Tropical geometry over the tropical hyperfield
Oliver Lorscheid

TL;DR
This paper develops a new framework for tropical scheme theory by integrating tropical hyperfields with ordered blueprints, enabling a scheme-theoretic approach to tropicalization and linking it to Berkovich analytification.
Contribution
It introduces a novel scheme-theoretic tropicalization method using tropical hyperfields and ordered blueprints, connecting classical varieties with tropical geometry.
Findings
Characterizes Berkovich analytification via scheme-theoretic tropicalizations.
Shows Giansiracusa bend relations derive from scheme-theoretic tropicalization.
Provides a topological description of tropicalization and analytification.
Abstract
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety. Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.
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Tropical geometry over the tropical hyperfield
Oliver Lorscheid
Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Abstract.
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety.
Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.
The author thanks the Max Planck Institute for Mathematics that hosted and supported him during the preparation of this manuscript.
Contents
- 1 Ordered blueprints
- 2 Tropicalization as a base change to the tropical hyperfield
- 3 The Kajiwara-Payne tropicalization as a rational point set
- 4 The relation between the tropical hyperfield and the tropical semifield
- 5 Recovering the Giansiracusa bend
Introduction
History of the tropical hyperfield
While hyperrings were defined as early as 1956 by Krasner ([13]), the tropical hyperfield was introduced more recently in 2011 by Oleg Viro ([23]), with a reformulation of tropical geometry in mind. Around the same time Connes and Consani ([6]) recognized the relevance of hyperfields for absolute arithmetic and later found back Viro’s tropical hyperfield from their point of view ([7]). A closely related notion are Izhakian’s extended tropical numbers ([9]), which in fact were introduced before the tropical hyperfield, though the relation between these objects was understood only later (cf. Remark 1.8).
In 2013, the seminal paper [8] by Jeffrey and Noah Giansiracusa inaugurated tropical scheme theory, a new branch of tropical geometry that seeks for a scheme theoretic formulation of tropical geometry. Maclagan and Rincón ([18]) showed soon after that the weights of tropical varieties are encoded in the scheme structure, and the author ([15]) put this theory on a more sophisticated footing using ordered blueprints. This latter paper contains the observation that hyperrings are ordered blueprints, which provides a scheme theory for hyperrings as a byproduct. A variation of algebraic geometry over hyperrings was developed independently by Jun ([11], [12]) while exploring the relation to tropical geometry from an altered angle.
In 2016, Baker and Bowler ([2]) formulated matroid theory with coefficients in a hyperfield. In particular, matroids over the tropical hyperfield turn out to valuated matroids, aka tropical linear spaces following Speyer ([21]). A joint follow-up work of Baker and the author ([4]) uses scheme theory for ordered blueprints to construct moduli spaces of matroids. In particular, the moduli space of tropical linear spaces, aka the Dressian, is an object over the tropical hyperfield.
Intention and scope of this text
Our exposition is meant as a reader friendly introduction to tropical scheme theory from the particular perspective of the tropical hyperfield. While it is build on ideas and theories that were developed in the aforementioned works, this text explores a new mixture of hyperfields with algebraic geometry for ordered blueprints. As a result, we gain a frame work for tropical scheme theory that has certain advantages over previous approaches that are based on the tropical semifield.
Since a nonarchimedean absolute value can be interpreted as a morphism into the tropical hyperfield , we can define the scheme theoretic tropicalization of a classical variety literally as the base change along this morphism into . One of our main efforts in this text is to show that the Kaijiwara-Payne tropicalization emerges from the scheme theoretic tropicalization as the set of -rational points, including a characterization of its topology. A surprising insight to us is that the Giansiracusa bend relations appear naturally from the scheme theoretic tropicalization by enforcing the relation .
All this makes us believe that the tropical hyperfield is a promising tool for tropical scheme theory, and we hope that this text stimulates future developments in this direction.
The tropical hyperfield
Let us introduce the protagonist of our text, which acts out as a subtle variant of the tropical semifield . We begin with a description of , which appears in different incarnations in the literature: while the min-plus-algebra and the max-plus-algebra support the piecewise linear aspect of tropical varieties by using logarithmic coordinates, the Berkovich model is a simpler object from an algebraic perspective. It is this latter model of that we employ in our text. Its underlying set is , its multiplication is the usual multiplication of real numbers and its addition is defined by the rule where the maximum is taken with respect to the usual linear ordering of the real numbers.
The tropical hyperfield has the same underlying set and the same multiplication as , but the addition of gets replaced by the hyperaddition that associates with two elements the following subset a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b of :
[TABLE]
Advantages of the tropical hyperfield
In the following, we will list a number of advantages of the tropical hyperfield over the tropical semifield , which shall underline the potential of for tropical scheme theory.
Property 1**.**
The hyperaddition of is characterized by the property that for a field , the strict triangular inequality of a nonarchimedean absolute value is equivalent with the condition v(a+b)\in v(a){\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}v(b). This allows us to consider nonarchimedean absolute values as morphisms in a suitable category (Theorem 2.2). Thus we can consider the base change of a variety over to .
Property 2**.**
The hyperaddition of provides us with a notion of additive inverses: for every element , there is a unique element such that 0\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b, namely . More generally, we have 0\in a_{1}{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}\dotsb{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}a_{n} if and only if the maximum occurs twice among the summands (Lemma 1.7). This allows us to reformulate the corner locus of a tropical polynomial as the set of points such that (Remark 3.3).
Property 3**.**
A tropical linear space is the geometric realization of a valuated matroid or a -matroid in the language of [2]. Therefore it seems natural to use as a basis for tropical geometry. In particular, the moduli space of tropical linear spaces is an object defined over , as explained in [4].
Property 4**.**
In contrast to the polynomial ring , the ambient semiring of the free algebra over is a domain, i.e. the multiplication by nonzero elements defines an injective map. This might be helpful to establish a theory of prime ideals and to formulate a criterion for the irreducibility of a tropical scheme. For some details on , cf. [3, Appendix A].
Ordered blueprints
While hyperfields and hyperrings are coming short of certain properties that are required for tropical geometry, such as free objects and tensor products, the more ample notion of ordered blueprints has proven to be a suitable tool for tropical scheme theory (cf. [4] and [15]). Therefore we will refrain from spelling out the axiomatic of hyperfields, but we rather consider , along with other algebraic objects of interest, as ordered blueprints.
In this text, all semirings are commutative with [math] and . An ordered blueprint is a triple where is a semiring, is a multiplicatively closed set of generators of that contains [math] and and is a partial order on that is additive and multiplicative, i.e. implies and for all . We say that is generated by a set of relations if it is the smallest additive and multiplicative partial order that contains the relations .
We can realize the tropical hyperfield as the following ordered blueprint : its underlying monoid is the set of nonnegative real numbers together with the usual multiplication, its ambient semiring is the monoid semiring of finite formal sums of positive real numbers , and its partial order is generated by the relations for which c\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b.
Let and be ordered blueprints. A morphism is a map with , and for all that extends (necessarily uniquely) to an order-preserving semiring homomorphism . This defines the category of ordered blueprints.
Valuations as morphisms
Let be a field and a nonarchimedean absolute value, i.e. , , and whenever .
Using the identification , the relation is equivalent with where is the partial order of .
We associate with the ordered blueprint where is the multiplicative monoid of , is the group semiring generated by and is generated by all relations (considered as elements of ) for which in .
