Proving Universal Identities in Faithful Models

Faithful Models and Universal Identities

Some universal algebraic identities become easier to prove in a model with additional structure. For matrix identities, working over the complex numbers gives access to eigenvalues, diagonalization, and algebraic geometry. An identity between integer polynomial expressions established this way holds over every commutative ring, including rings with zero divisors and rings of positive characteristic.

The mechanism becomes clear when a theory is regarded as a category and its models as functors. A faithful model detects equality in the theory. Once an equation has been established there, every other model preserves it.

This principle applies both to Lawvere theories, whose models preserve finite products, and to PROPs, whose models preserve symmetric monoidal structure.

Faithful models reflect equations

Let $\mathcal{T}$$\mathcal T$ be a category of formal operations, with equality understood modulo the defining axioms. Suppose

$$F:\mathcal{T}⟶\mathcal{C}$$

is a faithful interpretation. For any parallel morphisms $f,g:X\to Y$$f,g:X\to Y$, faithfulness means

$$F(f)=F(g)⟹f=g.$$

Consequently, for any other interpretation $G:\mathcal{T}\to \mathcal{D}$$G:\mathcal T\to\mathcal D$,

$$F(f)=F(g)⟹f=g⟹G(f)=G(g).$$

The first implication uses faithfulness; the second uses functoriality. There need be no functor between the target categories. The two interpretations are connected by their common source.

When the source is a presented theory, its morphisms are expressions modulo the equations generated by the axioms. Equality in this category therefore means that the equation follows from those axioms. A faithful model satisfies exactly the equations of the theory, with no additional identifications between parallel morphisms.

There are two separate tasks: express the proposed identity as parallel morphisms, and prove that their interpretations agree in a faithful model. Merely writing an equation in categorical notation does not prove it. The useful mathematics lies in finding a suitable model, establishing its faithfulness, and carrying out the calculation there.

The interpretation need not be full. Even faithfulness on all hom-sets is more than a particular application needs: injectivity on the relevant hom-set suffices. A jointly faithful family of models also works, provided the equation holds in every member of that family.

Lawvere theories and algebraic terms

A Lawvere theory $\mathcal{T}$$\mathcal T$ is a category with specified finite products whose objects are the finite powers of a distinguished object. Label these objects by natural numbers. The object $n$$n$ is the $n$$n$-fold power of the generating object $1$$1$, so products of objects are given by addition and $0$$0$ is terminal.

A morphism $r\to s$$r\to s$ is an $s$$s$-tuple of terms in $r$$r$ variables. Composition substitutes terms into terms. A model in a category $\mathcal{C}$$\mathcal C$ with finite products is a finite-product-preserving functor

$$M:\mathcal{T}⟶\mathcal{C}.$$

Writing $A=M(1)$$A=M(1)$, we have canonical identifications

$$M(r)\cong A^r,M(f):A^r⟶A^s.$$

The cartesian structure accounts for the usual treatment of variables. Projections select variables, diagonals repeat them, and maps to the terminal object discard them. Product preservation ensures that these formal constructions acquire their expected meanings in every model.

Thus an equality of terms detected by one faithful model becomes an equality in every model, including models internal to other categories with finite products.

PROPs and monoidal diagrams

A PROP is a strict symmetric monoidal category $\mathcal{P}$$\mathcal P$ whose objects are natural numbers, with tensor product given by addition. A model in a symmetric monoidal category $\mathcal{C}$$\mathcal C$ is a strong symmetric monoidal functor

$$F:\mathcal{P}⟶\mathcal{C}.$$

For $V=F(1)$$V=F(1)$, its interpretation of a morphism has the form

$$f:r⟶s\mapsto F(f):V^{\otimes r}⟶V^{\otimes s},$$

using the structural isomorphisms of the functor.

Composition joins diagrams in sequence; tensor product places them side by side. For example, a multiplication $m:2\to 1$$m:2\to1$ is associative precisely when

$$m\circ (m\otimes {\mathrm{id}}_1)=m\circ ({\mathrm{id}}_1\otimes m)\text{in }\mathcal{P}(3,1).$$

If a faithful model identifies two parallel diagrams, they are equal in the PROP and hence have equal interpretations in every model.

