Quadratic and Symmetric Bilinear Forms

on Modules with Unique Base over a Semiring

Zur Izhakian, Manfred Knebusch, Louis Rowen

Received: September 4, 2015 Revised: December 3, 2015

Communicated by Ulf Rehmann

Abstract. We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt’s Cancelation Theorem. Also, the tensor product of an indecomposable bilinear module (U, γ) with an indecomposable quadratic module (V, q) is indecomposable, with the exception of one case, where two indecomposable components arise.

2010 Mathematics Subject Classification: Primary 15A03, 15A09, 15A15, 16Y60; Secondary 14T05, 15A33, 20M18, 51M20 Keywords and Phrases: Semirings, (semi)modules, bilinear forms, quadratic forms, symmetric forms, orthogonal decomposition.

1. Overview

This paper is part of a program to understand the theory of quadratic forms over the max-plus algebra and related semirings that arise in several mathematical contexts. Our motivation comes from two sources, tropical mathematics and real algebra, which interact with each other. Since the first area is still in its nascent stage, for the reader’s convenience, we provide a short overview of this mathematics and related subjects. Consider the field 𝕂 of Puiseux series over an algebraically closed field F of characteristic 0. The elements of 𝕂 are of the form

f = ∑_{τ∈ℚ} c_{τ} t^{τ} ,

where c_{τ} ∈ F and the powers of t are taken over well-ordered subsets of ℚ. (In the literature one often takes ℝ instead of ℚ.) Define the order valuation v : 𝕂 → ℚ by

v(f) := min{τ ∈ ℚ_{≥0} : c_{τ} ≠ 0}

Quadratic and Symmetric Bilinear Forms on Modules 775

for which the dominant term in f becomes c_{v(f)}t^{v(f)} as t → 0. Then v is a valuation, with residue field F, with respect to which 𝕂 is complete and thus Henselian. By Hensel’s lemma, 𝕂 also is algebraically closed, and thus elementarily equivalent to F. Applying v takes us from 𝕂 to the ordered group ℚ, which can be viewed as a “max-plus” semiring (taking −v instead of v), whose operations are “+” for multiplication and “sup” for addition. This process, called tropicalization, is explained in [15, 29]. The point of tropicalization is to simplify the combinatorics in algebraic geometry and linear algebra, and there has been considerable success in this direction in enumerative geometry. One can tropicalize structures arising in linear algebra, such as quadratic forms, simply by replacing the classical addition and multiplication by the max-plus operations respectively, but then the classical theory does not go through because our new addition (max) does not have negatives. Other important (non-tropical) semirings, where our below theory is relevant, occur in real algebra, such as the positive cone of an ordered field [4, p. 18] or a partially ordered commutative ring [5, p. 32]. A further application can be found in the algebra of groups over a splitting field, as described briefly at the end of this overview. Recall that a (commutative) semiring is a set R equipped with addition and multiplication, such that both (R, +, 0) and (R, ·, 1) are abelian monoids with elements 0 = 0_{R} and 1 = 1_{R} respectively, and multiplication distributes over addition in the usual way. In other words, R satisfies all the properties of a commutative ring except the existence of negation under addition. We call a semiring R a semifield, if every nonzero element of R is invertible; hence R\{0} is an abelian group. As in the classical theory, one considers bilinear and quadratic forms defined on (semi)modules over a semiring R, often a “supersemifield,” in order to obtain more sophisticated “trigonometric” information, cf. [24, §2, §3]. On one hand, these semirings lack negation, thereby playing havoc even with the notion of the underlying bilinear form of a quadratic form. On the other hand, they have the pleasant property that free modules have “unique base,” cf. Definition 1.2. Thus, our overall object is to classify quadratic forms over free modules having unique base, with applications to the supertropical setting. For the reader’s convenience, we recall some terminology and results from [22, §1-§4]. A module V over R (sometimes called a semimodule) is an abelian monoid (V, +, 0_{V}) equipped with a scalar multiplication R × V → V, (a, v) ↦ av, such that exactly the same axioms hold as customary for modules over a ring: a_{1}(bv) = (a_{1}b)v, a_{1}(v + w) = a_{1}v + a_{1}w, (a_{1} + a_{2})v = a_{1}v + a_{2}v, 1_{R} · v = v, and 0_{R} · v = 0_{V} = a_{1} · 0_{V} for all a_{1}, a_{2}, b ∈ R, v, w ∈ V. We write 0 for both 0_{V} and 0_{R}, and 1 for 1_{R}. When considering modules over semifields, one encounters several versions of “base,” as studied in depth in [21, §4 and §5.3]. Here we take the standard categorical version, and call an R-module V free, if there exists a family (ε_{i} | i ∈ I) in V such that every x ∈ V has a unique presentation x = ∑_{i∈I} x_{i}ε_{i} with scalars x_{i} ∈ R and only finitely many x_{i} nonzero, and we call (ε_{i} | i ∈ I) a base of the R-module V. Any free module with a base of n elements is clearly isomorphic

to R^{n}, under the map ∑_{i=1}^{n} x_{i}ε_{i} ↦ (x_{1}, …, x_{n}).

Bilinear forms on V are defined in the obvious way, [21].

Definition 1.1. For any module V over a semiring R, a quadratic form on V is a function q : V → R with

(1.1)  q(ax) = a^{2}q(x)

for any a ∈ R, x ∈ V, together with a symmetric bilinear form b : V × V → R (not necessarily uniquely determined by q) such that for any x, y ∈ V

(1.2)  q(x + y) = q(x) + q(y) + b(x, y).

Every such bilinear form b will be called a companion of q, and the pair (q, b) will be called a quadratic pair on V. We also call V a quadratic module.

In this generality, it is difficult to describe quadratic forms adequately on free modules over an arbitrary semiring. However, our task becomes more manageable when we introduce the following condition.

Definition 1.2. An R-module with unique base is a free R-module V in which any two bases 𝔅, 𝔅′ are projectively the same, i.e., we obtain the elements of 𝔅′ from those of 𝔅 by multiplying by units of R.

Although this never happens for free modules of rank ≥ 2 over a ring, it turns out to be quite common in the context of tropical algebra (and also often in real algebra, as noted in Example 2.4.d). Our main result, in §5, is an analog of Witt’s cancelation theorem:

Theorem 5.9.If W_{1}, W′_{1}, W_{2}, W′_{2} are finitely generated quadratic or bilinear modules with unique base such that W_{1} ≅ W′_{1} and W_{1} ⊥ W_{2} ≅ W_{2} ⊥ W′_{2}, then W_{2} ≅ W′_{2} (where ≅ means “isometric”). It actually is given in more general terms, where W_{2} needs not be finitely generated. When R is a ring, then a quadratic form q has just one companion, namely,

b(x, y) := q(x + y) − q(x) − q(y),

but if R is a semiring that cannot be embedded into a ring, this usually is not the case, and it is a major concern of quadratic form theory over semirings to determine all companions of a given quadratic form q : V → R. The first step in classifying quadratic forms is [22, Propositions 4.1 and 4.2], which lets us write a quadratic form q as the sum q = κ + ρ, where κ is quasilinear (and unique) in the sense that κ(x + y) = κ(x) + κ(y), and ρ is rigid in the sense that it has a unique companion. Quasilinearity of a quadratic form q implies that, for any vector x = ∑_{i∈I} x_{i}ε_{i} in V,

(1.3)  q(x) = ∑_{i∈I} x_{i}^{2}q(ε_{i}),

i.e., q has diagonal form with respect to the base (ε_{i} : i ∈ I). Quasilinear forms follow aspects of the classical theory of quadratic forms, and satisfy a Cauchy-Schwartz inequality given in [24]. On the other hand, by [22, Theorem 3.5], the rigid forms are precisely those with q(ε_{i}) = 0 for all i ∈ I. Our ultimate object being to classify quadratic forms over free modules with unique base, in this paper we study quadratic forms in terms of orthogonal decompositions of such forms into indecomposable forms, and then build them up again via tensor products of two symmetric bilinear forms and of a symmetric bilinear form with a quadratic form. Let us turn now to the tools needed in proving Theorem 5.9.

1.1. Partial quasilinearity. We seldom require quasilinearity in its entirety, but the following partial version plays a major role in our consideration of orthogonal decompositions of quadratic modules.

Definition 1.3. Given subsets S and T of V, we say that q is quasilinear on S × T if

q(x + y) = q(x) + q(y).

for all x ∈ S, y ∈ T.

The following helpful fact is a special case of [22, Lemma 1.18]. (We write S + S′ for {s + s′ : s ∈ S, s′ ∈ S′}.)

Lemma 1.4. Let S, S′, T be subsets of V. If q is quasilinear on S × T, S′ × T and S × S′, then q is quasilinear on (S + S′) × T.

1.2. Disjoint orthogonality. In §3 we develop the notion of (disjoint) orthogonality of two given submodules W_{1} and W_{2} of a quadratic R-module (V, q) (endowed with a fixed quadratic form q), which means that W_{1} ∩ W_{2} = {0} and q is partially quasilinear on W_{1}×W_{2}. (Note that there is no direct reference to an underlying symmetric bilinear form.) When V has unique base, we look for orthogonal decompositions V = W_{1} ⊥ W_{2}, and more generally V = ⊥_{i∈I} W_{i},

where the W_{i} are basic submodules of V, i.e., are generated by subsets of a base 𝔅 of V. We can choose a companion b of q (called “quasiminimal” companion) adapted to the notion of disjoint orthogonality, and then have an equivalence relation on the set 𝔅 at hands, which is generated by the pairs (ε, ε′) in 𝔅 with ε ≠ ε′, b(ε, ε′) ≠ 0. By the use of this equivalence relation the indecomposable basic submodules of V (in the sense of disjoint orthogonality) can be described as follows.

Theorem 3.8.Let {𝔅_{k} | k ∈ K} denote the set of equivalence classes in 𝔅 and, for every k ∈ K, let W_{k} denote the submodule of V having base 𝔅_{k}.

(a) Then every W_{k} is an indecomposable basic submodule of V and

V = ⊥_{k∈K} W_{k}.

(b) Every indecomposable basic submodule U of V is contained in W_{k}, for some k ∈ K uniquely determined by U.

(c) The modules W_{k}, k ∈ K, are precisely all the indecomposable basic orthogonal summands of V.

In §4 we develop the analogous notion of disjoint orthogonality in a bilinear R-module (V, b) with respect to a fixed symmetric bilinear form b on V, and we show:

Theorem 4.9. If b is a quasiminimal companion of a a quadratic module (V, q), then the indecomposable components of (V, q) coincide with the indecomposable components of (V, b). In §5, these decomposition theories yield the desired analog (Theorem 5.9) of Witt’s cancelation theorem.

1.3. Tensor products. The last two sections of the paper are devoted to tensor products. Whereas tensor products of modules over general semirings can be carried out in analogy with the usual classical construction over rings, it requires the use of congruences, resulting in some technical issues dealt with in [7, Chap. 16], for example. But for free modules with unique base the construction can be carried out easily, since then one does not need to worry about well-definedness. In §6 we construct the tensor product of two free bilinear R-modules over any semiring R, in analogy to the case where R is a ring, cf. [8, §2], [26, I, §5]. We then take the tensor product of a free bilinear R-module U = (U, γ) with a free quadratic R-module V = (V, q). A new phenomenon occurs here, in contrast to the theory over rings. It is necessary first to choose a so-called balanced companion b of q, which always exists, cf. [22, §1], but which usually is not unique. We then define the tensor product U ⊗_{b} V, depending on b, by choosing a so-called expansion B : V × V → R of the quadratic pair (q, b) which is a (not necessarily symmetric) bilinear form B with

B(x, x) = q(x), B(x, y) + B(y, x) = b(x, y)

for all x, y ∈ V, cf. [22, §1] and then proceed essentially as in the case of rings, e.g. [26, Definition 1.51], [8, p. 51]^{1}. The resulting quadratic form γ ⊗_{b} q does not depend on the choice of B but often depends on the choice of b. This is apparent already in the case γ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), where the matrix b is stored in the quadratic polynomial γ ⊗_{b} q, cf. Example 6.8 below. In §7 we turn to the indecomposability of tensor products. For convenience, we assume that R \ {0} is closed under multiplication and addition, implying by Theorem 2.3 that all free R-modules have unique base. After obtaining partial results along the way, we arrive at the main result of this section, Theorem 7.16, which states that, discarding trivial situations and excluding some pathological semirings, the tensor product of an indecomposable bilinear module (U, γ) with an indecomposable quadratic module (V, q) is again indecomposable, with the exception of one case, where two indecomposable components arise.