Under these identifications, defines a multiplicative map that extends to an order-preserving semiring homomorphism . In other words, the nonarchimedean absolute value corresponds to a morphism of ordered blueprints (Theorem 2.2).
Tropicalization as a base change
For an ordered blueprint , we denote by the corresponding object of the dual category of and call it an affine ordered blue scheme. Note that the category contains tensor products , this is, the colimits of diagrams of the form .
Given a morphism of ordered blueprints, we define the scheme theoretic tropicalization of along as \operatorname{Spec}\big{(}B\otimes_{{\mathbf{k}}}{\mathbf{T}}\big{)}, which can be thought of as the base change of from to . Note that the tensor product comes with a canonical morphism .
The Kajiwara-Payne tropicalization
Let be an affine -scheme with coordinate ring . While the Berkovich analytification of is defined intrinsically as the set of seminorms that extend to , the tropicalization of requires an additional choice of coordinates, e.g. in form of a closed immersion into an affine toric variety over . Pulling back global sections defines a map . The Kajiwara-Payne tropicalization of is defined as the image of the map
[TABLE]
See Remark 3.2 for the relation with more common definition in terms of the bend locus.
The Kajiwara-Payne tropicalization as a rational point set
Jun observes in [11] that the Berkovich analytification of corresponds to the hyperring morphism from , considered as a hyperring, into the tropical hyperfield . We transfer this approach to ordered blueprints, which allows us to recover both the analytification and the tropicalization of as -rational point sets of the following scheme theoretic tropicalizations.
We associate with the following ordered blueprint . Its underlying monoid is the multiplicative monoid of . Its ambient semiring is the monoid semiring modulo the identification of with the empty sum. Its partial order is generated by the relation (considered as elements of ) for which in . We define .
Let be the surjection that pulls back global sections along the closed immersion . Let be the following ordered blueprint. Its underlying monoid is the submonoid of . Its ambient semiring is the monoid semiring modulo the identification of with the empty sum. Its partial order is generated by the relation (with ) for which in . We define .
Note that the inclusion induces a -linear morphism . We define the sets of -linear morphisms
[TABLE]
Composing with defines a map .
Theorem A**.**
There are natural bijections and such that the diagram
[TABLE]
commutes.
In fact, the bijections in this statement are homeomorphisms with respect to topologies that stem from the Euclidean topology of ; cf. section 2.4 for details.
The Giansiracusa bend
Jeff and Noah Giansiracusa introduce in [8] the bend relation for tropical polynomials, which allows them to prove an analogous result to Theorem A for scheme theoretic tropicalizations over the tropical semifield . The following result shows that the scheme theoretic tropicalization over recovers the Giansiracusa bend in a natural way.
Recall the context of the Kajiwara-Payne tropicalization: is a field with nonarchimedean absolute value and is a -scheme together with a closed immersion into a toric -variety of the form .
The Giansiracusa bend of (along with respect to ) is defined as the spectrum of the quotient of the free -algebra of finite -linear combinations of elements by the relations of the form
[TABLE]
for and with in .
We consider the field with one element as the ordered blueprint {{\mathbb{F}}_{1}}=\big{(}\{0,1\},{\mathbb{N}},=\big{)} and the Boolean semifield as the ordered blueprint {\mathbb{B}}=\big{(}\{0,1\},\{0,1\},=\big{)} where “” stand for the trivial partial order and the addition of is characterized by the equation .
Theorem B**.**
There is a canonical isomorphism .
In fact, we prove a stronger version of Theorem B, in which we express the refinement of as a blueprint in terms of . While this latter result can be found as Theorem 5.6 in this text, Theorem B appears as Corollary 5.7.
The guiding example
We illustrate the main concepts and results of this paper at the appropriate positions in the case of the standard plane line defined by the polynomial . We summarize these explanations in the following in order to exemplify Theorems A and B.
Let be a field with nonarchimedean absolute value . Let be the closed subscheme of the affine plane over that is defined by the polynomial , which comes with a closed immersion and coordinate ring .
The set theoretic tropicalization
The tropicalization of is
[TABLE]
which is also called the bend locus of ; cf. section 3.2 for details and Example 2.9 for an illustration.
The scheme theoretic tropicalization
We turn to a description of the scheme theoretic tropicalization of with respect to its embedding into the affine plane over . Let be the morphism associated with , cf. Theorem 2.2. The associated ordered blueprint is as follows: its ambient semiring is the polynomial algebra
[TABLE]
where is the group semiring of finite formal sums of elements of . Its underlying monoid consists of all terms of the form where and . Its partial order is generated by the relations
[TABLE]
By Lemma 3.6, the tropicalization of has the following explicit description: the association defines an isomorphism
[TABLE]
of semirings that identifies the underlying monoid of with the submonoid of that consists of all terms of the form with and . The partial order of coincides with the partial order of that is generated by the defining relations of the partial order of together with the relations
[TABLE]
cf. Example 3.7 for details.
Recovering the set theoretic tropicalization
Theorem A asserts that the tropicalization equals the set of -linear homomorphisms . Mapping to \big{(}f(T_{1}),f(T_{2})\big{)} defines a bijection of with
[TABLE]
By Lemma 1.7, each of the four defining relations on is equivalent with the condition that the maximum among , and occurs twice. Thus this latter set is precisely , as claimed in Theorem A.
The Giansiracusa bend
We turn to Theorem B, which exhibits the bend relations in terms of the scheme theoretic tropicalization. In our example, the Giansiracusa bend is the semiring
[TABLE]
cf. Example 5.1. On the other hand, we have the identifications
[TABLE]
which show that holds for every in , cf. section 4.2. Under the identification , we obtain that
[TABLE]
(using ) and that
[TABLE]
(using ) holds in . Thus we gain the equality in . Repeating the same argument with the roles of , and exchanged yields the defining relations
[TABLE]
of . This illustrates Theorem B in our example.
Remark on variations
Let be as before. Whenever we have a morphism into some ordered blueprint , we can consider the base change \operatorname{Spec}\big{(}B\times_{\mathbf{k}}C\big{)} of an affine ordered blue -scheme to along this morphism, which should be thought of as the scheme theoretic tropicalization of over .
Tropicalizations along the following morphisms might produce interesting theories. First of all, we can consider higher rank valuations (where we use the exponential notation) as a morphism where is the ordered blueprint with underlying monoid , with ambient semiring and with the partial order that is generated by relations of the form
[TABLE]
for which there is an such that for and such that the maximum among , and appears twice.
Another interesting example is the sign map , which can be interpreted as a morphism where is the ordered blueprint associated with and where is the sign hyperfield, which has the following shape as an ordered blueprint. Its underlying monoid is , its ambient semiring is (where has to be understood as a symbol and not as an additive inverse of ) and its partial order is generated by the relations
[TABLE]
Tropicalizations along might be useful to study real algebraic varieties. In particular, this might bring new insights to questions around the (disproven) Macphersonian conjecture; cf. [19] and [14].
A variation of the sign map is the phase map , which assigns to a nonzero complex number its argument on the unit circle . This map can be realized as a morphism where is associated with and where is the phase hyperfield. We omit a description of , but refer to section 2.1 in [1] for details and further variations.
