Non Reduced Theory of Quadratic Forms Over Rings

TL;DR

This paper develops a non-reduced abstract quadratic forms theory over rings using multirings and multifields, expanding the foundational understanding of quadratic forms beyond traditional reduced frameworks.

Contribution

It introduces a novel non-reduced quadratic forms framework over rings, leveraging multiring and multifield structures, which broadens the theoretical landscape.

Findings

Establishment of a non-reduced quadratic forms theory

Application of multiring and multifield machinery

Extension of quadratic forms theory to general coefficients

Abstract

Based on the works of M. Marshall on multirings, we propose the fundamentals for a \textbf{non reduced} abstract quadratic forms theory in general coefficients on rings, with the machinery of multirings and multifields.

Equations

$$a\in D^t(b,c)\iff a\in D(b,c)\wedge -b\in D(-a,c)\wedge -c\in D(b,-a).$$

$$d\in a{x}^2+b{y}^2$$

$$\stackrel{QT3}{\Rightarrow }d\in a{d}^2+b{d}^2\Rightarrow d\in D(a,b).$$

$$\mathcal{R}\mathcal{S}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$M(A):=A{/}_{m}S\text{mbox}(TheMarshal{l}^{\prime }sQuotient).$$

$$\overline{a}(\alpha )=\{\begin{array}{l}1\text{mbox}ifa\in \alpha \setminus (-\alpha )\\ 0\text{mbox}ifa\in \alpha \cap -\alpha \\ -1\text{mbox}ifa\in (-\alpha )\cap \alpha .\end{array}$$

Full text

1. 1 Introduction
2. 2 Connections Between ARS, Realsemigroups and Multirings
3. 3 Quadratically Tuned Multirings
4. 4 Examples
5. 5 Quadratic Forms over QT-Multirings

1 Introduction

Quadratic forms over fields (of characteristc not 2) is a large and succesfull subject of study in mathematics. The book [4] covers almost all introductory aspects of the theory as been as [3] covers the relationship between orderings, valuations and quadratic forms, in the theory that will be known as Reduced Theory of Quadratic Forms. My master degree thesis [8] establishes a path between the both classic and reduced theory with the abstract ones.

However, the generalization of the theory to general coefficients in rings is a hard step. The book [2] cover some basic aspects in the most general setting possible, and we have Marshall’s theory of abstract real spectrum ([6]) and its algebraic counterpart, the realsemigroups of Dickmann and Petrovich ([1]) given a nice approach for the reduced theory of quadratic forms on rings, but, most of the relevant aspects of quadratic forms, like Witt rings, Pfister forms and etc, are uncovered.

Based on the works of M. Marshall in the paper [5], we propose the fundamentals for a non reduced abstract quadratic forms theory in general coefficients on rings, with the machinery of multirings and multifields.

2 Connections Between ARS, Realsemigroups and Multirings

Definition 2.1** (Abstract Real Spectra).**

An abstract real spectrum or space of signs, abreviatted to ARS, is a pair $(X,G)$ satisfying:

AX1 -

$X$ is a non-empty set, $G$ is a submonoid of $\{-1,0,1\}^{X}$, $G$ contais the constants functions $-1,0,1$, and $G$ separates points in $X$.

If $a,b\in G$, the value set $D(a,b)$ is defined to be the set of all $c\in G$ such that, for all $x\in X$, either $a(x)c(x)>0$ or $b(x)c(x)>0$ or $c(x)=0$. The value set $D^{t}(a,b)$ is defined to be the set of all $c\in G$ such that, for all $x\in X$, either $a(x)c(x)>0$ or $b(x)c(x)>0$ or $c(x)=0$ and $b(x)=-a(x)$. Note that $c\in D^{t}(a,b)\Rightarrow c\in D(a,b)$. Conversely, $c\in D(a,b)\Rightarrow c\in D^{t}(ac^{2},bc^{2})$.

AX2 -

If $P$ is a submonoid of $G$ satisfying $P\cup-P=G$, $-1\notin P$, $a,b\in P\Rightarrow D(a,b)\subseteq P$ and $ab\in P\cap-P\Rightarrow a\in P\cap-P$ or $b\in P\cap-P$, then there exists $x\in X$ (necessarily unique) such that $P=\{a\in G:a(x)\leq 0\}$.

AX3a (Weak Associativity) -

For all $a,b,c\in G$, if $p\in D(a,r)$ for some $q\in D(b,c)$ then $p\in D(r,c)$ for some $r\in D(a,b)$.