1When R is a ring the “b” in the tensor product is not specified since q has only one companion.

1.4. Applications. The remainder of this introduction discusses how quadratic forms over modules with unique base over semirings arise naturally in various contexts in mathematics. (The reader could skip directly on to the main theoretical results of this paper.)

1.4.1. Quadratic forms over rings. Supertropical semirings, to be defined below (cf. [25, 22]), establish a class of semirings over which every free module has a unique base. There is a way to pass from a quadratic form on a free module over a (commutative) ring R to quadratic forms on free modules over a supertropical semiring U. To explain this, we sketch the notion of supertropicalization of a quadratic form q : V → R, obtained by a so-called supervaluation φ : R → U. An m-valuation (= monoid valuation) on a ring R is a map v : R → M from R to a totally ordered abelian monoid M = (M, · , ≤), containing an absorbing element 0 = 0_{M} (0 · x = x · 0 = 0) with 0 ≤ x for all x ∈ M, which satisfies the following rules:

v(0) = 0, v(1) = 1, v(xy) = v(x)v(y),

and

(1.4)  v(x + y) ≤ max{v(x), v(y)}

for all x, y ∈ M. When Γ := M \ {0} is a group, we call the m-valuation v : R → M a valuation. These are exactly the valuations as defined by Bourbaki [3] and studied, e.g., in [14] and [27, Ch. I], except that for Γ we have chosen the multiplicative notation instead of the additive notation. In this case v^{−1}(0) is a prime ideal of R [loc. cit.]. When R is a field this forces v^{−1}(0) = {0}, and we return to Krull valuations. Given an m-valuation v : R → M, we equip M with the additive operation defined as

a + b := max{a, b},

which makes M a bipotent semiring, i.e., a semiring M′ in which a + b ∈ {a, b} for all a, b ∈ M′. Conversely any bipotent semiring M′ has a natural total order given by

a < b ⇔ a + b = b,

and can be viewed as a totally ordered abelian monoid with an absorbing element 0_{M′}. Therefore, totally ordered monoids M with zero can be referred to as bipotent semirings (or bipotent semifields when M \ {0} is a group). Viewed in this way, rule (1.4) reads

(1.5)  v(x + y) ≤ v(x) + v(y).

This brings us into the realm of semirings. A semiring U is called supertropical if the following conditions hold:

• e := 1_{U} + 1_{U} is idempotent (i.e., 2 × 1 = 4 × 1),

• the ghost ideal M = eU is a bipotent semiring,

• addition is defined in terms of the ghost map a ↦ ea and the ordering of M, as follows:

(1.6)  a + b = \{ \begin{matrix}a & if ea < eb; \\ b & if eb < ea; \\ ea & if ea = eb. \end{matrix}

In particular ea = 0 implies a = 0 (take b = 0 in (1.6)). The elements of eU are called ghost elements and those of U \ eU are called tangible elements. The zero element is regarded both as tangible and ghost. See [17, 18, 25] for the ideas behind this terminology. A supervaluation on a ring R is a multiplicative map φ : R → U sending R into a supertropical semiring, such that φ(0) = 0, φ(1) = 1, and

eφ(x + y) ≤ eφ(x) + eφ(y)

for all x, y ∈ R. The map v := eφ : R → M, x ↦ eφ(x), is then an m-valuation, which as we say is covered by φ. For any given m-valuation v : R → M, there usually is an extended hierarchy of supervaluations φ : R → U covering v (with U ⊃ M, eU = M, U varying) studied in [17, 18]. The supertropicalizations of a quadratic form q : V → R on a free R-module V are constructed by using a supervaluation φ : R → U as follows. We choose an ordered base L of V, say L = {v_{i} : i ∈ I} with I = {1, …, n}, and write q as a homogenous polynomial of degree 2

(1.7)  q( ∑_{i=1}^{n} x_{i}v_{i}) = ∑_{i=1}^{n} α_{i}x_{i}^{2} + ∑_{i<j} β_{ij}x_{i}x_{j},

with α_{i} = q(v_{i}), β_{ij} = b(v_{i}, v_{j}), where b is the (unique) companion of q. We denote by U^{n} the free U-module consisting of all n-tuples in U. Let {ε_{1}, …, ε_{n}} be the standard base of U, where each ε_{i} has i-th coordinate 1 and all other coordinates 0. Using a new set of variables λ_{1}, …, λ_{n}, we define

(1.8)  q^{φ}( ∑_{i=1}^{n} λ_{i}ε_{i}) := ∑_{i=1}^{n} φ(α_{i})λ_{i}^{2} + ∑_{i<j} φ(β_{ij})λ_{i}λ_{j}

by applying φ to the coefficients of the polynomial (1.7). We write (U^{(I)}, q^{φ}), or (Ṽ, q̃) for short, for the supertropicalization of the quadratic module (V, q) with respect to the base L. Since every U-module has a unique base, cf. §2, the base {ε_{i} : i ∈ I} of Ṽ is unique up to permuting the ε_{i} and multiplying them by units of U (which are the invertible tangible elements of U). That is, the base L of V becomes “frozen” in the free quadratic module (Ṽ, q̃) obtained from (V, q) by a kind of “degenerate scalar extension” φ : R → U. {φ is multiplicative, but respects addition only weakly.} This central fact motivates our interest in supertropicalization. One reason that we work with m-valuations in general, instead of just valuations covered by supervaluations, is that m-valuations which are not valuations often arise naturally in the context of commutative algebra as described in the paper [11] of Harrison and Vitulli. They construct so-called “V -valuations” (there named “formally finite” V -valuations). This construction has been complemented later by D. Zhang with somewhat dual “V^{0}-valuations” [33]. These constructions have been revised in [19, §1-§3], showing that any m-valuation on a ring can be coarsened both to a V -valuation and to a V^{0}-valuation, and also to a valuation in a minimal way. In [12] Harrison and Vitulli, pursuing their idea of “infinite primes” (in the sense of classical number theory) from [11], construct ℂ-valued places on a field by a somewhat similar method. This construction has been extended by Valente and Vitulli in [31] to “preplaces” on a ring R, which are interpreted in [19] as multiplicative maps χ : R → R′ to a bipotent semiring R′ such that χ(0) = 0, χ(1) = 1, and

χ(x + y) ≤ c(χ(x) + χ(y))

for all x, y ∈ R, where c is a unit of R′. Such a map χ provides various supervaluations φ : R → U that cover V -valuations v : R → eU [19, §4]. Since the multiplicative monoids eU \ {0} are cancellative, these V -valuations are true valuations. By a related method, supervaluations arise that cover V^{0}-valuations, which again are true valuations. Although not all supervaluations can be constructed in this way, at least we gain a rich stock of m-valuations and supervaluations on a ring. Facing a problem on quadratic forms over a ring R, it may be a piece of art to address an appropriate supervaluation which fits best the supertropical framework. Much space is left for further study in this research direction.

1.4.2. A surprise. In an earlier version of this paper we considered quadratic forms over supertropical semirings, knowing already from [22, Theorem 0.9] that a free module over these semirings has unique base, and we obtained the results in §3-§7 for such quadratic forms. Only later did we realize that these results go through for any semiring R over which all free modules have unique base. As a consequence, supertropical semirings hardly appear explicitly in §3-§7. This paves the way for an extra application, which we now describe. Namely, take an algebra A with a bilinear form, whose orthogonal base generates a natural proper semiring of A.

1.4.3. Table algebras. A classical example is the set of characters of a finite group G over a field whose characteristic does not divide |G|; since the sum (resp. product) of characters is the character of the direct sum (resp. tensor product) of their underlying representations, we can restrict to the semiring of characters, which is a free module over ℕ_{0}. A similar situation arises for the center of the group algebra, which is a free module whose base is comprised of the sums of elements from conjugacy classes. These algebras have been generalized by Hoheisel [13] and Arad-Blau [1] as explained in the fine survey by Blau [2], where he defines Hoheisel algebras and table algebras. These have a distinguished base L that spans the sub-semialgebra A^{+} that they generate over ℝ^{+}, so again A^{+} is a free module over ℝ^{+} (with unique base L), and a natural framework in which to build quadratic forms.

2. R-modules with unique base and their basic submodules

We assume throughout this paper that V is a free R-module with unique base 𝔅. Accordingly, we begin by examining this property.

Remark 2.1. Any change of base of the free module R^{n} is attained by multiplication by an invertible n × n matrix, so having unique base is equivalent to every invertible matrix in M_{n}(R) being a generalized permutation matrix.

Our interest in these modules stems from the following key fact.

Theorem 2.2 ([21, Corollary 5.25] and [22, Theorem 0.9] ). If R is a supertropical semiring, then every free R-module has unique base.

More generally, one may ask, “What conditions on the semiring R guarantee that R^{n} has unique base, or equivalently, that every invertible matrix is a generalized permutation matrix?” The matrix question was answered in [30] and [6]. In their terminology, an “antiring” is a semiring R such that R \ {0} is closed under addition. We prefer the terminology “lacks zero sums,” since this property holds also for sums of squares in a real closed field, and “antiring” does not seem appropriate in that context. Tan and Dolz̆na-Oblak classify the invertible matrices over these rings lacking zero sums. These are just the generalized permutation matrices when R \ {0} also is closed under multiplication, which they call “entire” (the case in tropical mathematics), and more generally by [6, Theorem 1] (as interpreted in Theorem 2.5) when R is indecomposable, i.e., not isomorphic to a direct product R_{1} × R_{2} of semirings.

Theorem 2.3 (cf. [6, §2, Corollary 3], an alternative proof given below). If the set R \ {0} is closed under addition and multiplication (i.e., a + b = 0 ⇒ a = b = 0, a · b = 0 ⇒ a = 0 or b = 0), then every free R-module has unique base.

In view of Remark 2.1, Theorem 2.3 follows from Dolz̆an and Oblak [6, §2, Corollary 3] using matrix arguments within a wider context extending work of Tan [30, Proposition 3.2], which in turn relies on Golan’s book on semirings [9, Lemma 19.4].

Example 2.4. Here are some instances where R\{0} is closed under addition and multiplication.

a) The “Boolean semifield” 𝔹 = {−∞, 0} (and thus subalgebras of algebras that are free modules over 𝔹). This shows that our results pertain to “F_{1}-geometry.”

b) Rewriting the Boolean semifield instead as 𝔹 = {0, 1} where 1 + 1 = 1, one can generalize it to {0, 1, …, q} L = [1, q] := {1, 2, …, q} the “truncated semiring without 0” of [23, Example 2.14], where “a + b′′ is defined to be the minimum of their sum and q. c) Function semirings, polynomial semirings, and Laurent polynomial

semirings over these semirings.

d) If F is a formally real field, i.e. −1 is not a sum of squares in F, then the subsemiring R = ΣF^{2}, consisting of all sums of squares in F, lacks zero sums. In fact R is a semifield; the inverse of a sum of squares

a = x_{1}^{2} + ⋯ + x_{r}^{2} is a^{−1} = (\frac{x_{1}}{a} ) + ⋯ + (\frac{x_{r}}{a} )^{2} .

Other than the trivial fact that every free R-module of rank 1 has unique base, all examples known to us of modules with unique base stem from Theorem 2.5, which is essentially [6, Theorem 1]:

Theorem 2.5 ([6, Theorem 1]). Assume that R is an indecomposable semiring lacking zero sums. Then every free R-module has unique base.

We now reprove Theorem 2.3 by a simple matrix-free argument in preparation for a reproof of the more general Theorem 2.5.