Divergence in notation
In this text, we aim for a simplified account of scheme theoretic tropicalization, in contrast to the broader context of [15]. The specific situation of this paper—namely, the restriction to affine ordered blue schemes and the fixed tropicalization base —allows us to simplify the exposition of our results considerably. In the following, we point out the major differences to [4], [15] and [17] in order to avoid confusion when comparing these writings.
Most notably, we are using different incarnations of the tropical numbers in this text, whose notations deviate from that used in [4], [15] and [17]. Namely, in [15] and [17] the symbol is used for both the tropical semifield, which is denoted by in this text, and the associated ordered blueprint , which agrees with the notation in this text. In [4], we use for the tropical hyperfield, which is denoted by in this text and by in [17].
The reason for us to use in [15] and [17] the symbol for both the tropical semifield and the associated blueprint is that we identify a semiring with the associated blueprint . Since for this text a different realization of semiring as ordered blueprints stays in the foreground, we make a clear distinction between these different objects.
This distinction has the advantage that there is no ambiguity between the free semiring over a semiring , whose elements are polynomials, and the free ordered blueprint over the associated ordered blueprint , whose elements are monomials with coefficients in .
Content overview
In section 1, we review the definition of and some basic facts for ordered blueprints. In section 2, we introduce scheme theoretic tropicalizations, which includes the interpretation of absolute values as morphisms of ordered blueprints, as well as some results on the natural topology for -rational point sets. In section 3, we exhibit the Kajiwara-Payne tropicalization as the -rational point set of a scheme theoretic tropicalization. In section 4, we explain the relation between the tropical hyperfield and the tropical semifield. In section 5, we recover the Giansiracusa bend from the scheme theoretic tropicalization.
Acknowledgements
The author thanks Matt Baker and Sam Payne for their help with preparing this text.
1. Ordered blueprints
In this section, we introduce ordered blueprints and some basic constructions such as free algebras, quotients and tensor products. We will explain how we can consider monoids and semirings as ordered blueprints and finally introduce the tropical hyperfield in its incarnation as an ordered blueprint. For more details on ordered blueprints we refer to [4], [15] and [17].
1.1. Basic definitions
In this text, a semiring is always commutative and with [math] and , i.e. both and are commutative monoids, multiplication distributes over addition and for all .
An ordered blueprint is a triple where is a semiring, is a subset of and is a partial order on such that
- (i)
is closed under multiplication, contains [math] and and generates as a semiring; 2. (ii)
is additive and multiplicative, i.e. implies and for all .
We call the underlying monoid, the ambient semiring and the partial order of the ordered blueprint .
We typically denote the elements of by , with , and , and the elements of by , , and or , , and where we assume that the , , and are elements of . Note that every element of is indeed a sum of elements in .
We consider as the underlying set of the ordered blueprint , and we say that is an element of if .
A morphism of ordered blueprints is a multiplicative map with and that extends to an order preserving semiring homomorphism . Note that is uniquely determined by since generates as a semiring. This defines the category of ordered blueprints.
In the following, we will introduce several constructions and subclasses of ordered blueprints as well as several explicit examples of ordered blueprints.
1.2. Free algebras
Let be an ordered blueprint. An ordered blue -algebra is an ordered blueprint together with a morphism , which we call the structure map of . We often refer to an ordered blue -algebra by without mentioning the structure map explicitly. A -linear morphism between two ordered blue -algebras and is a morphism of ordered blueprints that commutes with the structure maps of and .
Let be an ordered blueprint and be a commutative and multiplicatively written monoid. We define the free ordered blue -algebra in as the following ordered blue -algebra . Its ambient semiring is the semiring
[TABLE]
whose addition is defined componentwise and its multiplication extends the multiplication of linearly. We consider as a submonoid of and we write for the element with and for . In particular, we consider as an element of .
The underlying monoid of is defined as the subset
[TABLE]
and its partial order is generated by the relations of the form with whenever in .
The association defines a morphism of ordered blueprints , which endows with the structure of an ordered blue -algebra. It satisfies the universal property that every monoid morphism into the underlying monoid of an ordered blue -algebra extends uniquely to a morphism of ordered blue -algebras.
Example 1.1**.**
In the case that consists of the monomials in , we write , which we think of as the “polynomial algebra” over . Its ambient semiring is the usual polynomial semiring and its underlying monoid consists of the monomials with coefficient .
Another example of importance for tropical geometry are Laurent polynomial algebras, which are the free algebras associated with the group of Laurent monomials. In this case, we write . Its ambient semiring is the usual semiring of Laurent polynomials and its underlying monoid consists of the Laurent monomials with coefficient .
1.3. Algebraic blueprints
Let be a semiring. The trivial partial order on is the partial order with only if for all . We will refer to the trivial partial order by . An algebraic blueprint, or simply blueprint, is an ordered blueprint whose partial order is trivial.
Let be an ordered blueprint. Then its algebraic core is the blueprint where we replace the partial order of by the trivial partial order . The identity map on defines a morphism of ordered blueprints.
Example 1.2**.**
The Boolean semifield is the semiring whose multiplication is determined by the axioms for [math] and and whose addition is determined by the rule . We identify with the algebraic blueprint and call by abuse of language the Boolean semifield.
1.4. Monoids with zero
A monoid with zero is a multiplicatively written commutative monoid with a distinguished element [math], called the zero of , that satisfies for all . We define as the quotient of by the relation that identifies the zero [math] of with the additively neutral element of .
Note that every element of can uniquely written as a sum of nonzero elements . In particular, the map embeds as a submonoid of . Therefore is a blueprint, which we call the blueprint associated with .
This allows us to associate the algebraic blueprint with a monoid with zero .
Note further that the semiring satisfies the universal property that every multiplicative map into a semiring with and extends uniquely to a semiring morphism .
Example 1.3**.**
The field with one element is the ordered blueprint , which is associated with the monoid . It is an initial object in , which means that there is a unique morphism into any other ordered blueprint .
1.5. Semirings
Let be a semiring. We denote the multiplicative monoid of by . The associated monomial ordered blueprint is the ordered blueprint where is generated by the (left) monomial relations for which in . Note that the underlying set of is itself.
This association is functorial in the sense that a homomorphism of semirings is tautologically a morphism between the associated monomial ordered blueprints, which we denote by . This embeds the category of semirings as a full subcategory into .
We sometimes denote the associated monomial ordered blueprint by a boldface letter, e.g. by and where is typically a field.
Note that this construction differs from the realization of a semiring as the algebraic blueprint from [15, section 2.10]; we will encounter this latter construction, applied to , in section 4.1 of this text.
Remark 1.4**.**
At first sight, the definition of might seem unmotivated. To give some intuition, we explain its consistency with the association of rings with ordered blueprints passing through hyperrings. Namely, given a ring , one defines a hyperaddition by the rule a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b=\{a+b\}, which turns into a hyperring.
Given a hyperring , the relation c\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b is not symmetric, but monomial in the argument on the left hand side. This leads to the realization of the hyperring as the ordered blueprint where is the multiplicative monoid of and is generated by the monomial relations for which c\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b in . Also cf. Remark 2.8 in [15].
1.6. Quotients by relations
Given an ordered blueprint and a set of relations with , we define the ordered blueprint as the following triple . Let be the smallest preorder on that contains and and that is closed under multiplication and addition. We write if and . Then is an equivalence relation on , and we define as , which inherits naturally the structure of an ordered blueprint since is closed under multiplication and addition. The preorder induces a partial order on , which turns into an ordered semiring. The multiplicative subset is defined as the image of under the quotient map .