AX3b -

For all $a,b\in G$, $D^{t}(a,b)\neq\emptyset$.

We hasten to point out that AX3a and AX3b combined are equivalent to the simgle axiom AX3 below:

AX3 (Strong Associativity) -

For all $a,b,c\in G$, if $p\in D^{t}(a,r)$ for some $q\in D^{t}(b,c)$ then $p\in D^{t}(r,c)$ for some $r\in D^{t}(a,b)$.

Definition 2.2**.**

A ternary semigroup (abbreviated TS) is a struture $(S,\cdot,1,0,-1)$ with individual constants $1,,0,-1$ and a binary operation “$\cdot$” such that:

TS1 -

$(S,\cdot,1)$ is a commutative semigroup with unity;

TS2 -

$x^{3}=x$ for all $x\in S$;

TS3 -

$-1\neq 1$ and $(-1)(-1)=1$;

TS4 -

$x\cdot 0=0$ for all $x\in S$;

TS5 -

For all $x\in S$, $x=-1\cdot x\Rightarrow x=0$.

We shall write $-x$ for $(-1)\cdot x$. The semigroup verifying conditions [TS1] and [TS2] (no extra constants) will be called 3-semigroups.

Here, we will enrich the language $\{\cdot,1,0,-1\}$ with a ternary relation $D$. In agreement with 2.1, we shall write $a\in D(b,c)$ instead of $D(a,b,c)$. We also set:

[TABLE]

The relations $D$ and $D^{t}$ are called representation and transversal representation respectivel.

Definition 2.3**.**

A real semigroup (abbreviated RS) is a ternary semigroup $(G,1,0,-1)$ together with a ternary relation $D$ satisfying:

RS0 -

$c\in D(a,b)$ if and only if $c\in D(b,a)$.

RS1 -

$a\in D(a,b)$.

RS2 -

$a\in D(b,c)$ implies $ad\in D(bd,cd)$.

RS3 (Strong Associativity) -

If $a\in D^{t}(b,c)$ and $c\in D^{t}(d,e)$, then there exists $x\in D^{t}(b,d)$ such that $a\in D^{t}(x,e)$.

RS4 -

$e\in D(c^{2}a,d^{2}b)$ implies $e\in D(a,b)$.

RS5 -

If $ad=bd$, $ae=be$ and $c\in D(d,e)$, then $ac=bc$.

RS6 -

$c\in D(a,b)$ implies $c\in D^{t}(c^{2}a,c^{2}b)$.

RS7 (Reduction) -

$D^{t}(a,-b)\cap D^{t}(b,-a)\neq$ implies $a=b$.

RS8 -

$a\in D(b,c)$ implies $a^{2}\in D(b^{2},c^{2})$.

A pre-real semigroup(abbreviated PRS) is a ternary semigroup $(G,1,0,-1)$ together with a ternary relation $D$ satisfying [RS0]-[RS6], [RS8] and

RS7’ -

$x\in D^{t}(0,a)\Leftrightarrow x=a$.

Definition 2.4**.**

A pre real semigroup (abbreviated PRS) is a ternary semigroup $(G,1,0,-1)$ together with a ternary relation $D^{t}$ satisfying:

DT0 -

$a\in D^{t}(b,c)$ if and only if $a\in D^{t}(c,b)$.

DT1 -

$a\in D^{t}(b,c)$ implies $-b\in D^{t}(-a,c)$.

DT2 -

$1\in D^{t}(1,a)$ for all $a\in G$.

DT3 -

$a\in D^{t}(b,c)$ implies $ad\in D^{t}(bd,cd)$.

DT4 (Strong Associativity) -

If $a\in D^{t}(b,c)$ and $c\in D^{t}(d,e)$, then there exists $x\in D^{t}(b,d)$ such that $a\in D^{t}(x,e)$.

DT5 -

If $ad=bd$, $ae=be$ and $c\in D^{t}(c^{2}d,c^{2}e)$, then $ac=bc$.

DT6 -

$e\in D^{t}(c^{2}e^{2}a,d^{2}e^{2}b)$ implies $e\in D^{t}(e^{2}a,e^{2}b)$.

DT7 -

$c\in D(a,b)$ implies $c\in D^{t}(c^{2}a,c^{2}b)$.

DT8 -

$a\in D^{t}(a^{2}b,a^{2}c)$ implies $a^{2}\in D^{t}((ab)^{2},(ac)^{2})$.

DT9 -

$x\in D^{t}(0,a)\Leftrightarrow x=a$.