Proof of Theorem 2.3. Let V be a free R-module and 𝔅 a base of V. If x ∈ V \ {0} is given, we have a presentation

x = ∑_{i=1}^{r} λ_{i}x_{i}

with x_{i} ∈ 𝔅 and λ_{i} ∈ R\{0}. We call the set {x_{1}, …, x_{r}} ⊂ 𝔅 the support of x with respect to 𝔅 and denote this set by supp_{𝔅}(x). Note that if x, y ∈ V \{0}, then x + y ≠ 0 and

(2.1)  supp_{𝔅}(x + y) = supp_{𝔅}(x) ∪ supp_{𝔅}(y)

due to the assumption that λ + μ ≠ 0 for any λ, μ ∈ R \ {0}. Also

(2.2)  supp_{𝔅}(λx) = supp_{𝔅}(x)

for x ∈ V \ {0}, λ ∈ R \ {0}, due to the assumption that for λ, μ ∈ R \ {0} we have λμ ≠ 0. Now assume that 𝔅′ is a second base of V. Given x ∈ 𝔅, we have a presentation

x = λ_{1}y_{1} + ⋯ + λ_{r}y_{r}

with λ_{i} ∈ R \ {0} and distinct y_{i} ∈ 𝔅′. It follows from (2.1) and (2.2) that

{x} = supp_{𝔅}(x) = supp_{𝔅}(y_{1}) ∪ ⋯ ∪ supp_{𝔅}(y_{r}).

This forces

(2.3)  {x} = supp_{𝔅}(y_{1}) = ⋯ = supp_{𝔅}(y_{r}).

¿From this, we infer that r = 1. Indeed, suppose that r ≥ 2. Then y_{1} = μ_{1}x, y_{2} = μ_{2}x with μ_{1}, μ_{2} ∈ R \ {0}. But this implies μ_{2}y_{1} = μ_{1}y_{2}, a contradiction since y_{1}, y_{2} are different elements of a base of V. Thus {x} = supp_{𝔅}(y) for a unique y ∈ 𝔅′, which means y = λx with λ ∈ R \ {0}. By symmetry we have a unique z ∈ 𝔅 and μ ∈ R \ {0} with x = μz. Then x = λμz, whence x = z and λμ = 1. Thus λ, μ ∈ R^{∗} and x ∈ R^{∗}y, y ∈ R^{∗}x. Of course, y runs through all of 𝔅′ if x runs through 𝔅, since both 𝔅 and 𝔅′ span the module V. □

Proof of Theorem 2.5. Assume that 𝔅 and 𝔅′ are bases of V. Given x ∈ 𝔅, we write again

(2.4)  x = λ_{1}y_{1} + ⋯ + λ_{r}y_{r}

with different y_{i} ∈ 𝔅′, λ_{i} ∈ R \ {0}. But now, instead of (2.3) we can only conclude that

(2.5)  {x} = supp_{𝔅}(λ_{1}y_{1}) = ⋯ = supp_{𝔅}(λ_{i}y_{i}).

Thus we have scalars μ_{i} ∈ R \ {0} such that

(2.6)  λ_{i}y_{i} = μ_{i}x  for 1 ≤ i ≤ r.

Suppose that r ≥ 2. Then we have for all i, j ∈ {1, …, r} with i ≠ j.

μ_{j}λ_{i}y_{i} = μ_{j}μ_{i}x = μ_{i}μ_{j}x = μ_{i}λ_{j}y_{j}.

Since the y_{i} are elements of a base, this implies μ_{i}λ_{j} = μ_{j}λ_{i} = 0 for i ≠ j and then

(2.7)  μ_{i}μ_{j} = 0  for i ≠ j.

On the other hand, we obtain from (2.4) and (2.6) that

x = μ_{1}x + μ_{2}x + ⋯ + μ_{r}x,

and then

(2.8)  1 = μ_{1} + μ_{2} + ⋯ + μ_{r}.

Multiplying (2.8) by μ_{i} and using (2.7), we obtain

(2.9)  μ_{i}^{2} = μ_{i}.

Thus

R ≅ Rμ_{1} × ⋯ × Rμ_{r}.

This contradicts our assumption that R is indecomposable. We have proved that r = 1. Thus for every x ∈ 𝔅 there exist unique y ∈ 𝔅′ and λ ∈ R with x = λy. By the same argument as in the end of proof of Theorem 2.3, we conclude that 𝔅 is projectively unique. □

Of course, if R \ {0} is closed under multiplication, i.e., R has no zero divisors, then R is indecomposable. This also holds when R is supertropical (cf. [25, §3], [22, Definition 0.3]), since then for any two elements μ_{1}, μ_{2} of R with μ_{1}+μ_{2} = 1 either μ_{1} = 1 or μ_{2} = 1. Thus, Theorem 2.5 generalizes both Theorems 2.2 and 2.3. The following example reveals that Theorem 2.5 is the best we can hope for, in order to guarantee that every free R-module has unique base, as long as we stick to the natural assumption that R is a semiring lacking zero sums.

Example 2.6. If R_{0} is a semiring lacking zero sums, then R := R_{0} × R_{0} also lacks zero sums. Put μ_{1} = (1, 0), μ_{2} = (0, 1). These are idempotents in R with μ_{1}μ_{2} = 0 and μ_{1} + μ_{2} = 1. Now let V be a free R-module with base 𝔅 = {ε_{1}, ε_{2}, …, ε_{n}}, n ≥ 2, choose a permutation π ∈ S_{n}, π ≠ 1, and define

ε′_{i} := μ_{1}ε_{i} + μ_{2}ε_{π(i)} (1 ≤ i ≤ n).

We claim that 𝔅′ := {ε′_{1}, …, ε′_{n}} is another base of V. Indeed, V is a free R_{0}-module with base (μ_{i}ε_{j} | 1 ≤ i ≤ 2, 1 ≤ j ≤ n). We have

μ_{1}ε′_{i} = μ_{1}ε_{i}, μ_{2}ε′_{i} = μ_{2}ε_{π(i)},

and thus (μ_{i}ε′_{j} | 1 ≤ i ≤ 2, 1 ≤ j ≤ n) is a permutation of this base over R_{0}, i.e., regarded as a set, the same base. Thus certainly 𝔅′ spans V as R-module.

Given x ∈ V, let x = ∑_{1}^{n} a_{i}ε′_{i} with a_{i} ∈ R. We have

a_{i} = a_{i1}μ_{1} + a_{i2}μ_{2}  with  a_{i1} ∈ R_{0}, a_{i2} ∈ R_{0},

whence

x = ∑_{i=1}^{n} a_{i1}(μ_{1}ε_{i}) + ∑_{i=1}^{n} a_{i2}(μ_{2}ε_{π(i)}).

This shows that the coefficients a_{i1}, a_{i2} ∈ R_{0} are uniquely determined by x, whence the coefficients a_{i} ∈ R are also uniquely determined by x. Our claim is proved. Since supp_{𝔅}(ε′_{i}) has two elements if π(i) ≠ i, 𝔅′ differs projectively from 𝔅. The base 𝔅 of the R-module V is not unique.

3. Orthogonal decompositions of quadratic modules with unique base

Assume that V is an R-module equipped with a fixed quadratic form q : V → R. We then call V = (V, q) a quadratic R-module.

Definition 3.1.

(a) Given two submodules W_{1}, W_{2} of the R-module V, we say that W_{1} is disjointly orthogonal to W_{2}, if W_{1}∩W_{2} = {0} and q(x+y) = q(x)+q(y) for all x ∈ W_{1}, y ∈ W_{2}, i.e., q is quasilinear on W_{1} × W_{2}. (We say “orthogonal” for short, when it is clear a priori that W_{1} ∩ W_{2} = {0}.)

(b) We write V = W_{1} ⊥ W_{2} if V = W_{1} ⊕ W_{2} (as R-module) with W_{1} disjointly orthogonal to W_{2}. We then call W_{1} an orthogonal summand of W, and W_{2} an orthogonal complement of W_{1} in V.

Caution. If V = W_{1} ⊥ W_{2}, we may choose a companion b of q such that b(W_{1}, W_{2}) = 0, but note that it could well happen that the set of all x ∈ V with b(x, W_{1}) = 0 is bigger than W_{2}, even if R is a semifield and q|W_{1} is anisotropic (e.g., if q itself is quasilinear). Our notion of orthogonality does not refer to any bilinear form.

We now also define infinite orthogonal sums. This seems to be natural, even if we are originally interested only in finite orthogonal sums. Indeed, even if R is a semifield, a free R-module with finite base often has many submodules which are not finitely generated.

Definition 3.2. Let (V_{i} | i ∈ I) be a family of submodules of the quadratic module V. We say that V is the orthogonal sum of the family (V_{i}), and then write

V = ⊥_{i∈I} V_{i},

if for any two different indices i, j the submodule V_{i} is disjointly orthogonal to V_{j}, and moreover V = ⨁_{i∈I} V_{i}.

N.B. Of course, then for any subset J ⊂ I, the module V_{J} = ∑_{i∈J} V_{i} is the

orthogonal sum of the subfamily (V_{i} | i ∈ J); in short,

V_{J} = ⊥_{i∈J} V_{i}.

We state a fact which, perhaps contrary to first glance, is not completely trivial.

Proposition 3.3. Assume that we are given an orthogonal decomposition V = ⊥_{i∈I} V_{i}. Let J and K be two disjoint subsets of I. Then the submodule

V_{J} = ⊥_{i∈J} V_{i} of V is disjointly orthogonal to V_{K} = ⊥_{i∈K} V_{i}, and thus

V_{J∪K} = V_{J} ⊥ V_{K}.

Proof. It follows from Lemma 1.4 above that for any three different indices i, j, k the form q is quasilinear on V_{i} × (V_{j} + V_{k}), and thus V_{i} is orthogonal to V_{j} ⊥ V_{k}. By iteration, we see that the claim holds if J and K are finite. In the general case, let x ∈ V_{J} and y ∈ V_{K}. There exist finite subsets J′, K′ of J and K with x ∈ V_{J′}, y ∈ V_{K′}, and thus q(x + y) = q(x) + q(y). This proves that V_{J} is orthogonal to V_{K}. □

In the rest of this section, we assume that V has unique base.

Definition 3.4. We call a submodule W of V basic, if W is spanned by 𝔅_{W} := 𝔅∩W, and thus W is free with base 𝔅_{W} . Note that then we have a unique direct decomposition V = W ⊕ U, where the submodule U is basic with base 𝔅 \ 𝔅_{W} . W and U again are R-modules with unique base. We call U the complement of W in V, and write U = W^{c}.

The theory of basic submodules of V is of utmost simplicity. All of the following is obvious.

Scholium 3.5.

(a) We have a bijection W ↦ 𝔅_{W} := 𝔅 ∩ W from the set of basic submodules of V onto the set of subsets of 𝔅.

(b) If W_{1} and W_{2} are basic submodules of V, then also W_{1}∩W_{2} and W_{1}+W_{2} are basic submodules of V, and

𝔅_{W_{1}∩W_{2}} = 𝔅_{W_{1}} ∩ 𝔅_{W_{2}}, 𝔅_{W_{1}+W_{2}} = 𝔅_{W_{1}} ∪ 𝔅_{W_{2}}.

(c) If W is a basic submodule of V, then as stated above,

𝔅_{W^{c}} = 𝔅 \ 𝔅_{W} .

(d) Finally, if W_{1} ⊂ W_{2} are basic submodules of V, then W_{1} is basic in W_{2} and W_{1}^{c} ∩ W_{2} is the complement of W_{1} in W_{2}.

Thus a basic orthogonal summand W of V has only one basic orthogonal complement, namely, W^{c}, equipped with the form q|W^{c}.

Definition 3.6. If the quadratic module V has a basic orthogonal summand W ≠ V, we call V decomposable. Otherwise we call V indecomposable. More generally, we call a basic submodule X of V decomposable if X is decomposable with respect to q|X, and otherwise we call X indecomposable.

Our next goal is to decompose the given quadratic module V orthogonally into indecomposable basic submodules. Therefore, we choose a base 𝔅 of V (unique up to multiplication by scalar units). We then choose a companion b of q such that b(ε, η) = 0 for any two different ε, η ∈ 𝔅 such that q is quasilinear on Rε × Rη, cf. [22, Theorem 6.3]. We call such a companion b a quasiminimal companion of q.

Comment. In important cases, e.g., if R is supertropical or more generally “upper bound” (cf. [22, Definition 5.1]), the set of companions of q can be partially ordered in a natural way. The prefix “quasi” here is a reminder that we do not mean minimality with respect to such an ordering.