The quotient comes with a canonical morphism that satisfies the universal property that for every morphism such that holds in for every in , there is a unique morphism such that ; cf. [17, Prop. 5.3.2] for a proof.
Example 1.5**.**
It is immediate that we have (cf. Example 1.2) where stands for and .
Another example is , which we will encounter again in section 4.3. Its underlying monoid is and its ambient semiring is . In order to determine the partial order of consider two natural numbers and and assume that is larger than , i.e. for some . Then the relation implies that (multiply by ) and (add to ). We conclude that the partial order of is the natural linear order of .
1.7. Tensor products
The category of ordered blueprints is complete and cocomplete. In particular, the tensor product of three ordered blueprints , and with respect to morphisms and exists and satisfies the universal property of a pushout of the diagram . The tensor product can be constructed as follows.
The semiring is the usual tensor product of commutative semirings, whose elements are classes of finite sums of pure tensors with respect to the usual identifications. The monoid is defined as the subset of all pure tensors of for which and . The partial order on is defined as the smallest partial order that is closed under addition and multiplication and that contains all relations of the forms
[TABLE]
for which in and in , respectively.
Example 1.6**.**
The tensor product satisfies the usual compatibilities with free algebras and quotients. For example, we have
[TABLE]
1.8. The tropical hyperfield
In this section, we shall introduce Viro’s tropical hyperfield in its incarnation as an ordered blueprint. As a hyperfield, it is defined as together with the usual multiplication and the hyperaddition
[TABLE]
where the maximum is taken with respect to the natural linear order of . In other words, c\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b if and only if the maximum among , and occurs twice.
Let be the multiplicative monoid of and ({\mathbb{R}}_{\geqslant 0}^{\bullet})^{\textup{alg}}=\big{(}{\mathbb{R}}_{\geqslant 0}^{\bullet},({\mathbb{R}}_{\geqslant 0}^{\bullet})^{+},=\big{)} the associated algebraic blueprint, cf. section 1.4. Intuitively, the ordered blueprint version of the tropical hyperfield results from a symbolic exchange of the relation c\in a{\,\raisebox{-1.1pt}{\larger[-0]{\boxplus}}\,}b by . Looking more closely at the ordered blueprint
[TABLE]
reveals that its underlying monoid is , its ambient semiring is and its partial order is defined by all relations for which the maximum occurs twice among , and . This latter fact generalizes to multiple term sums as follows.
Lemma 1.7**.**
For and , we have in if and only if the maximum occurs twice among the elements .
Proof.
We proceed by induction on . If , then obviously if and only if . If , then the claim follows from the definition of .
Let and . Since is generated by monomial relations with only two terms on the right hand side, there must be a partition of into smaller nonempty subsets with and , a relation and a relation for every . By the inductive hypothesis, the maximum occurs twice among and the (with varying ) and for every among and the (with varying ).
Thus there is some and such that is the maximum of and the (for ). If we have for some different from , then maximum among occurs twice.
If not, then . If , then maximum among occurs twice. If not, then for some different from . Then there is a such that . Thus also in the last case, the maximum among occurs twice, which verifies one implication of the claim of the lemma.
Conversely, assume that the maximum occurs twice among . First we consider the case that it occurs twice among . After relabeling the elements, we can assume that for all . By the inductive hypothesis, we have thus and . If , then the former relation implies that
[TABLE]
and if , then the latter relation implies that
[TABLE]
Thus our claim follows if the maximum occurs twice among . If is equal to the maximum of , say , then and by the inductive hypothesis. Adding to the former relation yields
[TABLE]
as desired. This concludes the proof of the lemma. ∎
Remark 1.8**.**
Izhakian’s extended tropical semiring ([9]) is closely related to the tropical hyperfield , as was explained to me by Stephane Gaubert. Namely, the multiplication and the hyperaddition of extends to the family of all singletons and all intervals (the ghost elements) by the rules
[TABLE]
which defines a semiring structure on \big{\{}\{a\},a^{\mynu}\big{|}a\in{\mathbf{T}}\big{\}}. This semiring is isomorphic to Izhakian’s extended tropical semiring.
2. Tropicalization as a base change to the tropical hyperfield
In this section, we explain the variant of scheme theoretic tropicalization over the tropical hyperfield. Roughly speaking, we interpret a nonarchimedean absolute value as a morphism into the tropical hyperfield and define the scheme theoretic tropicalization of a -variety as its base change to along this morphism.
As we will explain in section 3.3, the set theoretic tropicalization can be recovered as the set of -rational points from the scheme theoretic tropicalization. In last part of this section, we explain how inherits a topology from .
2.1. Nonarchimedean seminorms as morphisms
Let be a ring. A nonarchimedean seminorm on is a map that satisfies for all that
- (i)
and ; 2. (ii)
; 3. (iii)
where the relation in (iii) is the natural linear order of . Note that if is a field, then a nonarchimedean seminorm is the same as a nonarchimedean absolute value. Though the following fact is well-known, we include a proof for completeness.
Lemma 2.1**.**
Let be a ring and a nonarchimedean seminorm. Let in . Then the maximum occurs twice among .
Proof.
A simple induction shows that . Thus the maximum among is attend by for some . Without loss of generality, we can assume that . Since implies in , we have and . Axiom (iii) of a nonarchimedean seminorm applied to yields
[TABLE]
which implies that the maximum occurs twice among . ∎
Recall that the underlying set of the ordered blueprint is and that the underlying set of the tropical hyperfield is . Thus by definition, a morphism is a map .
That nonarchimedean seminorms can be interpreted as morphisms of hyperrings was observed Viro in [23]; also cf. [7] and [11]. In so far, the following theorem does not contain a novel mathematical fact, though its appearance in terms of ordered blueprints is new. Since it is a key fact for our theory, we include a short proof.
Theorem 2.2**.**
Let be a ring. The association defines a bijection
[TABLE]
Proof.
We begin with the verification that is well-defined, i.e. that is indeed a nonarchimedean seminorm. We denote by in the following. Axioms (i) and (ii) follow from the fact that is a morphism of monoids with zero. Thus we are left with verifying axiom (iii).
Let and . Then we have in and thus in . By Lemma 1.7, this means that the maximum among , and appears twice. This implies (iii) at once.
Conversely, consider a nonarchimedean seminorm and define as a map . By axioms (i) and (ii), is a morphism of monoids with zero. Since is the freely generated semiring, extends uniquely to a semiring homomorphism . We are left with verifying that is order-preserving, which can be verified on generators of the partial order of .
The relation means that in . By Lemma 2.1, the maximum among , and appears twice. By Lemma 1.7, this implies that , which shows that is a morphism of ordered blueprints. This verifies the claim of the lemma. ∎
Given a nonarchimedean seminorm , we shall call the morphism with the associated morphism in the following.
Corollary 2.3**.**
Let be a field and the associated monomial ordered blueprint. Let be a nonarchimedean absolute value and the associated morphism. Let be a -algebra and the associated -algebra where . Then the association defines a bijection
[TABLE]
Proof.
