η the stalk at η and by
K be the associated constant sheaf.
Definition 4.9. [FW14, §6]
(1) A Cartier divisor is a global section of the sheaf K × /OX× of abelian groups. We let Cart(X )
be the group of Cartier divisors on X .
×
(2) A principal Cartier divisor is a Cartier divisor which is represented by some a ∈ OX,
η . Let
P(X ) be the subgroup of Cart(X ) consisting of principal Cartier divisors.
(3) We let CaCl(X ) := Cart(X )/P(X ) be the group of Cartier divisors modulo principal Cartier
divisors.
Flores and Weibel prove the following.
Theorem 4.10. [FW14, Proposition 6.1.] Let X be a cancellative monoid scheme. Then
Pic(X ) ≃ CaCl(X ).
Now, as in the classical case, for any Cartier divisor D, one can associate an invertible sheaf L (D)
on X .
In fact, the same argument as in the proof of Theorem 4.10 shows the following:
Proposition 4.11. Let X be a cancellative semiring scheme. Then the map ϕ : CaCl(X ) → Pic(X )
sending [D] to [L (D)] is an isomorphism, where [D] is the equivalence class of D ∈ Cart(X ) and
[L (D)] is the equivalence class of L (D) in Pic(X ).
10

Proof. The argument is similar to [FW14, Propoisiton 6.1], however, we include a proof for the sake
of completeness.
We have the following short exact sequence of sheaves of abelian groups:
0 −→ OX× −→ K

×

−→ K × /OX× −→ 0.

(4)

×

×

Since X is cancellative, X is irreducible. Furthermore, K is a constant sheaf on X and hence K is
flasque. In particular, H1 (X , K × ) = 0. Therefore, the cohomology sequence induced by (4) becomes:
f

0 −→ OX× (X ) −→ K × (X ) −→ (K × /OX× )(X ) −→ H1 (X , OX× ) −→ 0.
But, we have that H1 (X , OX× ) = Pic(X ), (K × /OX× )(X ) = Cart(X ), and

f (K

(5)

× (X )) = P(X ) and hence

Pic(X ) ≃ CaCl(X ) as claimed.



Example 4.12. Let A := T[x1 , ..., xn ] be the polynomial semiring over T. Recall that A is integral, but
not cancellative. Consider B := A/ ∼, where ∼ is a congruence relation on A such that f (x1 , ..., xn ) ∼
g(x1 , ..., xn ) if and only if f and g are same as functions on Tn . It is well–known (see for example
[Jun17] or [JM17]) that in this case B is cancellative. Now let X = Spec B. It follows from Proposition
4.11 that Pic(X ) = CaCl(X ). But, we know from [Jun17, Corollary 4.23.] that Pic(X ) is the trivial
group and hence so is CaCl(X ). In tropical geometry, one may obtain a “reduced model” of a tropical
scheme as above, which could be used to compute CaCl(X ), see [Jun17, Remark 4.27.].
Remark 4.13. One can easily see (cf. [FW14]) that if X is a cancellative monoid scheme and K is a
field, then CaCl(X ) is isomorphic to CaCl(XK ), where CaCl(XK ) is the group of Cartier divisors of
the scheme XK modulo principal Cartier divisors.
Remark 4.14. In the case of set-theoretic tropicalizations of curves, there is a well-developed theory
of divisors. Let C be an algebraic curve defined over a valued field and G be the set-theoretic tropicalization of C. We remind the reader, that G is a graph. There exists a notion of a Picard group of
G (cf. [BN07]). Moreover, there is a well-defined map Pic(C) → Pic(G), which one may think of
as the tropicalization map. If C has genus 0 then the Picard groups Pic(C) and Pic(G) are equal and
isomorphic to Z. However, in general Pic(C) and Pic(G) are rarely the same. For example, consider
an elliptic curve E degenerating to a cycle of P1 ’s. Note that Pic0 (E) = E but Pic(G) = S1 , the unit
circle.
Example 4.15 (Computing the Picard group of the tropicalization of P1 × P1 ). We consider the
quadric surface XC = P1 × P1 , which is a toric variety. We compute explicitly the Picard group
of the tropical scheme XT . Note that the calculation is analogous to the classical one.
We can do the computation on P1 × P1 or identify it with its image under the Segre embedding into
3
P , i.e., XT is defined by a congruence generated by a single “bend relation” x0 x1 ∼ x2 x3 . The two
computations are analogous.
Let U = ∪4i=1Ui be an open cover for XT = (P1 × P1 )T , where
U1 = Spec T[x, y],
U2 = Spec T[x, y−1 ],
U3 = Spec T[x−1 , x−1 y−1 ],
U4 = Spec T[x−1 , xy].
Now we can see which sections over each Ui are units in Γ(U, OXT ), namely,
Ai := OX×T (Ui ) = R,
OX×T (U1 )