A Lawvere theory can be viewed as a cartesian PROP. The distinction matters: a general PROP does not automatically supply copying or discarding operations. Expressions that reuse an input require suitable additional structure. This affects which equations can be formulated, while the principle of reflection through a faithful functor remains unchanged.

Faithful models of commutative rings

Now let ${\mathbb{T}}_{\mathrm{CRing}}$$\mathbb T_{\mathrm{CRing}}$ be the Lawvere theory of commutative unital rings. Concretely,

$${\mathbb{T}}_{\mathrm{CRing}}(r,s)=\mathbb{Z}[x_1,\dots ,x_r{]}^s,$$

with composition given by polynomial substitution. Integer polynomials are the normal forms of terms modulo the commutative ring axioms.

Every commutative unital ring $R$$R$ determines a model

$$M_R:{\mathbb{T}}_{\mathrm{CRing}}⟶\mathbf{Set},M_R(r)=R^r,$$

which sends a tuple of polynomials to its evaluation function.

Proposition. The model $M_R$$M_R$ is faithful if and only if $R$$R$ has characteristic zero.

Proof. Faithfulness on constant terms requires injectivity of

$${\mathbb{T}}_{\mathrm{CRing}}(0,1)=\mathbb{Z}⟶\mathrm{Set}(R^0,R)\cong R,k⟼k{1}_R.$$

Thus faithfulness implies characteristic zero.

Conversely, suppose $\mathbb{Z}\hookrightarrow R$$\mathbb Z\hookrightarrow R$. If two integer polynomials induce the same function on $R^r$$R^r$, their difference $h$$h$ vanishes there. Restricting to tuples from the embedded copy of the integers gives

$$h(a_1,\dots ,a_r)=0\text{in }\mathbb{Z}\text{for every }(a_1,\dots ,a_r)\in {\mathbb{Z}}^r.$$

Such a polynomial is zero. For one variable this follows from the finite-root bound over $\mathbb{Q}$$\mathbb Q$. Inductively, write $h$$h$ as a polynomial in its last variable. Fixing integer values of the other variables shows that each coefficient vanishes on every integer tuple; induction makes every coefficient zero. Constant polynomials are handled by the injectivity of $\mathbb{Z}\to R$$\mathbb Z\to R$.

Applying this argument coordinatewise proves injectivity on every hom-set. $◻$$\square$

In particular, $\mathbb{Z}$$\mathbb Z$, $\mathbb{Q}$$\mathbb Q$, $\mathbb{R}$$\mathbb R$, and $\mathbb{C}$$\mathbb C$ are faithful models. So is any commutative ring of characteristic zero with zero divisors. The coefficients of the terms here belong to $\mathbb{Z}$$\mathbb Z$; the proposition does not assert injectivity for polynomial functions with arbitrary coefficients from $R$$R$.

The proposition gives us freedom to choose a faithful model with useful additional structure. Over $\mathbb{C}$$\mathbb C$, algebraic closedness and the geometry of affine space can simplify a proof substantially. Faithfulness controls equations in the original ring language; it places no restriction on the methods used to prove them. Once an integer polynomial identity has been established in this model, it holds in every commutative ring.

Faithfulness is relative to the chosen theory. For an arbitrary associative unital ring, the relevant theory has

$${\mathbb{T}}_{\mathrm{Ring}}(r,s)=\mathbb{Z}⟨x_1,\dots ,x_r{⟩}^s.$$

Its model is faithful exactly when no nonzero noncommutative polynomial with integer coefficients vanishes under every substitution in the ring. Characteristic zero is still necessary, but is no longer sufficient. For example, $\mathbb{Q}$$\mathbb Q$ is not faithful for this theory: the distinct terms $xy$$xy$ and $yx$$yx$ have the same interpretation.