By Theorem 2.2, the association defines a bijection between morphisms and nonarchimedean seminorms . A morphism is -linear if . This condition is evidently equivalent with , which proves our claim. ∎
2.2. Affine ordered blue schemes
For the purpose of this text, we define the category of affine ordered blue schemes as the dual category of . We denote the anti-equivalences between these categories by
[TABLE]
Typically, we say that is an affine ordered blue scheme where we assume implicitly that is an ordered blueprint. Given a morphism of ordered blueprints, we write for the dual morphism from to . Given a morphism , we denote its dual morphism by .
Let be an ordered blueprint. An affine ordered blue -scheme is an affine ordered blue scheme together with a morphism , which we call the structure morphism. Often we suppress the structure morphism from the notation and refer to as an affine ordered blue -scheme. A -linear morphism between affine ordered blue -schemes and is a morphism that commutes with the structure morphisms of and . Note that the morphism is the dual of an ordered blueprint morphism . This means that is a -algebra.
Example 2.4**.**
Let be an ordered blueprint. We define the -dimensional affine space over as the affine ordered blue -scheme
[TABLE]
and the -dimensional torus over as
[TABLE]
By the universal property of , a -linear morphism corresponds to the tuple \big{(}f(T_{1}),\dotsc,f(T_{n})\big{)} in . This establishes a canonical bijection
[TABLE]
which is analogous to the characterizing property of the -dimensional affine space in usual algebraic geometry. Similarly, we have a canonical bijection
[TABLE]
where is the group of invertible elements in .
Remark 2.5**.**
The reason that we restrict ourselves to affine ordered blue schemes is purely a matter of exposition. While it is possible to define affine ordered blue schemes as objects of the dual category of , the definition of ordered blue schemes requires a more sophisticated setup. The definition of an ordered blue scheme can be found in [4] and [15]. An alternative, but equivalent, definition uses relative schemes in the sense of Toën and Vaquié ([22]); see [16] for a proof of the equivalence in the case of algebraic blueprints.
2.3. Scheme theoretic tropicalization
Let be a field and . Let be a nonarchimedean absolute value and the associated morphism, cf. Theorem 2.2. Let be an affine ordered blue -scheme and the structure map. The scheme theoretic tropicalization of along is the affine ordered blue -scheme \operatorname{{Trop}}_{\mathbf{v}}(Y)=\operatorname{Spec}\big{(}B\otimes_{{\mathbf{k}}}{\mathbf{T}}\big{)}.
Note that the canonical inclusion endows with the structure of an ordered blue -scheme. Note further that since is a functor, is functorial in , i.e. a -linear morphism induces a -linear morphism .
Example 2.6**.**
Since , cf. Example 1.6, the scheme theoretic tropicalization of affine -space over is . Similarly, we have .
2.4. Topology for rational point sets
Let be an affine ordered blue -scheme. The set of -rational points is the set of -linear morphisms from to . The affine topology of is defined as the compact-open topology on with respect to the Euclidean topology of and the discrete topology for . In other words, has the coarsest topology such that the evaluation maps
[TABLE]
are continuous for all .
The affine topology for -rational point sets is very well-behaved, similar to the situation of rational point sets over topological fields.
Theorem 2.7**.**
The affine topology satisfies the following properties for affine ordered blue -schemes and .
- (T1)
The canonical bijection is a homeomorphism. 2. (T2)
The canonical bijection is a homeomorphism. 3. (T3)
A -linear morphism induces continuous map . If is an open (a closed) immersion, then is an open (a closed) topological embedding.
Proof.
Since is a topological ordered blueprint (i.e. the multiplication map is continuous) with open unit group (i.e. is open in and multiplicative inversion defines a continuous map ), everything follows from [15, Thm. 6.4] but the claim that a closed immersion induces a closed topological embedding. We will prove this claim in the following.
Let be a closed -linear immersion. By [15, Thm. 6.4 (F5)], is a topological embedding. All that is left to prove is that the image is closed in .
If for some set of relations on , then the image equals the subset
[TABLE]
of . In the following proof, we break down the relations in into more elementary terms and trace this back to an expression of as an intersection of finite unions of elementary closed sets of of the form where and is a closed subset. Note that is indeed closed since the complement of in is the elementary open subset of where the complement of is open in .
Our first observation is that
[TABLE]
which reduces us to the proof that a subset of the form \big{\{}f:B\to{\mathbf{T}}\big{|}\sum f(a_{i})\leqslant\sum f(b_{j})\big{\}} is closed in . Therefore let us focus on one fixed relation in the following where we specify the index sets and for reference.
Since is monomial, i.e. generated by relations of the form , there must be a partition of such that for all . This means that
[TABLE]
is the finite union over all partitions of of intersections of subsets of the form \big{\{}f:B\to{\mathbf{T}}\big{|}f(a_{i})\leqslant\sum_{j\in J_{i}}f(b_{j})\big{\}}. We have reduced the proof therefore to the situation of showing that a subset of the form \big{\{}f:B\to{\mathbf{T}}\big{|}f(a_{i})\leqslant\sum_{j\in J_{i}}f(b_{j})\big{\}} is closed in . For simplicity, we assume and set .
In the case that is empty, the relation in question is . Thus we could also equally assume that and . This allows us to rely on Lemma 1.7, which states that holds in if and only if the maximum occurs twice among , i.e. if we have for some and all . This means that \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{0})\leqslant\sum_{j=1}^{n}f(b_{j})\big{\}} equals
[TABLE]
i.e. a finite union of the intersection of subsets of the forms \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{k})=f(b_{l})\big{\}} and \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{l})\geqslant f(b_{j})\big{\}} of . This reduces our proof to the study of these two particular types of subsets.
We begin with \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{k})=f(b_{l})\big{\}}, which can be expressed as \big{\{}f:B\to{\mathbf{T}}\big{|}(f(b_{k}),f(b_{l}))\in\Delta\big{\}} where is the diagonal. Since is Hausdorff, is closed in and can thus be written as an intersection of finite unions of basic closed subsets, i.e.
[TABLE]
where and are index sets, with finite for every , and where and are closed subsets of . Thus
[TABLE]
where U_{b_{j},V_{p,q,j}}=\big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{j})\in V_{p,q,j}\big{\}} is a basic closed subset of for . This shows that \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{k})=f(b_{l})\big{\}} is a closed subset of .
We continue with the remaining case \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{l})\geqslant f(b_{j})\big{\}}, which equals the set \big{\{}f:B\to{\mathbf{T}}\big{|}(f(b_{k}),f(b_{j})\in\nabla\big{\}} where is the subset of all with . Since is closed in , an analogous argument as in the preceding case shows that \big{\{}f:B\to{\mathbf{T}}\big{|}f(b_{l})\geqslant f(b_{j})\big{\}} is a closed subset of . This concludes the proof that is a closed topological embedding. ∎
Remark 2.8**.**
In [15], the affine topology for is extended to possibly non-affine ordered blue -schemes . In this more general situation, the topology on is called the fine topology. By [15, Thm. 6.4], we conclude that for every affine open covering of , we have as sets and that the inclusions are open topological embeddings, which determines the fine topology of in terms of the affine topology for the . In particular the affine and the fine topology agree on if is affine.
Example 2.9**.**
Since , we have . Thus by (T1) and (T2) of Theorem 2.7, we have
[TABLE]
as topological spaces, where we consider with respect to the product topology.
Let and . By (T3) of Theorem 2.7,
[TABLE]
is a closed subspace of . Since if and only if the maximum occurs twice, we see that is the standard plane tropical line, which can be depicted as
[TABLE]
where the illustration on the left hand side uses the natural coordinates in and the illustration on the right hand side follows the more common convention of double-logarithmic coordinates in \big{(}{\mathbb{R}}\cup\{-\infty\}\big{)}^{2}.