A real semigroup (abbreviated RS) is a pre-real semigroup satisfying

DT10 (Reduction) -

$D^{t}(a,-b)\cap D^{t}(b,-a)\neq\emptyset$ implies $a=b$.

Theorem 2.5** (The Duality Theorem).**

There is a functorial duality between the category $\mathcal{RS}$ of real semigroups with RS-morphisms and $\mathcal{ARS}$ of abstract real spectra with ARS-morphisms. Moreover, the duality establishes an equivalence between the categories $\mathcal{RS}$ and $\mathcal{ARS}^{op}$, the opposite category of $\mathcal{ARS}$.

Let $A$ be a multiring with $-1\notin\sum A^{2}$ and consider $T$ a preorder. We denote by $Q_{T}(A)$ the image of $A$ in $Q_{2}^{X_{T}}$, where $X_{T}=\{P\in\mbox{Sper}(A):T\subseteq P\}$.

Denote the image of $A$ in $Q_{2}^{X_{T}}$ by $Q_{T}(A)$. Addition on $Q_{T}(A)$ is defined by $\overline{a}+\overline{b}:=\{\overline{c}:c\in a+b\}$, $\overline{a}\overline{b}:=\overline{ab}$, $-\overline{a}:=\overline{-a}$. The zero element of $Q_{T}(A)$ is $\overline{0}$.

Proposition 2.6** (Local-Global principle).**

Let $A$ be a multiring with $-1\notin\sum A^{2}$ and $T$ a proper preordering of $A$. Then:

$Q_{T}(A)$ is a multiring. 2. 2.

$Q_{T}(A)$ is strong embedded in $Q_{2}^{X_{T}}$.

We denote $Q_{\sum A^{2}}(A)$ by $Q_{red}(A)$ which we refer to as the real reduced multiring associated to $A$.

Proposition 2.7**.**

For a multiring $A$ with $-1\notin\sum A^{2}$, the map $a\mapsto\overline{a}$ from $A$ onto $Q_{red}(A)$ is an isomorphism if and only if $A$ satisfies the following properties:

1. a -

$a^{3}=a$. 2. b -

$a+ab^{2}=\{a\}$. 3. c -

$a^{2}+b^{2}$ contains a unique element.

Definition 2.8**.**

A multiring satisfying $-1\notin\sum A^{2}$ and the equivalent conditions of proposition 2.7 will be called real reduced multiring. A morphism of real reduced multirings is just a morphism of multirings. The category of real reduced multirings will be denoted by $\mathcal{MR}_{red}$.

Corollary 2.9**.**

A multiring $A$ is real reduced if and only if the following properties holds for all $a,b,c,d\in A$:

1. i -

$1\neq 0$; 2. ii -

$a^{3}=a$; 3. iii -

$c\in a+ab^{2}\Rightarrow c=a$; 4. iv -

$c\in a^{2}+b^{2}$ and $d\in a^{2}+b^{2}$ implies $c=d$.

This implies that the morphism $a\mapsto\overline{a}$ from $A$ to $Q_{\mbox{red}}(A)$ is an isomorphism. In particular, follow by the local-global principle 2.6 that for any real reduced multiring $A$, $c\in a+b\subseteq F$ if and only if, for every $\sigma:A\rightarrow Q_{2}$, $\sigma(c)\in\sigma(a)+\sigma(b)$.

Proposition 2.10**.**

Let $A$ be a real reduced multiring. Then we have the following:

1. i -

$x\in ax^{2}+bx^{2}$ if and only if $x\in aA^{2}+bA^{2}$; 2. ii -

$x\in a+b$ if and only if $x\in ax^{2}+bx^{2}$, $-a\in ba^{2}-xa^{2}$ and $-b\in ab^{2}-xb^{2}$; 3. iii -

If $ax=bx$, $ay=by$ and $z\in xz^{2}+yz^{2}$, then $az=bz$; 4. iv -

If $x\in ax^{2}+bx^{2}$, then $x^{2}\in a^{2}x^{2}+b^{2}x^{2}$.

Theorem 2.11**.**

Let $(X,G)$ an abstract real spectra and define $a+b=\{d\in G:d\in D^{t}(a,b)\}$. Then $(G,+,\cdot,-,0,1)$ is a real reduced multiring.

Theorem 2.12**.**

Let $A$ be an real reduced multiring and consider the strong embedding $i:A\rightarrow Q_{2}^{\mbox{Sper}(A)}$ given by $i(a)=\hat{a}:\mbox{Sper}(A)\rightarrow Q_{2}$ when $\hat{a}(\sigma)=\sigma(a)$. Define $\hat{A}=i(A)$. Then $(\mbox{Sper}(A),\hat{A})$ is an abstract real spectra.