Adjugates and Cayley–Hamilton

Fix $n\ge 2$$n\ge2$ and put $N=n^2$$N=n^2$. Arrange $N$$N$ formal variables into a matrix. Matrix multiplication, the adjugate, and the determinant define morphisms in ${\mathbb{T}}_{\mathrm{CRing}}$$\mathbb T_{\mathrm{CRing}}$:

$$\begin{array}{rlrl}\mu & :2N⟶N,& & (A,B)⟼AB,\\ a& :N⟶N,& & A⟼\mathrm{adj}(A),\\ d& :N⟶1,& & A⟼\det \left(A\right).\end{array}$$

These are polynomial operations in the matrix entries. Define also

$$\begin{array}{rlrl}s& :1+N⟶N,& & (\lambda ,A)⟼\lambda A,\\ p_k& :1⟶1,& & x⟼x^k,\\ \tau & :2N⟶2N,& & (A,B)⟼(B,A).\end{array}$$

Here $k\ge 0$$k\ge0$, and $p_0$$p_0$ is the constant polynomial $1$$1$.

The product identity is an equation from $2N$$2N$ to $N$$N$:

$$a\circ \mu =\mu \circ (a\times a)\circ \tau .$$

The double-adjugate identity is an equation from $N$$N$ to $N$$N$:

$$a\circ a=s\circ ⟨p_{n-2}\circ d,{\mathrm{id}}_N⟩.$$

The determinant identity is an equation from $N$$N$ to $1$$1$:

$$d\circ a=p_{n-1}\circ d.$$

The pairing in the double-adjugate equation uses the same matrix input both to compute a scalar and to supply the matrix multiplied by that scalar. This is precisely where the cartesian structure enters.

Cayley–Hamilton belongs alongside these equations. Write

$${\chi }_A(t)=\det (tI-A)=t^n+c_1(A)t^{n-1}+\cdots +c_n(A).$$

Each coefficient is an integer polynomial in the entries of $A$$A$. Therefore the expression

$$h_n(A)=A^n+c_1(A)A^{n-1}+\cdots +c_n(A)I$$

defines a morphism $h_n:N\to N$$h_n:N\to N$, and Cayley–Hamilton is the equation

$$h_n=0_{N,N},$$

where the right-hand side is the tuple of $N$$N$ zero polynomials.

We have now expressed four proposed identities in the theory. Their familiar proofs over $\mathbb{Q}$$\mathbb Q$ or $\mathbb{C}$$\mathbb C$ establish equalities of the interpreted functions in a faithful model, and therefore establish equations in ${\mathbb{T}}_{\mathrm{CRing}}$$\mathbb T_{\mathrm{CRing}}$. Below we will give a geometric proof of Cayley–Hamilton over $\mathbb{C}$$\mathbb C$ to show how the choice of model can simplify the argument.

Every ring model then preserves them. Under the identifications

$$M_R(N)\cong M_n(R),M_R(2N)\cong M_n(R)\times M_n(R),$$

the interpreted operations are

$$\begin{array}{rl}{M}_R(\mu )& :M_n(R)\times M_n(R)⟶M_n(R),\\ M_R(a)& :M_n(R)⟶M_n(R),\\ M_R(d)& :M_n(R)⟶R.\end{array}$$

They are exactly multiplication, adjugation, and determinant over $R$$R$. For example, putting ${\mu }_R=M_R(\mu )$$\mu_R=M_R(\mu)$, $a_R=M_R(a)$$a_R=M_R(a)$, and ${\tau }_R=M_R(\tau )$$\tau_R=M_R(\tau)$, functoriality and product preservation give

$$\begin{array}{rl}{a}_R\circ {\mu }_R& =M_R(a\circ \mu )\\ & =M_R(\mu \circ (a\times a)\circ \tau )\\ & ={\mu }_R\circ (a_R\times a_R)\circ {\tau }_R.\end{array}$$