3. The Kajiwara-Payne tropicalization as a rational point set
We begin this section with a review of Berkovich spaces and the Kajiwara-Payne tropicalization before we explain how to recover them as rational point sets of a scheme theoretic tropicalization.
3.1. The Berkovich analytification
Let be a field together with a nonarchimedean absolute value . Let be a -algebra and . As a topological space, the Berkovich space of is the set
[TABLE]
together with the compact-open topology with respect to the discrete topology for and the Euclidean topology of . In other words, the topology of is the coarsest topology such that the maps
[TABLE]
are continuous for all . Thus the topology of is generated by the open subsets of the form
[TABLE]
where and is an open subset.
Example 3.1**.**
Berkovich analytifications tend to be rather involved topological spaces. In the case of curves, one can find a description in Berkovich’s book [5]. For the purpose of illustration, we will give the description of an easy case of a Berkovich analytification, which occurs for the trivial absolute value with for all and the affine line .
The nonarchimedean seminorms extending the trivial absolute value are classified as follows:
- •
the trivial norm with whenever ;
- •
for every irreducible polynomial and every the -adic norm with whenever is not divisible by ;
- •
for every irreducible polynomial the seminorm with if is divisible by and if not;
- •
for every the -adic norm with for .
The analytification can be depicted as
[TABLE]
where and are irreducible polynomials in and where one has to imagine an infinite number of rays emerging from the center . The empty circle at the end of the ray of indicates the missing point at infinity. The topology of is generated by open subsets of the following forms:
- •
where is irreducible and is open or and is open;
- •
where is irreducible and is closed or and is closed.
3.2. The Kajiwara-Payne tropicalization
The tropicalization of an affine -scheme along a nonarchimedean absolute value requires an additional choice of coordinates. Such a choice is given by a closed -linear immersion into an affine toric variety , which is an affine -scheme of the form for a suitable multiplicative and commutative monoid . To be precise, is a toric variety if satisfies the following conditions:
- (i)
is finitely generated as a monoid; 2. (ii)
is integral, i.e. can be embedded as a submonoid in a group ; 3. (iii)
is saturated, i.e. if is the subgroup of generated by and for some and , then .
The closed immersion corresponds to a surjective -algebra homomorphism . The ideal of definition for is the ideal of . The restriction of to yields a morphism of multiplicative monoids.
Consider an element in where and . We define as a finite formal -linear combinations of elements of . We define .
Let be the set of maps with and for all . It comes together with the compact-open topology with respect to the discrete topology for and the Euclidean topology of . It is generated by the open subsets of the form
[TABLE]
where and is an open subset.
Let be a finite -linear combination. The zero set or bend locus of is the subset
[TABLE]
of . It is a closed subset of since it can be written as the intersection of closed analytic subsets of . We consider together with the subspace topology.
The tropicalization of along with respect to is the closed subspace
[TABLE]
of .
The Kajiwara-Payne tropicalization of along with respect to is the map
[TABLE]
Note that the composition is obviously in . If in where and , then by Lemma 2.1 the maximum among the is assumed twice. Thus lies indeed in . Note further that the map is continuous, which is immediately clear from the definitions of the respective topologies as compact-open topologies.
Remark 3.2**.**
The Kajiwara-Payne tropicalization is moreover proper and surjective, due to the following argument that was explained to the author by Sam Payne. In the case that is trivial or that algebraically closed and complete with respect to , then this is proven in [20].
If is arbitrary with non-trivial absolute value , then we can reduce the claim to the corresponding claim of a suitable field extension of by the following arguments. We write for the base change of to a field extension of . If is the completion of with respect to and is the canonical extension of to , then the Berkovich spaces and agree since every nonarchimedean seminorm extending extends uniquely to a nonarchimedean seminorm on that extends .
Since is complete, the absolute value extends uniquely to the algebraic closure of . As shown in [5], is the quotient of by the action of the Galois group of over . Completing yields an algebraically closed and complete field with , to which Payne’s result from [20] applies. Thus we obtain a proper and surjective map
[TABLE]
Clearly this implies that trop is surjective. Given a compact subset of , then its inverse image in is compact as the inverse image under a proper map. Since the projection is continuous, the image of is compact in is compact. Since , this shows that trop is proper.
Remark 3.3**.**
For the following reinterpretation of , we identify as sets. Let and . By Lemma 1.7, the maximum among the terms occurs twice if and only if as elements of . Thus the identification leads to an identification
[TABLE]
This expression for the zero set of stays in direct analogy to the zero set of a function over a field . One can find a similar expression for using Izhakian’s extended tropical semiring in [10]; also cf. Remark 1.8.
Example 3.4**.**
We explain the Kajiwara-Payne tropicalization in the example of the trivial absolute value and the line in given by , i.e. with where the ideal of definition is .
Then is isomorphic to for which we have calculated the Berkovich analytification in Example 3.1. The tropicalization is the standard plane tropical line as illustrated in Example 2.9.
Let . Note that the map is injective. Since the classes of and in are irreducible, it follows immediately from the definition of the seminorm in Example 3.1 that the restriction of to is trivial unless is one of , or . Thus the Kajiwara-Payne tropicalization can be illustrated as
[TABLE]
where all thin rays of are contracted to the central point of and the three thick rays of are mapped bijectively to the three rays of .
3.3. Recovering the analytification and tropicalization as rational point sets
We continue with the context of the previous sections and define , and . Let be the morphism that maps to where and . We define
[TABLE]
whose underlying monoid is the submonoid of , whose ambient semiring is and whose partial order is generated by the relations with and for which in . We define . Note that the inclusion defines a -linear morphism .
Theorem 3.5**.**
There are canonical homeomorphisms and such that the diagram
[TABLE]
commutes.
Proof.
As a first step, we define the canonical map . Let be a seminorm in and the associated morphism from Corollary 2.3. We define the -linear morphism as the morphism induced by and the identity . It sends an element to .
Conversely, given a -linear morphism , we define as the seminorm associated with the composition . Since the composition is equal to , the restriction of to is . This defines a map . It is clear that the maps and are mutually inverse bijections.
We continue with showing that both and are open. This can be verified on the generators of the topologies of and . Consider such a generator of the topology of where and is open. Then we can consider as an element of and as an open subset of and find that
[TABLE]
which is a generator of the topology of . Conversely, consider a generator of the topology of where and is open.
Since is -linear, we have , and is equivalent with . Since the multiplication in is continuous, is open in . Considering as an element in and as an open subset of yields that
[TABLE]
which is a generator of the topology of . This concludes the proof that is a homeomorphism.
We turn to the definition of the homeomorphism . Consider a map in . We define by the rule for , and .
This association is well-defined as a map since
[TABLE]
for all , and . Clearly, is multiplicative with and . Since the semiring is freely generated by the underlying monoid with zero , the monoid morphism extends uniquely to a semiring morphism . Thus we are left with showing that is order-preserving in order to show that is a morphism of ordered blueprints. This can be verified on the generators of the partial order of , which stem from the relations in and .
Since is monomial, its partial order is generated by relations of the form with and . Note that this corresponds to the relation of . The relation implies that as elements of where is the ideal of definition for . In other words, is an element of . Since , we conclude that the maximum occurs twice in . By Lemma 1.7, this means that in .