Theorem 2.13**.**

Let $(G,\cdot,1,0,-1,D)$ be a realsemigroup and define $+:G\times G\rightarrow\mathcal{P}(G)\setminus\{\emptyset\}$, $a+b=\{d\in G:d\in D^{t}(a,b)\}$ and $-:G\rightarrow G$ by $-(g)=-1\cdot g$. Then $(G,+,\cdot,-,0,1)$ is a real reduced multiring.

Theorem 2.14**.**

Let $A$ be a real reduced multiring. Then $(A,\cdot,1,0,-1,D)$ is a realsemigroup, where $d\in D(a,b)\Leftrightarrow d\in d^{2}a+d^{2}b$.

3 Quadratically Tuned Multirings

Definition 3.1**.**

A multiring is said to be quadratically tuned if for all $a,b,c,d,e\in A$:

QT 0 -

$a^{3}=a$.

QT 1 -

$1\in 1+a$.

QT 2 -

If $ad=bd$, $ae=be$ and $c\in c^{2}d+c^{2}e$ then $ac=bc$.

QT 3 -

If $e\in(ce)^{2}a+(de)^{2}b$ then $e\in e^{2}a+e^{2}b$.

QT 4 -

If $a\in a^{2}b+a^{2}c$ then $a^{2}\in(ab)^{2}+(ac)^{2}$.

QT 5 -

$e\in a+b$ if and only if $e\in ae^{2}+be^{2}$, $-a\in ba^{2}-ea^{2}$ and $-b\in ab^{2}-eb^{2}$.

A morphism between two quadratically tuned multirings is just a multiring morphism. The category of quadratically tuned multirings will be denoted by $\mathcal{MR}_{qt}$.

Proposition 3.2**.**

Every real reduced multiring is a quadratically tuned multiring. In particular, every real reduced multifield is a quadratically tuned multiring

Proposition 3.3**.**

Every special multifield is a quadratically tuned multiring.

Set $d\in D(a,b)\Leftrightarrow d\in ad^{2}+bd^{2}$.

Proposition 3.4**.**

Let $A$ be a qt-multiring and $a,b,c,d,e\in A$.

1. i -

$d\in D(a,b)$ iff $d\in aA^{2}+bA^{2}$.

Proof.

1. i -

($\Rightarrow$) is just the definition of $D$. For ($\Leftarrow$), let $d\in ax^{2}+by^{2}$. Then

[TABLE]

∎

Theorem 3.5**.**

Let $(G,\cdot,1,0,-1,D)$ be a pre-real semigroup and define $+:G\times G\rightarrow\mathcal{P}(G)\setminus\{\emptyset\}$, $a+b=\{d\in G:d\in D^{t}(a,b)\}$ and $-:G\rightarrow G$ by $-(g)=-1\cdot g$. Then $(G,+,\cdot,-,0,1)$ is a quadratically tuned multiring.

Corollary 3.6**.**

The correspondence above can be extended to a functor $M:\mathcal{PRS}\rightarrow\mathcal{MR}_{qt}$.

Theorem 3.7**.**

Let $A$ be a quadratically tuned multiring. Then $(A,\cdot,1,0,-1,D)$ is a pre-real semigroup, where $d\in D(a,b)\Leftrightarrow d\in d^{2}a+d^{2}b$.

Corollary 3.8**.**

The correspondence above can be extended to a functor $R:\mathcal{MR}_{qt}\rightarrow\mathcal{PRS}$.

Corollary 3.9**.**

The following diagram commute:

[TABLE]

4 Examples

We already see that special multifields, and real reduced multifields and multirings are quadratically tuned multirings (3.2, 3.3). By the functors established in [7], we have in particular, that special groups and real semigroups are quadratically tuned multirings. Now, in the view of [9], we have a more “algebraically tasted” example below.