Lemma 3.7. Let W and W′ be basic submodules of V with W ∩ W′ = {0}. If b is any quasiminimal companion of q, then W is (disjointly) orthogonal to W′ iff b(W, W′) = 0.

Proof. If b(W, W′) = 0, then q(x + y) = q(x) + q(y) for any x ∈ W and y ∈ W′, which means by definition that W is orthogonal to W′. (This holds for any companion b of q.) Conversely, if W is orthogonal to W′, then for base vectors ε ∈ 𝔅_{W} , η ∈ 𝔅_{W′} the form q is quasilinear on Rε × Rη and thus b(ε, η) = 0. This implies that b(W, W′) = 0. □

We now introduce the following equivalence relation on the set 𝔅. We choose a quasiminimal companion b of q. Given ε, η ∈ 𝔅, we put ε ∼ η, iff either ε = η, or there exists a sequence ε_{0}, ε_{1}, …, ε_{r} in 𝔅, r ≥ 1, such that ε = ε_{0}, η = ε_{r}, and ε_{i} ≠ ε_{i+1}, b(ε_{i}, ε_{i+1}) ≠ 0 for i = 0, …, r − 1.

Theorem 3.8. Let {𝔅_{k} | k ∈ K} denote the set of equivalence classes in 𝔅 and, for every k ∈ K, let W_{k} denote the submodule of V having base 𝔅_{k}.

(a) Then every W_{k} is an indecomposable basic submodule of V and

V = ⊥_{k∈K} W_{k}.

(b) Every indecomposable basic submodule U of V is contained in W_{k}, for some k ∈ K uniquely determined by U.

(c) The modules W_{k}, k ∈ K, are precisely all the indecomposable basic orthogonal summands of V.

Proof. (a): Suppose that W_{k} has an orthogonal decomposition W_{k} = X ⊥ Y with basic submodules X ≠ 0, Y ≠ 0. Then 𝔅_{k} is the disjoint union of the non-empty sets 𝔅_{X} and 𝔅_{Y} . Choosing ε ∈ 𝔅_{X} and η ∈ 𝔅_{Y}, there exists a sequence ε_{0}, ε_{1}, …, ε_{r} in 𝔅_{k} with ε = ε_{0}, η = ε_{r} and b(ε_{i−1}, ε_{i}) ≠ 0, ε_{i−1} ≠ ε_{i}, for 1 ≤ i ≤ r. Let s denote the last index in {1, …, r} with ε_{s} ∈ 𝔅_{X}. Then s < r and ε_{s+1} ∈ 𝔅_{Y}. But b(X, Y) = 0 by Lemma 3.7 and thus b(ε_{s}, ε_{s+1}) = 0, a contradiction. This proves that W_{k} is indecomposable. Since 𝔅 is the disjoint union of the sets 𝔅_{k}, we have

V = ⨁_{k∈K} W_{k}.

Finally, if k ≠ ℓ, then b(W_{k}, W_{ℓ}) = 0 by the nature of our equivalence relation. Thus

V = ⊥_{k∈K} W_{k}.

(b): Given an indecomposable basic submodule U of V, we choose k ∈ K with 𝔅_{U} ∩ 𝔅_{k} ≠ ∅. Then U ∩ W_{k} ≠ 0. ¿From V = W_{k} ⊕ W_{k}^{c}, we conclude that U = (U ∩ W_{k}) ⊕ (U ∩ W_{k}^{c}), and then have U = (U ∩ W_{k}) ⊥ (U ∩ W_{k}^{c}) because W_{k} is orthogonal to W_{k}^{c}. Since U is indecomposable and U ∩ W_{k} ≠ 0, it follows that U = U ∩ W_{k}, i.e., U ⊂ W_{k}. Since W_{k} ∩ W_{ℓ} = 0 for k ≠ ℓ, it is clear that k is uniquely determined by U.

(c): If U is an indecomposable basic orthogonal summand of V, then V = U ⊥ U^{c}. We have U ⊂ W_{k} for some k ∈ K, and obtain W_{k} = U ⊥ (U^{c} ∩ W_{k}), whence W_{k} = U. □

Definition 3.9. We call the submodules W_{k} of V occurring in Theorem 3.8 the indecomposable components of the quadratic module V.

The following facts are easy consequences of the theorem.

Remark 3.10.

(i) If U is a basic orthogonal summand of V, then the indecomposable components of the quadratic module U = (U, q|U) are the indecomposable components of V contained in U.

(ii) If U is any basic submodule of V, then

U = ⊥_{k∈K} (U ∩ W_{k}) ,

and every submodule U ∩ W_{k} ≠ {0} is an orthogonal sum of indecomposable components of U.

4. Orthogonal decomposition of bilinear modules with unique base

We now outline a theory of symmetric bilinear forms analogous to the theory for quadratic forms given in §3. The bilinear theory is easier than the quadratic theory due the fact that, in contrast to quadratic forms, on a free module we do not need to distinguish between “functional” and “formal” bilinear forms cf. [22, §1]. As before, R is a semiring. Assume in the following that V is an R-module equipped with a fixed symmetric bilinear form b : V ×V → R. We then call V = (V, b) a bilinear R-module. If X is a submodule of V, we denote the restriction of b to X × X by b|X.

Definition 4.1.

(a) Given two submodules W_{1}, W_{2} of the R-module V, we say that W_{1} is disjointly orthogonal to W_{2}, if W_{1} ∩W_{2} = {0} and b(W_{1}, W_{2}) = 0, i.e., b(x, y) = 0 for all x ∈ W_{1}, y ∈ W_{2}.

(b) We write V = W_{1} ⊥ W_{2} if W_{1} is disjointly orthogonal to W_{2} and moreover V = W_{1}⊕W_{2} (as R-module). We then call W_{1} an orthogonal summand of V and W_{2} an orthogonal complement of W_{1} in V.

Definition 4.2. Let (V_{i} | i ∈ I) be a family of submodules of the bilinear module V. We say that V is the orthogonal sum of the family (V_{i}), and then write

V = ⊥_{i∈I} V_{i},

if for any two different indices i, j the submodule V_{i} is disjointly orthogonal to V_{j}, and moreover V = ⨁_{i∈I} V_{i}.

In contrast to the quadratic case, the exact analog of Proposition 3.3 is now a triviality.

Proposition 4.3. Assume that V = ⊥_{i∈I} V_{i}. Let J and K be disjoint subsets

of I. Then V_{J} = ⊥_{i∈J} V_{i} is disjointly orthogonal to V_{K} = ⊥_{i∈K} V_{i}, and

V_{J∪K} = V_{J} ⊥ V_{K}.

In the following, we assume again that V has unique base. Then again a basic orthogonal summand W of V has only one basic orthogonal complement in V, namely, W^{c} equipped with the bilinear form b|W^{c}. For X a basic submodule of V, we define the properties “decomposable” and “indecomposable” in exactly the same way as indicated by Definition 3.6 in the quadratic case. We start with a definition and description of the “indecomposable components” of V = (V, b) in a similar fashion as was done in §3 for quadratic modules. We choose a base 𝔅 of V and again introduce the appropriate equivalence relation on the set 𝔅, but now we adopt a more elaborate terminology than in §3. This will turn out to be useful later on.

Definition 4.4. We call the symmetric bilinear form b alternate if b(ε, ε) = 0 for every ε ∈ 𝔅.

Comment. Beware that this does not imply that b(x, x) = 0 for every x ∈ V. The classical notion of an alternating bilinear form is of no use here since in the semirings under consideration here (cf. §2) α + β = 0 implies α = β = 0, whence b(x + y, x + y) = 0 implies b(x, y) = 0. An alternating bilinear form in the classical sense would be identically zero.

Definition 4.5. We associate to the given symmetric bilinear form b an alternate bilinear form b_{alt} by the rule

b_{alt}(ε, η) = \begin{cases}b(ε, η) & if ε ≠ η \\ 0 & if ε = η \end{cases}

for any ε, η ∈ 𝔅.

Lemma 4.6. Let W and W′ be basic submodules of V with W ∩ W′ = {0}. Then W is (disjointly) orthogonal to W′ iff b_{alt}(W, W′) = 0.

Proof. This can be seen exactly as with the parallel Lemma 3.7. Just replace in its proof the quasiminimal companion of q by b_{alt}. □

Definition 4.7.

(a) A path Γ in V = (V, b) of length r ≥ 1 in 𝔅 is a sequence ε_{0}, ε_{1}, …, ε_{r} of elements of 𝔅 with

b_{alt}(ε_{i}, ε_{i+1}) ≠ 0 (0 ≤ i ≤ r − 1).

In essence this condition does not depend on the choice of the base 𝔅, since 𝔅 is unique up to multiplication by units, and so we also say that Γ is a path in V. We say that the path runs from ε := ε_{0} to η := ε_{r}, or that the path connects ε to η. A path of length 1 is called an edge. This is just a pair (ε, η) in 𝔅 with ε ≠ η and b(ε, η) ≠ 0.

(b) We define an equivalence relation on 𝔅 as follows. Given ε, η ∈ 𝔅, we declare that ε ∼ η if either ε = η or there runs a path from ε to η.

It is now obvious how to mimic the theory of indecomposable components from the end of §3 in the bilinear setting.

Scholium 4.8. Theorem 3.8 and its proof remain valid for the present equivalence relation on 𝔅. We only have to replace the quasiminimal companion b of q there by b_{alt} and to use Lemma 4.6 instead of Lemma 3.7. Again we denote the set of equivalence classes of 𝔅 by {𝔅_{k} | k ∈ K} and the submodule of V with base 𝔅_{k} by V_{k}, and again we call the V_{k} the indecomposable components of V. Also the analog to Remark 3.10 remains valid.

We state a consequence of the parallel between the two decomposition theories.

Theorem 4.9. Assume that (V, q) is a quadratic module with unique base and b is a quasiminimal companion of q. The indecomposable components of (V, q) coincide with the indecomposable components of (V, b).

Proof. The equivalence relation used in Theorem 3.8 is the same as the equivalence relation in Definition 4.7. □

We add an easy observation on bilinear modules.

Proposition 4.10. Assume that (V, b) is a bilinear R-module with unique base. A basic submodule W of V is indecomposable with respect to b, iff W is indecomposable with respect to b_{alt}.

Proof. The equivalence relation on 𝔅 just defined (Definition 4.7) does not change if we replace b by b_{alt}. □

5. Isometries, isotypical components, and a cancelation theorem

Let R be any semiring.

Definition 5.1.

(a) For quadratic R-modules V = (V, q) and V′ = (V′, q′), an isometry σ : V → V′ is a bijective R-linear map with q′(σx) = q(x) for all x ∈ V. Likewise, if V = (V, b) and (V′, b′) are bilinear R-modules, an isometry is a bijective R-linear map σ : V → V′ with b′(σx, σy) = b(x, y) for all x, y ∈ V.

(b) If there exists an isometry σ : V → V′, we call V and V′ isometric and write V ≅ V′. We then also say that V and V′ are in the same isometry class.

In the following we study quadratic and bilinear R-modules with unique base on an equal footing. It would not hurt if we supposed that the semiring R satisfies the conditions in Theorem 2.5, so that every free R-module has unique base, but the simplicity of all of the arguments in the present section becomes more apparent if we do not rely on Theorem 2.5.

Notation/Definition 5.2.

(a) Let (V_{λ}^{0} | λ ∈ Λ) be a set of representatives of all isometry classes of indecomposable quadratic (resp. bilinear) R-modules with unique base of rank bounded by the cardinality of V, in order to avoid set-theoretical complications.

(b) If W is such an R-module, where W ≅ V_{λ}^{0} for a unique λ ∈ Λ, we say that W has type λ (or: W is indecomposable of type λ).

(c) We say that a quadratic (resp. bilinear) module W ≠ 0 with unique base is isotypical of type λ, if every indecomposable component of V has type λ.

(d) Finally, given a quadratic (resp. bilinear) R-module with unique base, we denote the sum of all indecomposable components of V of type λ by V_{λ} and call the V_{λ} ≠ 0 the isotypical components of V.

The following is now obvious from §3 and §4 (cf. Theorem 3.8 and Scholium 4.8).