Evaluating at $(A,B)$$(A,B)$ gives the first identity. All four become

$$\begin{array}{rl}\mathrm{adj}(AB)& =\mathrm{adj}(B)\mathrm{adj}(A),\\ \mathrm{adj}(\mathrm{adj}(A))& =\det {\left(A\right)}^{n-2}A,\\ \det (\mathrm{adj}(A))& =\det {\left(A\right)}^{n-1},\\ {\chi }_A(A)& =0.\end{array}$$

No homomorphism from $\mathbb{Q}$$\mathbb Q$ to $R$$R$ is needed. The argument passes through their common theory:

$$M_{\mathbb{Q}}(f)=M_{\mathbb{Q}}(g)⟹f=g⟹M_R(f)=M_R(g).$$

For each fixed matrix size these are finite-arity equations. Varying the size gives a family of equations.

Using geometry inside a faithful model

Over $\mathbb{C}$$\mathbb C$, polynomial identities can be tested on Zariski-dense subsets of affine space. For two theory morphisms $f,g:r\to s$$f,g:r\to s$ and a Zariski-dense subset $D\subseteq {\mathbb{C}}^r$$D\subseteq\mathbb C^r$, we have

$${M_{\mathbb{C}}(f)|}_D={M_{\mathbb{C}}(g)|}_D⟹M_{\mathbb{C}}(f)=M_{\mathbb{C}}(g)⟹f=g.$$

Indeed, each coordinate of the difference is a polynomial whose zero set is closed and contains $D$$D$. Density makes it vanish everywhere; faithfulness then reflects the equality into the theory. The set $D$$D$ serves as a testing set and need not itself be a model.

For Cayley–Hamilton, consider the affine space of matrices and the discriminant polynomial

$$M_n(\mathbb{C})\cong {\mathbb{A}}^{n^2}(\mathbb{C}),\mathrm{\Delta }(A)={\mathrm{disc}}_t\det (tI-A).$$

The discriminant is polynomial in the characteristic coefficients, hence in the matrix entries. Its nonvanishing locus

$$\mathrm{\Omega }=\{A\in M_n(\mathbb{C}):\mathrm{\Delta }(A)\ne 0\}$$

consists of matrices with distinct eigenvalues. It contains $\mathrm{diag}(1,\dots ,n)$$\operatorname{diag}(1,\ldots,n)$, so it is nonempty. Since affine space is irreducible, this open subset is Zariski dense.

Every matrix in $\mathrm{\Omega }$$\Omega$ has an eigenbasis. For an eigenvector $v_i$$v_i$ with eigenvalue ${\lambda }_i$$\lambda_i$,

$${\chi }_A(A)v_i={\chi }_A({\lambda }_i)v_i=0.$$

Thus ${\chi }_A(A)$$\chi_A(A)$ vanishes on $\mathrm{\Omega }$$\Omega$. Each entry of ${\chi }_A(A)$$\chi_A(A)$ is polynomial in the entries of $A$$A$, so it vanishes on the entire matrix space. Faithfulness of the complex-number model then gives

$$M_{\mathbb{C}}(h_n)=M_{\mathbb{C}}(0_{N,N})⟹h_n=0_{N,N}⟹M_R(h_n)=M_R(0_{N,N})$$

for every commutative ring $R$$R$.

This proof uses diagonalization only for matrices with distinct eigenvalues, whose eigenvectors are linearly independent without invoking Cayley–Hamilton.

The adjugate identities admit the same treatment. On the dense open subset ${\mathrm{GL}}_n(\mathbb{C})$$\mathrm{GL}_n(\mathbb C)$, one can use

$$\mathrm{adj}(A)=\det \left(A\right)A^{-1}$$

and the usual rules for inverses and determinants. For the product identity, work on the dense open subset

$${\mathrm{GL}}_n(\mathbb{C})\times {\mathrm{GL}}_n(\mathbb{C})\subseteq M_n(\mathbb{C})\times M_n(\mathbb{C}).$$

Polynomiality extends the resulting equations to all complex matrices; faithfulness makes them universal.

Algebraic closedness, topology, and matrix inversion are available during these proofs. The final equations involve only the original ring operations, which is what allows them to be interpreted in every commutative ring.