Since the composition is the identity on , it is clear that preserves all relations of coming from and that is -linear. This shows that is a -linear morphism and thus an element of .
The inverse map maps a -linear morphism to the map that is defined by . The map is clearly multiplicative with and thus an element of .
In order to show that lies indeed in the subset of , consider an element in the ideal of definition where and . Then we have in and thus in . Note that . Since is -linear, we obtain
[TABLE]
in . By Lemma 1.7, this means that the maximum occurs twice among the elements . Since corresponds to under the identification , this shows that is indeed an element of .
It is evident that the maps and are mutually inverse bijections. In order to show that is a homeomorphism, we begin with a reduction step to the case of a toric variety .
Recall that the topology of is by definition the subspace topology induced from . Since the quotient map is surjective, the induced map is surjective as well, which means that the morphism is a closed immersion of ordered blue schemes. By Theorem 2.7 (T3), is closed topological subspace . Thus it suffices to prove that is a homeomorphism for and . This can be verified on generators of the respective topologies of and .
The topology of is generated by open subsets of the form where and is open. We have that
[TABLE]
which is an open subset of where we identify with the corresponding subset of .
Conversely, the topology of is generated by open subsets of the form with , , and open. We have . Since the multiplication of is continuous, the subset is open as well. Since if and only if , we find that
[TABLE]
which is an open subset in where we identify with the corresponding open subset of . This completes the proof that is a homeomorphism.
The last step of the proof concerns the commutativity of the diagram
[TABLE]
Consider an element of , which is a seminorm that extends . Then and maps to , which is the same as .
On the other side, maps to . The image of under is the morphism that maps to . This shows that and that the diagram commutes. ∎
The following description of is useful for the calculation of explicit examples.
Lemma 3.6**.**
The association ca\otimes t\mapsto\big{(}v(c)t\big{)}\cdot a defines an -linear isomorphism
[TABLE]
of ordered blueprints where and .
Proof.
Let us define for the sake of this proof
[TABLE]
As a first step, we observe that
[TABLE]
which shows the independence under the action of on tensors. Since is the identity on , it is clear that all relation coming from are preserved and that is -linear. Thus we are left with showing that preserves the relations coming from , which can be verified on generators, which are of the form where in . This means that
[TABLE]
as desired.
In order to show that is an isomorphism, we will prove that the association with and defines a -linear morphism , which is obviously inverse to . It is clear that this association defines a -linear multiplicative map . We are left with showing that it preserves the additive relations, which can be verified on generators of the form for which in . By the definition of , this means that in and thus
[TABLE]
This concludes the proof of the lemma. ∎
Example 3.7**.**
We explain Theorem 3.5 in the case of the standard plane tropical line from Example 3.4.
Using the context of Theorem 3.5, let be the affine plane over and , together with its natural closed embedding as the zero set of . Note that here .
In this case, we have that the partial order of
[TABLE]
is generated by the relations
[TABLE]
Using Lemma 3.6 and the fact that , we derive an isomorphism
[TABLE]
Sending a -linear morphism to \big{(}f(T_{1}\otimes 1),f(T_{2}\otimes 1)\big{)}\in{\mathbf{T}}^{2} yields thus an identification
[TABLE]
By Lemma 1.7, each of the four relations on and is equivalent to the condition that the maximum among , and occurs twice. This equals the bend locus of , which is exactly , cf. Examples 2.9 and 3.4.
4. The relation between the tropical hyperfield and the tropical semifield
In this section, we will explain how to recover the tropical semifield from the tropical hyperfield using some functorial constructions for ordered blueprints. The precise relation is formulated in Proposition 4.3, which is a key fact for our description of the Giansiracusa bend in terms of the scheme theoretic tropicalization in section 5.
All except for Proposition 4.3 is covered already in [15] and [17], but we present an independent and streamlined exposition, including a shortened proof of Lemma 4.1.
4.1. The tropical semifield as an algebraic blueprint
Let be the tropical semifield whose addition is characterized by the rule and whose multiplication is the usual multiplication of real numbers. We associate with the algebraic blueprint .
4.2. Idempotent ordered blueprints
An ordered blueprint is idempotent if is an idempotent semiring, i.e. if . Given an ordered blueprint , we define its associated idempotent ordered blueprint as . The quotient map defines a canonical morphism .
The construction of is functorial. Given a morphism of ordered blueprints, the composition factors uniquely through a morphism by the universal property of the quotient map ; cf. section 1.6.
Our main example of an idempotent ordered blueprint is . Note that and thus for every ordered blueprint , which reinforces the functoriality of .
4.3. Totally positive ordered blueprints
An ordered blueprint is totally positive if holds in . Multiplying this relation by yields for every . Given an ordered blueprint , we define its associated totally positive blueprint as . The quotient map defines a canonical morphism .
The construction of is functorial: given a morphism of ordered blueprints, the composition factors uniquely through a morphism by the universal property of the quotient map ; cf. section 1.6.
Looking back at Example 1.5, we see that is indeed the totally positive ordered blueprint associated with . More generally, we have for every ordered blueprint , which reinforces the functoriality of .
Lemma 4.1**.**
If is algebraic and idempotent, then is a bijection between the respective underlying monoids and in if and only if in . The canonical morphism is an isomorphism of ordered blueprints.
Proof.
This follows at once from Proposition 2.12 and Lemma 2.15 in [15]. For completeness, we give an independent proof in the following.
Let be the relation on that is defined by the rule that if and only if . We claim that is an additive and multiplicative partial order on .
The relation is reflexive since the idempotent relation implies . It is antisymmetric since the relations and imply . It is transitive since if and , then and , and thus , which yields as desired. To prove additivity and multiplicativity, consider , i.e. , and let . Then and , and henceforth and , as claimed. This completes the proof that is an additive and multiplicative partial order on .
Since , the relation contains . On the other hand, is generated by the relation for the following reason. Consider an equality . Multiplying by yields , and therefore , as claimed.
We conclude that is the smallest additive and multiplicative partial order on that contains . This means that is the partial order of , i.e.
[TABLE]
From this it is obvious that the quotient map is a bijection between the respective underlying monoids and that is equal to , i.e. the canonical morphism is an isomorphism. ∎
Example 4.2**.**
As an immediate consequence of the characterization of the partial order of for algebraic and idempotent , we see that the partial order of is the natural linear order of the nonnegative real numbers. Moreover, we have .
4.4. Recovering the tropical semifield from the tropical hyperfield
In this section, we explain the relation between the tropical hyperfield and the tropical semifield . In particular, we see that results from under a functorial construction. In this sense, we can consider as a refinement of .
Proposition 4.3**.**
The identity map is a morphism of ordered blueprints, which induces an isomorphism . Taking algebraic cores yields an isomorphism .
Proof.
We begin with the verification that the identity map is a morphism . Clearly is multiplicative and and . Since is freely generated by the monoid and , it extends uniquely to a semiring homomorphism .
That is order-preserving can be verified on the generators of the partial order of , which are of the form and where with larger than (as real numbers). Then we have in and thus in , which is the first desired relation. As we have seen in Example 4.2, we have in , and multiplying by yields in . Thus in , which is the second desired relation. This concludes the proof that is a morphism.