Proposition 5.3. If V is a quadratic or bilinear R-module with unique base, then

V = ⊥_{λ∈Λ′} V_{λ}

with Λ′ = {λ ∈ Λ | V_{λ} ≠ 0}.

Since our notion of orthogonality for basic submodules of V is encoded in the linear and quadratic, resp. bilinear, structure of V, the following fact also is obvious, but in view of its importance will be dubbed a “theorem”.

Theorem 5.4. Assume that V and V′ are quadratic (resp. bilinear) R-modules with unique bases and σ : V → V′ is an isometry. Let {V_{k} | k ∈ K} denote the set of indecomposable components of V.

(a) {σ(V_{k}) | k ∈ K} is the set of indecomposable components of V′.

(b) If V_{k} has type λ, then σ(V_{k}) has type λ, and so σ(V_{λ}) = V′_{λ} for every λ ∈ Λ.

Also in the remainder of the section, we assume that the quadratic or bilinear modules have unique base.

Definition 5.5. Let O(V) denote the group of all isometries σ : V → V (i.e., automorphisms) of (V, q), resp. (V, b). As usual, we call O(V) the orthogonal group of V.

Theorem 5.4 has the following immediate consequence.

Corollary 5.6. Every σ ∈ O(V) permutes the indecomposable components of V of fixed type λ, and so σ(V_{λ}) = V_{λ} for every λ ∈ Λ. We have a natural isomorphism

\xymatrix{O(V) \ar[r]^{1:1} & ∏_{λ∈Λ′} O(V_{λ}),}

sending σ ∈ O(V) to the family of its restrictions σ|V_{λ} ∈ O(V_{λ}).

Definition 5.7.

(a) Let λ ∈ Λ. We denote the cardinality of the set of indecomposable components of V_{λ} by m_{λ}(V), and we call m_{λ}(V) the multiplicity of V_{λ}. {N.B. m_{λ}(V) can be infinite or zero.}

(b) If m_{λ} ∈ ℕ_{0} for every λ ∈ Λ, we say that V is isotypically finite.

Theorem 5.8. If V and V′ are quadratic or bilinear R-modules with unique bases, then V ≅ V′ iff m_{λ}(V) = m_{λ}(V′) for every λ ∈ Λ.

Proof. This follows from Proposition 5.3 and Theorem 5.4. □

We are ready for a main result of the paper.

Theorem 5.9. Assume that W_{1}, W_{2}, W′_{1}, W′_{2} are quadratic or bilinear modules with unique base and that W_{1} is isotypically finite. Assume furthermore that W_{1} ≅ W′_{1} and that W_{1} ⊥ W_{2} ≅ W′_{1} ⊥ W′_{2}. Then W_{2} ≅ W′_{2}.

Proof. For every λ ∈ Λ, clearly m_{λ}(V) = m_{λ}(W_{1}) + m_{λ}(W_{2}) and m_{λ}(V′) = m_{λ}(W′_{1}) + m_{λ}(W′_{2}). Since V ≅ V′, the multiplicities m_{λ}(V) and m_{λ}(V′) are equal, and since W_{1} ≅ W′_{1}, the same holds for the multiplicities m_{λ}(W′_{1}). Since m_{λ}(W_{1}) = m_{λ}(W′_{1}) is finite, it follows that m_{λ}(W_{2}) = m_{λ}(W′_{2}). By Theorem 5.8 this implies that W_{2} ≅ W′_{2}. □

Remark 5.10. If the free R-module W_{1} has finite rank, then certainly W_{1} is isotypically finite. Thus Theorem 5.9 may be viewed as the analog of Witt’s cancellation theorem from 1937 [32] proved for quadratic forms over fields.

The assumption of isotypical finiteness in Theorem 5.9 cannot be relaxed. Indeed if m_{λ}(W_{1}) is infinite for at least one λ ∈ Λ, then the cancelation law becomes false. This is evident by Theorem 5.8 and the following example.

Example 5.11. Assume that V is the orthogonal sum of infinitely many copies V_{1}, V_{2}, … of an indecomposable quadratic or bilinear module V_{0} with unique base. Consider the following submodules of V:

\begin{matrix}W_{1} := & V_{2} ⊥ V_{3} ⊥ ⋯,  W_{2} := V_{1}, \\ W′_{1} := & V_{3} ⊥ V_{4} ⊥ ⋯,  W′_{2} := V_{1} ⊥ V_{2}. \end{matrix}

Then W_{1} ⊥ W_{2} = V = W′_{1} ⊥ W′_{2}, and W_{1} ≅ W′_{1}. But W_{2} is not isometric to W′_{2}.

6. Expansions and tensor products

Let q : V → R be a quadratic form on an R-module V. We recall from [22, §1] that, when V is free with base (ε_{i} : i ∈ I), then q admits a (not necessarily unique) balanced companion, i.e., a companion b : V × V → R such that b(x, x) = 2q(x) for all x ∈ V, and that it suffices to know for this that b(ε_{i}, ε_{i}) = 2q(ε_{i}) for all i ∈ I [22, Proposition 1.7]. Balanced companions are a crucial ingredient in our definition below of a tensor product of a free bilinear module and a free quadratic module. They arise from “expansions” of q, defined as follows, cf. [22, Definition 1.9].

Definition 6.1. A bilinear form B : V × V → R (not necessarily symmetric) is an expansion of a balanced pair (q, b) if B + B^{t} = b, i.e.,

(6.1)  B(x, y) + B(y, x) = b(x, y)

for all x, y ∈ V, and

(6.2)  q(x) = B(x, x)

for all x ∈ V. If only the form q is given and (6.2) holds, we say that B is an expansion of q.

As stated in the [22, §1], every bilinear form B : V ×V → R gives us a balanced pair (q, b) via (6.1) and (6.2), and, if the R-module V is free, we obtain all such pairs (q, b) in this way. But we will need a description of all expansions of (q, b) in the free case.

Construction 6.2. Assume that V is a free R-module and (ε_{i} | i ∈ I) is a base of V. When (q, b) is a balanced pair on V, we obtain all expansions B : V × V → R of (q, b) as follows. Let α_{i} := q(ε_{i}), β_{ij} := b(ε_{i}, ε_{j}) for i, j ∈ I. We have β_{ij} = β_{ji}. We choose a total ordering on I and for every i < j two elements χ_{ij}, χ_{ji} ∈ R with

β_{ij} = χ_{ij} + χ_{ji}, (i < j).

We furthermore put

χ_{ii} := α_{i},

and define B by the rule

B(ε_{i}, ε_{j}) = χ_{ij}

for all (i, j) ∈ I × I.

In practice one usually chooses χ_{ij} = β_{ij}, χ_{ji} = 0 for i < j, i.e., takes the unique “triangular” expansion B of (q, b), cf. [22, §1], but now we do not want to depend on the choice of a total ordering of the base (ε_{i} | i ∈ I). We used such an ordering above only to ease notation.

Tensor products over semirings in general require the use of congruences [10], but for free modules the basics can be done precisely as over rings, and we leave the formal details to the interested reader. We only state here that, given two free R-modules V_{1} and V_{2}, with bases 𝔅_{1} and 𝔅_{2}, the R-module V_{1} ⊗_{R} V_{2} “is” the free R-module with base 𝔅_{1} ⊗ 𝔅_{2}, which is a renaming of 𝔅_{1} × 𝔅_{2}, writing ε ⊗ η for (ε, η) with ε ∈ 𝔅_{1}, η ∈ 𝔅_{2}. If

𝔅_{1} = {ε_{i} | i ∈ I}, 𝔅_{2} = {η_{j} | j ∈ J}

and x = ∑_{i∈I} x_{i}ε_{i} ∈ V_{1} and y = ∑_{j∈J} y_{j}η_{j} ∈ V_{2}, we define, as common over rings,

(6.3)  x ⊗ y := ∑_{(i,j)∈I×J} x_{i}y_{j}(ε_{i} ⊗ y_{j}),

and this vector is independent of the choice of the bases 𝔅_{1} and 𝔅_{2}. If B_{1} and B_{2} are bilinear forms on V_{1} and V_{2} respectively, we have a well defined bilinear form on V_{1} ⊗_{R} V_{2}, denoted by B_{1} ⊗ B_{2}, such that for any x_{i} ∈ V_{1}, y_{j} ∈ V_{2} (i, j ∈ {1, 2})

(6.4)  (B_{1} ⊗ B_{2})(x_{1} ⊗ x_{2}, y_{1} ⊗ y_{2}) = B_{1}(x_{1}, y_{1})B_{2}(x_{2}, y_{2}).

If b_{1} and b_{2} are symmetric bilinear forms on V_{1} and V_{2} respectively, then b_{1} ⊗b_{2} is symmetric. Then we call the bilinear module (V_{1} ⊗_{R} V_{2}, b_{1} ⊗ b_{2}) the tensor product of the bilinear modules (V_{1}, b_{1}) and (V_{2}, b_{2}). We next define the tensor product of a free bilinear and a free quadratic module. The key fact which allows us to do this in a reasonable way is as follows.

Proposition 6.3. Let γ : U × U → R be a symmetric bilinear form and (q, b) a balanced quadratic pair on V. Assume that B and B′ are two expansions of (q, b). Then the bilinear forms γ ⊗ B and γ ⊗ B′ on U ⊗ V yield the same balanced pair (q̃, b̃) on U ⊗ V. We have b̃ = γ ⊗ b, whence for u_{1}, u_{2} ∈ U, v_{1}, v_{2} ∈ V,

(6.5)  b̃(u_{1} ⊗ v_{1}, u_{2} ⊗ v_{2}) = γ(u_{1}, u_{2})b(v_{1}, v_{2}).

Furthermore, for u ∈ U and v ∈ V,

(6.6)  q̃(u ⊗ v) = γ(u, u)q(v).

Proof. γ⊗B+(γ⊗B)^{t} = γ⊗B+γ^{t}⊗B^{t} = γ⊗B+γ⊗B^{t} = γ⊗(B+B^{t}) = γ⊗b. Also γ ⊗ B′ + (γ ⊗ B′)^{t} = γ ⊗ b. Furthermore,

\begin{aligned}(γ ⊗ B)(u ⊗ v, u ⊗ v) & = γ(u, u)B(v, v) \\ & = γ(u, u)q(v) & = (γ ⊗ B′)(u ⊗ v, u ⊗ v) \end{aligned}

for any u ∈ U, v ∈ V. Together these equations imply

(γ ⊗ B)(z, z) = (γ ⊗ B′)(z, z)

for any z ∈ U ⊗ V. □

Definition 6.4. We call q̃ the tensor product of the bilinear form γ and the quadratic form q with respect to the balanced companion b of q, and write

q̃ = γ ⊗_{b} q,

and we also write Ṽ = U ⊗_{b} V for the quadratic R-module Ṽ = (U ⊗ V, q̃).

Remark 6.5. If q has only one balanced companion, we may suppress the “b” here, writing q̃ = γ ⊗ q. Cases in which this happens are: q is rigid, V has rank one, R is embeddable in a ring.

Proposition 6.6. If U = (U, γ) has an orthogonal decomposition U = ⊥_{i∈I} U_{i},

then

U ⊗_{b} V = ⊥_{i∈I} U_{i} ⊗_{b} V.

Proof. It is immediate that (γ ⊗ b)(U_{i} ⊗ V, U_{j} ⊗ V) = 0 for i ≠ j. □

We proceed to explicit examples. For this we need notation from [22, §1] which we recall for the convenience of the reader. Assume that V is free of finite rank n and 𝔅 is a base of V for which we now choose a total ordering, 𝔅 = (ε_{1}, ε_{2}, …, ε_{n}). Then we identify a bilinear form B on V with the (n × n)-matrix

(6.7)  B = (\begin{matrix}β_{11} & β_{12} & ⋯ & β_{1n} \\ β_{21} & β_{22} & & β_{2n} \\ ⋮ & ⋮ & ... & ⋮ \\ β_{n1} & ⋯ & & β_{nn} \end{matrix}) ,

where β_{ij} = B(ε_{1}, ε_{j}). In particular, a bilinear R-module (V, β) is denoted by a symmetric (n × n)-matrix, namely its Gram matrix b = (β_{ij})_{1≤i,j≤n}, where β_{ij} = β_{ji} = b(ε_{i}, ε_{j}). Given a quadratic module (V, q), we choose a triangular expansion

(6.8)  B = (\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ 0 & α_{2} & ⋯ & α_{2n} \\ ⋮ & & ... & ⋮ \\ 0 & ⋯ & 0 & α_{n} \end{matrix})

of q and denote q by the triangular scheme

(6.9)  q = [\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ & α_{2} & ⋯ & α_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}] ,

so that q is given by the polynomial

q(x) = ∑_{i=1}^{n} α_{i}x_{i}^{2} + ∑_{i<j}^{n} α_{ij}x_{i}x_{j}.