Further polynomial identities

The same method applies whenever both sides are integer polynomial expressions in finitely many variables.

For an $m\times r$$m\times r$ matrix $A$$A$ and an $r\times m$$r\times m$ matrix $B$$B$, with $m\le r$$m\le r$, the Cauchy–Binet identity is

$$\begin{array}{c}\det \left(AB\right)=\sum _{\begin{array}{c}S\subseteq \{1,\dots ,r\}\\ |S|=m\end{array}}\det \left(A_{[:,S]}\right)\det \left(B_{[S,:]}\right).\end{array}$$

Here the selected rows and columns are ordered increasingly. Both sides define morphisms $2mr\to 1$$2mr\to1$. The square case gives multiplicativity of the determinant.

Newton's identities provide another example. For $r$$r$ variables, let $e_k$$e_k$ be the elementary symmetric polynomials, set $e_0=1$$e_0=1$, and write

$$q_i=\sum _{j=1}^r{x}_j^i.$$

Then

$$k{e}_k=\sum _{i=1}^k(-1{)}^{i-1}{e}_{k-i}{q}_i,1\le k\le r,$$

is an equality of morphisms $r\to 1$$r\to1$. Its integral form makes sense in every commutative ring; solving for $e_k$$e_k$ by division requires additional assumptions.

Plücker relations illustrate the importance of choosing the right inputs. For a formal $2\times 4$$2\times4$ matrix with rows $(x_i)$$(x_i)$ and $(y_i)$$(y_i)$, define

$$p_{ij}=x_i{y}_j-x_j{y}_i.$$

Its minors satisfy

$$p_{12}{p}_{34}-p_{13}{p}_{24}+p_{14}{p}_{23}=0.$$

If $P:8\to 6$$P:8\to6$ collects the six minors and $q:6\to 1$$q:6\to1$ is the displayed quadratic polynomial in six independent coordinates, the theory equation is

$$q\circ P=0_{8,1}.$$

The polynomial $q$$q$ itself is nonzero. The relation becomes an identity after substituting the minors as polynomial expressions in the matrix entries.

Universal elements and Yoneda

The generic-matrix argument expresses the same polynomial structure through representability. Let

$$U:\mathbf{CRing}⟶\mathbf{Set}$$

be the underlying-set functor. Free rings represent tuples:

$$U^r\cong {\mathrm{Hom}}_{\mathbf{CRing}}(\mathbb{Z}[x_1,\dots ,x_r],-).$$

Yoneda therefore gives

$$\begin{array}{rl}\mathrm{Nat}(U^r,U^s)& \cong U^s(\mathbb{Z}[x_1,\dots ,x_r])\\ & =\mathbb{Z}[x_1,\dots ,x_r{]}^s\\ & ={\mathbb{T}}_{\mathrm{CRing}}(r,s).\end{array}$$

Thus the terms of the theory are exactly the operations on commutative rings that are natural with respect to every unital ring homomorphism. Each operation is determined by its value at the universal tuple of variables.

For matrix operations, this tuple is the generic matrix

$$X=(x_{ij})\in M_n(\mathbb{Z}[x_{ij}]).$$

Every matrix over every commutative ring is obtained by applying the unique homomorphism sending each variable to the corresponding matrix entry. Binary operations use two independent sets of variables.

This separates two useful facts. Yoneda identifies natural operations with formal polynomials and detects their equality at a universal element. Faithfulness of $M_{\mathbb{Q}}$$M_{\mathbb Q}$ detects equality by evaluation on all rational tuples. The latter still requires the polynomial argument proved above.

Alternatively, one can verify a generic matrix identity after the injection

$$\mathbb{Z}[x_{ij}]\hookrightarrow \mathbb{Q}(x_{ij}),$$

reflect it back through injectivity, and then specialize to any matrix. This is the generic-matrix proof of Cayley–Hamilton recorded in the Stacks Project.

The categorical formulation identifies why such proofs yield consequences of the ring axioms. Their calculations may use the tools of a convenient faithful model; the resulting equations belong to the theory itself.