∀i = 1, 2, 3, 4.

(6)

For instance, for i = 1, we have that
is the group of (tropically) multiplicatively invertible
elements in OXT (U1 ) = T[x, y]. Since the only tropical polynomials, which are multiplicatively invertible, are nonzero constants, we have (6).
11

Let Ui j := Ui ∩U j . Now we have
A12 := OX×T (U12 ) = (T[x, y±1 ])× ∼
= R × Z,
and similarly A14 , A23 , A34 are isomorphic to R × Z. Note, however, that
A13 ∼
= A24 = T([x±1 , y±1 ])× ∼
= R × Z2 .
Similarly for Ui jk := Ui ∩U j ∩Uk one may also check that:
Ai jk := OX×T (Ui jk ) = (T[x±1 , y±1 ])× ∼
= R × Z2 .
In particular, we get that
Č0 (XT , O × ) ∼
= R4 ,
XT

Č1 (XT , OX×T ) ∼
= (R × Z)4 × (R × Z2)2 ,

Č2 (XT , OX×T ) ∼
= (R × Z2 )4 .

(7)

We can proceed with the computation using the usual Čech complex or we can do a “tropical”
computation as in [Jun17] considering the following cochain complex:
d0+
d0−

d1+

// A12 × A13 × A23 × A14 × A24 × A34

d2+

//
d1−

A123 × A124 × A134 × A234

//
d2−

···

(8)

The complex (8) was introduced to deal with the lack of subtraction in tropical geometry; instead
of one differential we consider pairs of morphisms (di+ , di− ). Also, instead of the difference of two
functions f − g, we will have a tuple ( f , g) and we replace the kernel condition f − g = 0 with
( f , g) ∈ ∆, where ∆ is the diagonal. However, since OX× is a sheaf of abelian groups, we can simply
use the classical cochain complex.
Either way, we can see that the image of d0 is generated by elements of the form ( f1 , f2 , f3 , f4 ), where
fi ∈ R for all i and ( f1 , f2 , f3 , f4 ) 6= λ (1, 1, 1, 1) for some λ 6= 0 and thus the image of d0 is isomorphic
to R3 . The kernel of d1 is generated by elements of the following form:
b
ac
a
( yk , bxl yk , cxk , axl , xl yk , yk ),
c
b
b

for a, b, c ∈ R and k, l ∈ Z.

(9)

Now, for the choice of ( f1 , f2 , f3 , f4 ) = (b, c, 1T , bc ), one can easily see that any element as in (9)
defines the same equivalence class as an element (yk , xl yk , xk , xl , xl yk , yk ) in Ȟ1 (XT , OX×T ). Now it is
easy to see that
H1 (XT , OX×T ) = Ȟ1 (XT , OX×T ) ∼
= Z2 .
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B INGHAMTON U NIVERSITY, B INGHAMTON , NY 13902, USA
E-mail address: jjun@math.binghamton.edu
YALE U NIVERSITY, N EW H AVEN , CT 06511, USA
E-mail address: kalina.mincheva@yale.edu
E-mail address: jeff.tolli@gmail.com

13