(Such triangular schemes have already been used in the literature when R is a ring, e.g. [28, I §2].) In the case that q is diagonal, i.e., all α_{ij} with i < j are zero, we usually write instead of (6.8) the single row

(6.10)  q = [α_{1}, α_{2}, …, α_{n}].

Analogously we use for a diagonal symmetric bilinear form b (i.e., b(ε_{i}, ε_{j}) = 0 for i ≠ j) the notation

(6.11)  b = ⟨β_{11}, β_{22}, …, β_{nn}⟩.

We note that the quadratic form (6.9) has the balanced companion

(6.12)  b = (\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ α_{12} & α_{2} & & α_{2n} \\ ⋮ & ⋮ & ... & ⋮ \\ α_{1n} & ⋯ & & α_{n} \end{matrix})

and (6.10), being diagonal, has the balanced companion

(6.13)  b = ⟨2α_{1}, 2α_{2}, …, 2α_{n}⟩.

Example 6.7. If a_{1}, …, a_{n}, c ∈ R, then

(6.14)  ⟨a_{1}, … a_{n}⟩ ⊗ [c] = [a_{1}c, …, a_{n}c].

This is evident from Proposition 6.6 and the rule ⟨a⟩ ⊗ [c] = [ac] for onedimensional forms which holds by (6.6). In particular

(6.15)  [a_{1}, …, a_{n}] = ⟨a_{1}, … a_{n}⟩ ⊗ [1].

Example 6.8. (As before, R is any semiring.) Assume that V = (V, q) has dimension n, and take a base η_{1}, …, η_{n} of V. Let

(U, γ) = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

with base ε_{1}, ε_{2}. We choose a balanced companion b of V, written as a symmetric (n × n)-matrix (b(η_{i}, η_{j})). We see by the use of the rules (6.5) and (6.6) that

(6.16)  (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ⊗_{b} q = [\frac{0 b}{0}]

written with respect to the base

ε_{1} ⊗ η_{1}, …, ε_{1} ⊗ η_{n}, ε_{2} ⊗ η_{1}, …, ε_{2} ⊗ η_{n}.

This example illustrates dramatically that in general the tensor product of γ and q depends on the chosen balanced companion b of q: tensoring q by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) produces the symmetric matrix of b.

Remark 6.9. If γ_{1} and γ_{2} are bilinear forms on the same free R-module U, then the rules (6.5) and (6.6) imply for any λ_{1}, λ_{2} ∈ R that

(6.17)  (λ_{1}γ_{1} + λ_{2}γ_{2}) ⊗_{b} q = λ_{1}(γ_{1} ⊗_{b} q) + λ_{2}(γ_{2} ⊗_{b} q).

Example 6.10. Using (6.17) with

γ_{1} = ⟨a_{1}, a_{2}⟩, γ_{2} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , λ_{1} = 1, λ_{2} = λ,

we obtain from Proposition 6.6 and Example 6.7 that

(6.18)  (\begin{matrix}a_{1} & λ \\ λ & a_{2} \end{matrix}) ⊗_{b} q = [\frac{a_{1}q λb}{a_{2}q}] .

Example 6.11. Let

q = [\begin{matrix}0 & a_{12} & ⋯ & a_{1n} \\ & ... & ... & ⋮ \\ & & & a_{n−1,n} \\ & & & 0 \end{matrix}]

with a_{ij} ∈ R (i < j). Then q is rigid (cf. [22, Proposition 3.4]; no assumption on R is needed here). Furthermore, let

γ = (\begin{matrix}γ_{11} & ⋯ & γ_{1m} \\ ⋮ & & ⋮ \\ γ_{m1} & ⋯ & γ_{mm} \end{matrix})

with γ_{ij} = γ_{ji} ∈ R. Then we obtain by the rules (6.5) and (6.6) that

(6.19)  γ ⊗ q = \begin{aligned}0 & a_{12}γ & ⋯ & a_{1n}γ \\ & 0 & & a_{2n}γ \\ & & ... & a_{n−1,n}γ \\ & & & 0 \end{aligned}

Documenta Mathematica 21 (2016) 773–808

More precisely, if the presentations of q and γ above refer to ordered bases (η_{1}, …, η_{n}) and (ε_{1}, …, ε_{m}), respectively, then (6.19) refers to the ordered base

(ε_{1} ⊗ η_{1}, …, ε_{m} ⊗ η_{1}, ε_{1} ⊗ η_{2}, …, ε_{m} ⊗ η_{n}).

We now consider the tensor product γ ⊗ [a] = γ ⊗_{b} [a], cf. Equation (6.10), where b is the unique balanced companion of [a], (6.13). Our starting point is a definition which makes sense for any semiring R and any R-module U.

Definition 6.12. Let γ : U × U → R be a symmetric bilinear form. The norm form of γ is the quadratic form n(γ) : U → R with

n(γ)(x) := γ(x, x)

for any x ∈ U.

Remark 6.13. The norm form n(γ) has the expansion γ : U × U → R and the associated balanced companion γ + γ^{t} = 2γ. The norm forms are precisely all the quadratic forms which admit a symmetric expansion. If U has a finite base ε_{1}, …, ε_{n}, then with respect to this base

(6.20)  n(γ) = [\begin{matrix}γ_{11} & 2γ_{12} & ⋯ & 2γ_{1m} \\ & γ_{22} \\ & & ... & ⋮ \\ & & & γ_{mm} \end{matrix}] ,

where γ_{ij} := γ(ε_{i}, ε_{j}).

Proposition 6.14. Assume that U = (U, γ) is a free bilinear R-module and a ∈ R. Then

(6.21)  U ⊗ [a] ≅ (U, a n(γ)).

Proof. We realize the form [a] as a quadratic module (V, q) with V = Rη free of rank 1 and q(η) = a. {q has the unique balanced companion b : V × V → R, with b(η, η) = 2a.} The form q̃:= γ ⊗ q = γ ⊗_{b} q is given by

q̃(x ⊗ η) = γ(x, x)a = (an(γ))(x).

The claim is obvious. □

Example 6.15. Assume that U has base ε_{1}, …, ε_{m}. Let γ_{ij} := γ(ε_{i}, ε_{j}). Then

γ ⊗ [a] ≅ (aγ) ⊗ [1],

and

(6.22)  γ ⊗ [1] = [\begin{matrix}γ_{11} & 2γ_{12} & ⋯ & 2γ_{1n} \\ & γ_{22} \\ & & ... & ⋮ \\ & & & γ_{mm} \end{matrix}] ,

where the right hand side refers to the base ε_{1} ⊗ η, ε_{2} ⊗ η, …, ε_{m} ⊗ η.

At a crucial point in §7, we will need an explicit description of the tensor products γ ⊗_{b} q with q indecomposable of rank 2. We start with a general fact.

Proposition 6.16. Assume that γ is a symmetric bilinear form on a free Rmodule U and q_{1}, q_{2} are quadratic forms on a free R-module V. Let b_{1}, b_{2} be balanced companions of q_{1} and q_{2}, respectively. Let q := λ_{1}q_{1} + λ_{2}q_{2} with λ_{1}, λ_{2} ∈ R. Then b := λ_{1}b_{1} + λ_{2}b_{2} is a balanced companion of q, and

(6.23)  γ ⊗_{b} q = λ_{1}(γ ⊗_{b_{1}} q_{1}) + λ_{2}(γ ⊗_{b_{2}} q_{2}).

This form has the balanced companion γ ⊗ b (as we know) and

(6.24)  γ ⊗ b = λ_{1}(γ ⊗ b_{1}) + λ_{2}(γ ⊗ b_{2}).

Proof. An easy check by use of (6.5) and (6.6). □

Example 6.17. We take a free module V with base η_{1}, η_{2}, and choose with respect to this base

q_{1} = \left[{a_{1} 0 \atop a_{2}}\right] = [a_{1}, a_{2}], q_{2} = \left[{0 c \atop 0}\right]

with a_{1}, a_{2}, c ∈ R, c ≠ 0, and the balanced companions

b_{1} = (\begin{matrix}2a_{1} & 0 \\ 0 & 2a_{2} \end{matrix}) , b_{2} = (\begin{matrix}0 & c \\ c & 0 \end{matrix}) .

Then

q := q_{1} + q_{2} = \left[{a_{1} c \atop a_{2}}\right]

has the balanced companion

b := b_{1} + b_{2} = (\begin{matrix}2a_{1} & c \\ c & 2a_{2} \end{matrix}) .

For

γ = (\begin{matrix}γ_{11} & ⋯ & γ_{1m} \\ ⋮ & & ⋮ \\ γ_{m1} & ⋯ & γ_{mm} \end{matrix})

on a free module U with to the base ε_{1}, …, ε_{m}, we get

γ ⊗_{b_{1}} q_{1} = [\frac{a_{1}n(γ)  0}{a_{2}n(γ)}] ,  γ ⊗_{b_{2}} \left[{0 c \atop 0}\right] = [\frac{0 cγ}{0}] ,

cf. (6.19), and finally

(6.25)  γ ⊗_{b} \left[{a_{1} c \atop a_{2}}\right] = [\frac{a_{1}n(γ) cγ}{a_{2}n(γ)}]

with respect to the base

ε_{1} ⊗ η_{1}, …, ε_{m} ⊗ η_{1}, ε_{1} ⊗ η_{2}, …, ε_{m} ⊗ η_{2}.

Remark 6.18. ¿From (6.25) and (6.18), we obtain the useful formula

(6.26)  γ ⊗_{b} \left[{a_{1} c \atop a_{2}}\right] = (\begin{matrix}a_{1} & c \\ c & a_{2} \end{matrix}) ⊗_{2γ} n(γ) ,

by use of Example 6.10 for the quadratic pair (n(γ), 2γ).

From now on, we assume that V has unique base. {We do not need that U has unique base.}

Definition 6.19. We call a companion b of q faithful if b is balanced and quasiminimal.

Proposition 6.20. Assume that b is a faithful companion of q, and that V = W_{1} ⊥ W_{2} is an orthogonal decomposition of V. Then, writing U ⊗_{b} W_{i} instead of U ⊗_{(b|W_{i})} W_{i}, we have

U ⊗_{b} V = U ⊗_{b} W_{1} ⊥ U ⊗_{b} W_{2}

for any bilinear R-module U.

Proof. b(W_{1}, W_{2}) = 0, since b is quasiminimal. It follows that

(γ ⊗ b)(U ⊗ W_{1}, U ⊗ W_{2}) = 0.

Thus, q̃ = γ ⊗_{b} q is quasilinear on (U ⊗ W_{1}) × (U ⊗ W_{2}). □

Example 6.21. Our assumption, that b is faithful, is necessary here. If V = W_{1} ⊥ W_{2}, and b is balanced, but b(W_{1}, W_{2}) ≠ 0, then

(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ⊗_{b} V = \left[{0 b \atop 0}\right]

is not the orthogonal sum of

(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ⊗_{b} W_{1} and (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ⊗_{b} W_{2}.

Example 6.22. Let q = [a_{1}, a_{2}, …, a_{n}] be a diagonal quadratic form. The diagonal symmetric bilinear form

b := ⟨2q_{1}, …, 2a_{n}⟩

is the unique faithful companion of q. For any bilinear R-module (U, γ), we have

(6.27)  γ ⊗_{b} q = γ ⊗ [a_{1}] ⊥ ⋯ ⊥ γ ⊗ [a_{n}].

Concerning the forms γ ⊗ [a_{i}], recall Proposition 6.14 and Example 6.15.