Example 4.1**.**

Let $A$ be a ring with $2\in\dot{A}$ and $T$ a preordering. Set $S=T\setminus\{0\}$ and

[TABLE]

Since $\overline{a}^{2}=\overline{1}\in M(A)$, is straghtforward verify that $M(A)$ is a quadratically tuned multiring. Now, as in [1], let the set $G_{T}(A)$ consist of all functions $\overline{a}:\mbox{Sper}_{T}(A)\rightarrow\bm{3}$, for $a\in A$, where

[TABLE]

with the operation induced by product in $A$. We have that $G_{T}(A)$ is a real semigroup. The map $a\mapsto\overline{a}$ from $M(A)$ to $G_{T}(A)$ is a multiring morphism. Since that for all $t\in T\setminus\{0\}$, $\overline{t}=\overline{1}\in G_{T}(A)$, by the universal property of Marshall’s quotient we have an induced (surjective) morphism $\varphi:M(A)\rightarrow G_{T}(A)$.

5 Quadratic Forms over QT-Multirings

Let $A$ be a QT-multiring which will remain fixed throughout this section.

A form of dimension $n\geq 1$ over $A$ is just an $n$-tuple $\varphi:=\langle a_{1},...,a_{n}\rangle$ where $a_{1},...,a_{n}\in A$. The dimension of $\varphi$ is denoted by $\mbox{dim}(\varphi)$. The discriminant of $f$ is defined to be $\mbox{disc}(\varphi):=a_{1}a_{2}...a_{n}\in A$. If $a\in A$, we can scale $\varphi$ by $a$ to obtain the form $a\varphi:=\langle aa_{1},...,aa_{n}\rangle$. The sum of $\varphi$ and a form $\psi=\langle b_{1},...,b_{m}\rangle$ is defined by $\varphi\oplus\psi=\langle a_{1},...,a_{n},b_{1},...,b_{m}\rangle$ and the tensor product of $\varphi$ and $\psi$ is defined by $\varphi\otimes\psi=\langle a_{1}b_{1},...,a_{i}b_{j},...,a_{n}b_{m}\rangle$.

Strong Isometry of one and two-dimensional forms is defined by $\langle a\rangle\equiv_{s}\langle b\rangle\Leftrightarrow a=b$ and $\langle a,b\rangle\equiv\langle c,d\rangle\Leftrightarrow$ $ab=cd$ and $a+b=c+d$. For forms of dimension $n\geq 3$ isometry is defined inductively by: $\langle a_{1},...,a_{n}\rangle\equiv_{s}\langle b_{1},...,b_{n}\rangle$ if and only if there are $x,y,z_{3},...,z_{n}\in A$ such that $\langle a_{1},x\rangle\equiv_{s}\langle b_{1},y\rangle$, $\langle a_{2},...,a_{n}\rangle\equiv_{s}\langle x,z_{3},...,z_{n}\rangle$ and $\langle b_{2},...,b_{n}\rangle\equiv_{s}\langle y,z_{3},...,z_{n}\rangle$.

The strong isometry is just the extension of the isometry concepts provenient of the already known abstract theories of quadratic forms (such as special groups and special multifields). However, in the multiring context, we have another “natural” isometry at hand, the weak isometry defined below:

Weak Isometry of one and two-dimensional forms is defined by $\langle a\rangle\equiv_{w}\langle b\rangle\Leftrightarrow a=b$ and $\langle a,b\rangle\equiv\langle c,d\rangle\Leftrightarrow$ $ab=cd$ and $a+b\cap c+d\neq\emptyset$. For forms of dimension $n\geq 3$ isometry is defined inductively by: $\langle a_{1},...,a_{n}\rangle\equiv_{w}\langle b_{1},...,b_{n}\rangle$ if and only if there are $x,y,z_{3},...,z_{n}\in A$ such that $\langle a_{1},x\rangle\equiv_{w}\langle b_{1},y\rangle$, $\langle a_{2},...,a_{n}\rangle\equiv_{w}\langle x,z_{3},...,z_{n}\rangle$ and $\langle b_{2},...,b_{n}\rangle\equiv_{w}\langle y,z_{3},...,z_{n}\rangle$.

Observe that $\varphi\equiv_{s}\psi\Rightarrow\varphi\equiv_{w}\psi$.

Proposition 5.1**.**

Let $A$ be a QT-multiring, $a,b,c,d,e\in A$ and $\varphi:=\langle a_{1},...,a_{n}\rangle$ and $\psi=\langle b_{1},...,b_{m}\rangle$ be forms over $A$. For $\sigma\in S_{n}$, write $\varphi^{\sigma}$ for the $n$-form $\varphi^{\sigma}=\langle a_{\sigma(1)},...,a_{\sigma(n)}\rangle$. Then

1. i -

$\langle a,b,c\rangle\equiv_{w}\langle a,b,c\rangle^{\sigma}$. 2. ii -

$\varphi\equiv_{w}\varphi^{\sigma}$.