The morphism induces a morphism where we use in the latter identification that is already idempotent. Since the composition of with the surjective quotient map is the bijective map , we conclude that is also bijective. Thus it suffices to show that is a semiring isomorphism and that the defining relation of occurs in .
We begin with the proof that the surjective semiring homomorphism is an isomorphism. It suffices to show that every equality in holds already in . This is so since in means that is larger than (as real numbers) and thus in . On the other side, we have in and thus in . This shows that in , as desired.
Finally, we observe that in , which concludes the proof that is an isomorphism of ordered blueprints. The last claim of the proposition follows at once from Lemma 4.1 and Example 4.2. ∎
5. Recovering the Giansiracusa bend
In their paper [8] on tropical scheme theory, Jeff and Noah Giansiracusa introduce the bend relations for a tropical variety. This approach finds a refinement in the author’s paper [15] that is based on ordered blueprints. We will see in this section that the tropicalization over the tropical hyperfield is a further refinement of the Giansiracusa tropicalization.
5.1. The Giansiracusa bend
We begin with a review of the Giansiracusa bend as a semiring. Let be the tropical semifield, as defined in section 4.1.
Let be a field with nonarchimedean absolute value , which we consider as a map into in the following. Let be an affine -scheme and a -linear closed immersion into an affine -scheme of the form for some commutative monoid . Let be the corresponding surjection of -algebras and the ideal of definition of inside .
In the definition of the Giansiracusa bend, we make use of congruences for semirings, which are additive and multiplicative equivalence relations. Congruences can be characterized as those equivalence relations on semirings for which addition and multiplication of representatives defines a semiring structure on the quotient set. For more details, we refer to [17, section 2.4].
The Giansiracusa bend of (with respect to and ) is the semiring
[TABLE]
where is the congruence on that is generated by the bend relations, which are relations of the form
[TABLE]
for which where and .
Example 5.1**.**
As an illustration, we calculate the Giansiracusa bend of the zero set of . In this case, is the affine -plane and the closed subscheme with coordinate algebra , together with the natural closed immersion .
It is easy to verify that the bend relation of the Giansiracusa bend
[TABLE]
is generated by the relations
[TABLE]
Remark 5.2**.**
To simplify our exposition, we omit the geometric counterpart that makes use of semiring schemes, but turn to the description of the bend as a blueprint right away. For details on a geometric description of the Giansiracusa bend as a semiring scheme, cf. [8] and [15].
5.2. The bend as a blueprint
The description of the Giansiracusa bend as a quotient of carries additional information that can be captured in terms of an algebraic blueprint whose ambient semiring is and whose underlying monoid is
[TABLE]
We call the blueprint the bend of (with respect to and ). As explained in section 1.2, the underlying monoid of a free algebra over an ordered blueprint consist of monomials, which yields the description
[TABLE]
of the bend of where is as in section 4.1.
The association defines a morphism of blueprints, which turns the bend of into a blue -algebra.
We define the bend of (with respect to and ) as the blue -scheme \operatorname{{Bend}}_{v,\myiota}(X)=\operatorname{Spec}\big{(}\operatorname{{Bend}}_{v,\myiota}(R)\big{)}.
Example 5.3**.**
We continue Example 5.1 where we described the Giansiracusa bend of the zero set of . The bend of is the blueprint
[TABLE]
where stands for and . More explicitly, the ambient semiring of is
[TABLE]
and its underlying monoid is
[TABLE]
5.3. The Kajiwara-Payne tropicalization from the bend
The key insight from [8] is that the bend of with respect to a closed immersion into a toric -variety recovers tropicalization of . At the same time we can recover the analytification of from the bend with respect to the closed immersion that is induced by the surjection where , is the underlying monoid of and sends a formal linear combination of elements of to its value as an element of .
Similarly to the case of -rational points of an ordered blue -scheme, we can endow the set of -rational points of a blue -scheme with a the compact-open topology with respect to the discrete topology of and the Euclidean topology of . The following is Theorem 9.1 in [15]; also cf. [8].
Theorem 5.4**.**
The Berkovich space is naturally homeomorphic to , the Kajiwara-Payne tropicalization is naturally homeomorphic to and the diagram
[TABLE]
of continuous maps commutes.
Example 5.5**.**
We continue Examples 5.1 and 5.3 where we consider the bend
[TABLE]
of . Let X=\operatorname{Spec}\big{(}\operatorname{{Bend}}_{v,\myiota}(R)\big{)}. Sending a -linear morphism to \big{(}f(T_{1}),f(T_{2})\big{)}\in{\mathbb{R}}_{\geqslant 0}^{2} defines an identification
[TABLE]
The condition
[TABLE]
is satisfied precisely for those for which the maximum occurs twice among , and . This set is the bend locus of , which equals ; also cf. Example 3.7.
5.4. Recovering the bend from the scheme theoretic tropicalization
In this section, we do not require any assumptions on the monoid . Therefore Theorem 5.6 applies to both the analytification, in which case we use the closed immersion from section 5.3, and the tropicalization, in which case we use a closed immersion into a toric variety .
Let and the associated morphism from Theorem 2.2. Let and the morphism that maps to . As in section 3.3, we define
[TABLE]
where and . Let .
Theorem 5.6**.**
There are canonical isomorphisms
[TABLE]
Proof.
The theorem follows at once from [15, Cor. 7.13], using Proposition 4.3. In the following, we give an independent proof.
Thanks to Lemma 4.1, the second claim of the theorem follows from the first claim. Thus it suffices to show that the association with and defines an isomorphism
[TABLE]
We begin with the proof that is well-defined as a morphism. Since the association defines a multiplicative map that maps [math] to [math] and to , we are left with showing that it respects all the defining relations of . Since the tensor product is compatible with taking the quotient by the relation and by Proposition 4.3, we have
[TABLE]
This shows that respects all relations coming from , which reduces our proof to the verification that the bend relations hold in . This can be verified on generators. For and , consider the relation
[TABLE]
in stemming from the element in the ideal of definition of in . Then we have in and
[TABLE]
in . Since is idempotent, this yields
[TABLE]
in , and since is totally positive, this yields
[TABLE]
in . Thus we have in as desired, which shows that is a well-defined -linear morphism of ordered blueprints.
In order to show that is an isomorphims of ordered blueprints, we need to show that all defining relations of are already contained in . Since is -linear, we can restrict ourselves to the relations coming from , which are of the form whenever in . In this case, is an element of and contains the bend relation . Thus we have
[TABLE]
in , as desired. This shows that is an isomorphism and concludes the proof of the theorem. ∎
Let and be as before. As an immediate consequence of Theorem 5.6, we find the following natural interpretation of the Giansiracusa bend .
Corollary 5.7**.**
We have a canonical identification . ∎
Example 5.8**.**
We illustrate Theorem 5.6 in the example of the zero set of in the affine plane over . We recall the notation from Example 3.7: let be the coordinate ring of and be the closed immersion as a subscheme. Let be the associated monomial ordered blueprint and the morphism associated with the given nonarchimedean absolute value . Let
[TABLE]
be the ordered blueprint associates with . As explained in Example 3.7, we have
[TABLE]
If we impose the additional relation on , then we conclude that
[TABLE]
(using ) and that
[TABLE]
(using ) in . Thus we gain the equality in . Repeating the previous argument with the roles of , and exchanged yields the relations
[TABLE]
in , which generate the bend relation on , cf. Examples 5.1 and 5.3.
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