7. Indecomposability in tensor products

In this section, we assume for simplicity that R \ {0} is an entire semiring lacking zero sums. So every free R-module has unique base (cf. Theorem 2.3), and R has no zero divisors. We discuss decomposability first in tensor products of (free) bilinear modules, later in tensor products of bilinear modules with quadratic modules. Let V_{1} = (V_{1}, b_{1}) and V_{2} = (V_{2}, b_{2}) be indecomposable free (symmetric) bilinear modules over R, and let V := V_{1} ⊗ V_{2} = (V_{1} ⊗ V_{2}, b) with b := b_{1} ⊗ b_{2}. We take bases 𝔅_{1} and 𝔅_{2} of the R-modules V_{1}, V_{2} respectively and then have the base

𝔅 = 𝔅_{1} ⊗ 𝔅_{2} := {ε ⊗ η | ε ∈ 𝔅_{1}, η ∈ 𝔅_{2}} of V. Our task is to determine the indecomposable components of V. First we discuss the “trivial” cases.

Remark 7.1. Assume that V_{1} has dimension (= rank) one, so V_{1} ≅ ⟨a⟩ with a ∈ R. If a ≠ 0, then V is clearly indecomposable. If a = 0, then b_{1} ⊗ b_{2} = 0, whence V is indecomposable only if also dim V_{2} = 1. Then V = ⟨0⟩.

In all the following, we assume that V_{1} ≠ ⟨0⟩, V_{2} ≠ ⟨0⟩. We resort to §4 to describe bases of the indecomposable components of V = (V, b) as the classes in

𝔅 = {ε ⊗ η | ε ∈ 𝔅_{1}, η ∈ 𝔅_{2}}

of an equivalence relation given by “paths”, cf. Definition 4.7. So a path of length r ≥ 1 in V, i.e., in 𝔅, is a sequence

(7.1)  Γ = (ε_{0} ⊗ η_{0}, ε_{1} ⊗ η_{1}, …, ε_{r} ⊗ η_{r})

with

(7.2)  b_{1}(ε_{i}, ε_{i+1})b_{2}(η_{i}, η_{i+1}) ≠ 0

and

(7.3)  ε_{i} ≠ ε_{i+1}  or  η_{i} ≠ η_{i+1}

for 0 ≤ i ≤ r − 1. Let us first assume that both b_{1} and b_{2} are alternate, whence also b = b_{1} ⊗ b_{2} is alternate. Now condition (7.3) is a consequence of (7.2) and thus can be ignored. We read off from (7.2) that

(7.4)  Γ_{1} = (ε_{0}, ε_{1}, …, ε_{r}), Γ_{2} = (η_{0}, η_{1}, …, η_{r})

are paths in V_{1} and V_{2} respectively of same length r. Conversely, given such paths Γ_{1} and Γ_{2}, they combine to a path Γ of length r in V, as written in (7.1). {Here we use the assumption that R has no zero divisors.} We write

(7.5)  Γ = Γ_{1} ⊗ Γ_{2}.

We will speak of “cycles” in 𝔅_{1}, 𝔅_{2}, 𝔅, in the following obvious way:

Definition 7.2. Let ℭ be a base of a free bilinear R-module W.

(a) We denote the length of a path Γ in ℭ by ℓ(Γ).

(b) A cycle Δ in W with base point ζ ∈ ℭ is a path (ζ_{0}, ζ_{1}, …, ζ_{r}) in ℭ with ζ_{0} = ζ_{r} = ζ. We say that the cycle Δ is even (resp. odd) if ℓ(Δ) is even (resp. odd). We say that Δ is a 2-cycle if ℓ(Δ) = 2, whence Δ = (ζ, ζ′, ζ) with (ζ, ζ′) an edge.

Lemma 7.3. Let ε, ε′ ∈ 𝔅_{1} and η, η′ ∈ 𝔅_{2}. Let Γ_{1} be a path from ε to ε′ of length r and Γ_{2} a path from η to η′ of length s, and assume that r ≡ s (mod 2). Then ε ⊗ η ∼ ε′ ⊗ η′.

Proof. Assume, without loss of generality, that s ≥ r, whence s = r + 2t with t ≥ 0. If t = 0, then Γ_{1} ⊗ Γ_{2} is a path from ε ⊗ η to ε′ ⊗ η′ in V. If t > 0, we replace Γ_{1} = (ε_{0}, ε_{1}, …, ε_{r}) by

Γ̃_{1} = (ε_{0}, ε_{1}, …, ε_{r}, ε_{r−1}, ε_{r}, …)

adjoining t copies of the 2-cycle (ε_{r}, ε_{r−1}, ε_{r}) to Γ_{1}. Now Γ̃_{1} ⊗ Γ_{2} runs from ε ⊗ η to ε′ ⊗ η′. □

Theorem 7.4. Assume that both b_{1} and b_{2} are alternate (and V_{1} ≠ ⟨0⟩, V_{2} ≠ ⟨0⟩, as always).

a) If V_{1} or V_{2} contains an odd cycle, then V_{1} ⊗ V_{2} is indecomposable.

b) Otherwise V_{1} ⊗ V_{2} is the orthogonal sum of two indecomposable components.

Proof. a): We assume that V_{1} contains an odd cycle Δ with base point δ. Let ε⊗η and ε′⊗η′ be different elements of 𝔅. We want to verify that ε⊗η ∼ ε′⊗η′. We choose a path Γ_{1} from ε to ε′ in V_{1} and a path Γ_{2} from η to η′ in V_{2}. If ℓ(Γ_{1}) ≡ ℓ(Γ_{2}) (mod 2), then we know by Lemma 7.3 that ε ⊗ η ∼ ε′ ⊗ η′. Now assume that ℓ(Γ_{1}) and ℓ(Γ_{2}) have different parity. We choose a new path Γ̃_{1} from ε to ε′ as follows: We first take a path H from ε to the base point δ of Δ, then we run through Δ, then we take the path inverse to H (in the obvious sense) from δ to ε, and finally we run through Γ_{1}. The length ℓ(Γ̃_{1}) has different parity than ℓ(Γ_{1}) and thus the same parity as ℓ(Γ_{2}). We conclude again that ε ⊗ η ∼ ε′ ⊗ η′.

b): Now assume that both V_{1} and V_{2} contain only even cycles. This means that both in V_{1} and V_{2} all paths from a fixed start to a fixed end have length of the same parity. Given ε ⊗ η and ε′ ⊗ η′ in 𝔅, every path Γ from ε ⊗ η to ε′ ⊗ η′ has the shape Γ_{1} ⊗ Γ_{2} with Γ_{1} running from ε to ε′, Γ_{2} running from η to η′, and ℓ(Γ_{1}) = ℓ(Γ_{2}). Thus, if the paths from ε to ε′ have length of different parity than those from η to η′, then ε ⊗ η cannot be connected to ε′ ⊗ η′ by a path. But ε ⊗ η can be connected to ε′ ⊗ η′′, where η′′ arises from η′ by adjoining an edge at the endpoint of η′. We fix some ε_{0} ∈ 𝔅_{1}, and η_{0}, η_{1} ∈ 𝔅_{2} with b_{2}(η_{0}, η_{1}) = 1. Then every element of 𝔅 can be connected by a path to ε_{0} ⊗ η_{0} or to ε_{0} ⊗ η_{1}, but not to both. V has exactly two indecomposable components. □

Remark 7.5. Assume again that b_{1} and b_{2} are alternate and 𝔅_{1} and 𝔅_{2} both contain only even cycles. Let ε, ε′ ∈ 𝔅_{1} and η, η′ ∈ 𝔅_{2}, and choose paths Γ_{1} from ε to ε′ and Γ_{2} from η to η′. As the proof of Theorem 7.4.b has shown, ε ⊗ η and ε′ ⊗ η′ lie in the same indecomposable component of V_{1} ⊗ V_{2} iff ℓ(Γ_{1}) and ℓ(Γ_{2}) have the same parity.

There remains the case that b_{1} or b_{2} is not alternate.

Theorem 7.6. Assume that b_{1} is not alternate and – as before – that V_{1} = (V_{1}, b_{1}) and V_{2} = (V_{2}, b_{2}) are indecomposable. Then (V_{1} ⊗ V_{2}, b_{1} ⊗ b_{2}) is indecomposable.

Proof. Every path in V := V_{1} ⊗ V_{2} with respect to (b_{1})_{alt} ⊗ (b_{2})_{alt} is also a path with respect to b_{1} ⊗ b_{2}, as is easily checked, and the paths in V_{i} with respect to b_{i} are the same as those with respect to (b_{i})_{alt} (i = 1, 2). Thus we are done by Theorem 7.4, except in the case that all cycles in V_{1} and in V_{2} are even. Then V has two indecomposable components W′, W′′ with respect to (b_{1})_{alt} ⊗ (b_{2})_{alt}. The base

𝔅 = 𝔅_{1} ⊗ 𝔅_{2} := (ε ⊗ η | ε ∈ 𝔅_{1}, η ∈ 𝔅_{2})

of V_{1} ⊗ V_{2} is the disjoint union of sets 𝔅′, 𝔅′′ which are bases of W′ and W′′. Any two elements of 𝔅′ are connected by a path with respect to (b_{1})_{alt}⊗(b_{2})_{alt}, hence by a path with respect to b_{1} ⊗ b_{2}, and the same holds for the set 𝔅′′. We choose some ρ ∈ 𝔅_{1} with b_{1}(ρ, ρ) ≠ 0 and an edge (η_{0}, η_{1}) in 𝔅′′. Since R has no zero divisors, it follows that (ρ ⊗ η_{0}, ρ ⊗ η_{1}) is an edge in 𝔅 with respect to b_{1} ⊗ b_{2}. Perhaps interchanging W′ and W′′, we assume that ρ ⊗ η_{0} ∈ 𝔅′. Suppose that also ρ⊗η_{1} ∈ 𝔅′. Then there exists a path Γ in 𝔅′ with respect to (b_{1})_{alt} ⊗ (b_{2})_{alt} running from ρ ⊗ η_{0} to ρ ⊗ η_{1}. Γ has the form Γ_{1} ⊗ Γ_{2}, with Γ_{1} a cycle in V_{1} with base point ρ, and Γ_{2} a path in V_{2} running from η_{0} to η_{1}. We have ℓ(Γ_{1}) = ℓ(Γ_{2}) and ℓ(Γ_{2}) is even. But there exists the path (η_{0}, η_{1}) from η_{0} to η_{1} of length 1. Since all paths in V_{2} from η_{0} to η_{1} have the same parity, we infer that ℓ(Γ_{2}) is odd, a contradiction. We conclude that ρ ⊗ η_{1} ∈ 𝔅′′. The elements ρ ⊗ η_{0} ∈ 𝔅′ and ρ ⊗ η_{1} ∈ 𝔅′′ are connected by a path with respect to b_{1} ⊗ b_{2}, and thus all elements of 𝔅 are connected by paths with respect to b_{1} ⊗ b_{2}. □

Turning to a study of indecomposable components of tensor products of bilinear and quadratic modules, we need some more terminology. Let V = (V, q) be a free quadratic R-module and 𝔅 a base of V. We focus on balanced companions of q.

Definition 7.7.

(a) We call a companion b of q faithful if b is balanced and quasiminimal (cf. §3 above), whence b(ε, ε) = 2q(ε) for all ε ∈ 𝔅 and b(ε, η) = 0 for ε ≠ η in 𝔅 such that q is quasilinear on Rε × Rη.

(b) Given a balanced companion b of q, we define a new bilinear form b_{f} on V by the rule that, for ε, η ∈ 𝔅,

b_{f}(ε, η) = \begin{cases}0 & if ε ≠ η and q is quasilinear on Rε × Rη, \\ b(ε, η) & else. \end{cases}

It is clear from [22, Theorem 6.3] that again b_{f} is a companion of q. By definition, this companion is quasiminimal. b_{f} is also balanced, since b_{f}(ε, ε) = b(ε, ε) = 2q(ε) for all ε ∈ 𝔅, cf. [22, Proposition 1.7], and so b_{f} is faithful. We call b_{f} the faithful companion of q associated to b.

Theorem 7.8. Assume that b is a balanced companion of q, and that W is a basic submodule of V. Then W is indecomposable with respect to q iff W is indecomposable with respect to b_{f}.

Proof. This is a special case of Theorem 4.9, since b_{f}|W = (b|W)_{f} is a quasiminimal companion of q|W. □

Definition 7.9.

(a) We say that q is diagonally zero if q(ε) = 0 for every ε ∈ 𝔅.

(b) We say that q is anisotropic if q(ε) ≠ 0 for every ε ∈ 𝔅.

Remarks 7.10.

(i) If q is diagonally zero, then q is rigid, cf. [22, Proposition 3.4]. Conversely, if q is rigid and the quadratic form [1] is quasilinear, i.e., (α + β)^{2} = α^{2} + β^{2} for any α, β ∈ R, then q is diagonally zero, as proved in [22, Theorem 3.5].

(ii) If q is anisotropic, then q(x) ≠ 0 for every x ∈ V \{0}. So our definition of anisotropy here coincides with the usual meaning of anisotropy for quadratic forms (which makes sense, say, for R a semiring without zero divisors and V any R-module).

Definition 7.11. In a similar vein, we call a symmetric bilinear form b on V anisotropic if b(ε, ε) ≠ 0 for every ε ∈ 𝔅, and then have b(x, x) ≠ 0 for every x ∈ V \ {0}.

Note that, if b is a balanced companion of q, then b is anisotropic iff q is anisotropic. Assume now that U := (U, γ) is a free bilinear module, V := (V, q) is a free quadratic module, and b is a balanced companion of q. Let

Ṽ := (Ṽ, q̃) := (U ⊗ V, γ ⊗_{b} q).

We want to determine the indecomposable components of Ṽ. Discarding trivial cases, we assume that U ≠ ⟨0⟩, V ≠ [0]. We choose bases 𝔅_{1} and 𝔅_{2} of the R-modules U and V, respectively, and introduce the subsets

𝔅_{1}^{+} := {ε ∈ 𝔅_{1} | γ(ε, ε) ≠ 0},
𝔅_{1}^{0} := {ε ∈ 𝔅_{1} | γ(ε, ε) = 0},
𝔅_{2}^{+} := {η ∈ 𝔅_{1} | q(η) ≠ 0},
𝔅_{2}^{0} := {η ∈ 𝔅_{1} | q(η) = 0},

of 𝔅_{1} and 𝔅_{2}, respectively, and furthermore the basic submodules U^{+}, U^{0}, V^{+}, V^{0} respectively spanned by these sets.

Lemma 7.12.

a) If ε ∈ 𝔅_{1}^{+}, then the indecomposable components of the basic submodule ε ⊗ V := (Rε) ⊗ V of U ⊗ V with respect to q̃ are the submodules ε ⊗ W with W running through the indecomposable components of V with respect to q. b) If η ∈ 𝔅_{1}^{+}, then the indecomposable components of U ⊗ η := U ⊗ (Rη) with respect to q̃ are the modules U ⊗ η with U′ running through the indecomposable components of U with respect to the norm form n(γ) of γ (cf. Definition 6.12).

Proof. This follows from the formulas q̃(ε ⊗ y) = γ(ε, ε)q(y) for y ∈ V and q̃(x ⊗ η) = γ(x, x)q(η) for x ∈ U (cf. (6.6)), since γ(ε, ε) ≠ 0, q(η) ≠ 0. □

In order to avoid certain pathologies concerning indecomposability in tensor products U ⊗_{b} V, we henceforth will assume that our semiring has the following property:

(NQL) For any a and c in R \ {0} there exists some μ ∈ R with a + μc ≠ a.

Clearly, this property means that every free quadratic module \left[{a c \atop 0}\right] with c ≠ 0 is not quasilinear on (Rη_{1})×(Rη_{2}), where (η_{1}, η_{2}) is the associated base, whence the label “NQL”.

Examples 7.13.

(a) In the important case that R is supertropical the condition (NQL) holds iff all principal ideals in eR are unbounded with respect to the total ordering of eR. In particular, the “multiplicatively unbounded supertropical semirings” appearing in [20, §7] have NQL.

(b) If R is any entire semiring lacking zero sums, then the polynomial ring R[t] in one variable (and so in any set of variables) has NQL.

(c) The polynomial function semirings over supersemirings appearing in [25, §4] have NQL.

Lemma 7.14. Assume that (V, q) is indecomposable. Let a, c ∈ R \ {0}. Then

(\begin{matrix}a & c \\ c & 0 \end{matrix}) ⊗_{b} V = [\frac{aq cb}{0}]

(cf. (6.19)) is indecomposable.

Proof. Let

(U, γ) = (\begin{matrix}a & c \\ c & 0 \end{matrix})

with respect to a base ε_{1}, ε_{2} and assume for notational convenience that V has a finite base η_{1}, …, η_{n}. By Lemma 7.12.a, we have

ε_{1} ⊗ η_{1} ∼ ε_{1} ⊗ η_{2} ∼ ⋯ ∼ ε_{1} ⊗ η_{n}.

For given ε_{1} ⊗ η_{i}, ε_{2} ⊗ η_{j} with i ≠ j, γ ⊗_{b} q has the value table

\left[{aq(η_{i}) cb(η_{i}, η_{j}) \atop 0}\right] .

Starting with ε_{2} ⊗ η_{j}, we find some η_{i}, i ≠ j, with b(η_{i}, η_{j}) ≠ 0, because (V, q) is indecomposable. Since R has NQL, it follows that R(ε_{i} ⊗ η_{i}) + R(ε_{j} ⊗ η_{j}) is indecomposable with respect to q̃, whence ε_{1} ⊗ η_{i} ∼ ε_{2} ⊗ η_{j}. Thus all ε_{k} ⊗ η_{ℓ} are equivalent. □

Lemma 7.15. Assume that (U, n(γ)) is indecomposable. Let a, c ∈ R \ {0}.

Then the tensor product U ⊗_{b} \left[{a c \atop 0}\right] , taken with respect to b = (\begin{matrix}2a & c \\ c & 0 \end{matrix}) , is

indecomposable.

Proof. By formula (6.26)

γ ⊗_{b} \left[{a c \atop 0}\right] = (\begin{matrix}a & c \\ c & 0 \end{matrix}) ⊗_{2γ} n(γ).

Now Lemma 7.14 with (V, q) := (U, n(γ)) gives the claim. □

We are ready for the main result of this section. Recall that U := (U, γ).

Theorem 7.16. Assume that R has NQL. Assume furthermore that both (U, n(γ)) and the quadratic free module V = (V, q) are indecomposable, and U ≠ ⟨0⟩, V ≠ [0]. Let b be a balanced companion of q. Then the quadratic module U ⊗_{b} V := (U ⊗V, γ ⊗_{b} q) is indecomposable, except in the case that γ is alternate, q is diagonally zero, U and V contain only even cycles with respect to γ and b. Then U ⊗_{b} V has exactly two indecomposable components, and these coincide with the indecomposable components of U ⊗ V with respect to γ ⊗ b, and also with respect to γ ⊗ b_{f}.

Proof. Of course, indecomposability of (U, n(γ)) implies indecomposability of (U, γ). As before, let q̃:= γ ⊗_{b} q. We distinguish three cases.

1) Assume that V^{+} ≠ {0}, i.e., there exist anisotropic base vectors in V. Our claim is that all elements of 𝔅_{1} ⊗ 𝔅_{2} are equivalent, whence U ⊗_{b} V is indecomposable. We choose η_{0} ∈ 𝔅_{2}^{+}. By Lemma 7.12.b, the module

(U ⊗ η_{0}, q̃) := (U ⊗ η_{0}, q̃ | U ⊗ η_{0})

is indecomposable, and thus all elements of 𝔅_{1} ⊗ η_{0} are equivalent. Let ε ⊗ η ∈ 𝔅_{1} ⊗ 𝔅_{2}. We verify the equivalence of ε ⊗ η with some element of 𝔅_{1} ⊗ η_{0}, and then will be done. If γ(ε, ε) ≠ 0, then by Lemma 7.12.a, all elements of ε ⊗ 𝔅_{2} are equivalent, whence ε ⊗ η ∼ ε ⊗ η_{0}. Assume now that γ(ε, ε) = 0. Since (U, γ) is indecomposable, there exists some ε′ ∈ 𝔅_{1} with c := γ(ε′, ε) ≠ 0. Let a := γ(ε′, ε′). We choose a base η_{1}, …, η_{n} of V, assuming for notational convenience that V has finite rank. By Example 6.10,

(Rε′ + Rε) ⊗_{b} V = \left[{aq cq \atop 0}\right]

with respect to the base ε′ ⊗ η_{1}, …, ε′ ⊗ η_{n}, ε ⊗ η_{1}, …, ε ⊗ η_{2}. Now Lemma 7.14 tells us that (Rε′ + Rε) ⊗_{b} V is indecomposable, whence all elements ε ⊗ η, ε′ ⊗ η′ with η, η′ ∈ 𝔅_{2} are equivalent. In particular, ε ⊗ η ∼ ε′ ⊗ η_{0}.

2) Assume that U^{+} ≠ {0}, i.e., there exist an anisotropic base vector in U with respect to n(γ). Our claim again is that all elements of 𝔅_{1} ⊗𝔅_{2} are equivalent, whence U ⊗_{b} V is indecomposable. We choose ε_{0} ∈ 𝔅_{1}^{+}, and then know by Lemma 7.12.a that all elements of ε_{0} ⊗ 𝔅_{2} are equivalent.

Let ε ⊗ η ∈ 𝔅_{1} ⊗ 𝔅_{2} be given. We verify equivalence of ε ⊗ η with some element of ε_{0} ⊗ 𝔅_{2}, and then will be done. If q(η) ≠ 0, then by Lemma 7.12.a all elements of 𝔅_{1} ⊗ η are equivalent, and thus ε ⊗ η ∼ ε_{0} ⊗ η. Hence, we may assume that q(η) = 0. Since (V, q) is indecomposable, there exists some η′ ∈ 𝔅_{2} with c := b(η, η′) ≠ 0. Let a := q(η′). Then

(Rη′ + Rη, q) = \left[{a c \atop 0}\right] .

Let b′ := b|(Rη′ + Rη) = (\begin{matrix}a & c \\ c & 0 \end{matrix}) . Then we see from (6.25) that

γ ⊗_{b′} \left[{a c \atop 0}\right] = [\frac{an(γ) cγ}{0}] .

By Lemma 7.15, this quadratic module is indecomposable, whence all elements ε ⊗ η, ε′ ⊗ η′ with ε, ε′ ∈ 𝔅_{1} are equivalent. In particular, ε ⊗ η ∼ ε_{0} ⊗ η′.

3) The remaining case: U = U^{0}, and V = V^{0}, i.e., γ is alternate and q is diagonally zero. Now (U ⊗ V, q̃) is rigid. By Theorem 7.8, the indecomposable components of (U ⊗V, q̃) coincide with those of (U ⊗V, (γ ⊗b)_{f}). But q̃ has only one companion, whence (γ ⊗ b)_{f} = γ ⊗ b = γ ⊗ b_{f}. Invoking Theorem 7.4, we see that the assertion of the theorem also holds in the case under consideration, where γ is alternate and b is diagonally zero. □

In general, let {U_{i} | i ∈ I} denote the set of indecomposable components of (U, n(γ)). Then

U ⊗_{b} V = ⊥_{i∈I} U_{i} ⊗_{b} V

by Proposition 6.6, whence, applying Theorem 7.16 to each summand U_{i} ⊗_{b} V, we obtain a complete list of all indecomposable components of U ⊗_{b} V. In particular, if q is not diagonally zero, or if (V, b) contains an odd cycle, then the U_{i} ⊗_{b} V themselves are the indecomposable components of U ⊗_{b} V.

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Zur Izhakian Institute of Mathematics University of Aberdeen AB24 3UE Aberdeen UK zzur@abdn.ac.uk; zzur@math.biu.ac.il Manfred Knebusch Department of Mathematics NWF-I Mathematik Universität Regensburg 93040 Regensburg Germany manfred.knebusch@

mathematik.uni-

regensburg.de

Louis Rowen

Department of Mathematics Bar-Ilan University Ramat-Gan 52900 Israel rowen@math.biu.ac.il