The Spectrum (Eigenvalue one) Functor on R-algebras and Its Application to Matrix Rings

Let $\mathcal{C}$$\mathcal{C}$ be the category of $R$$R$-algebras with base point. We use the notation $A_a$$A_a$ to refer the algebra with base point $a$$a$.

We define a functor:

$$\begin{array}{}\text{(1)}& \mathrm{sp}:{\mathcal{C}}^{\mathrm{op}}\to P(R)\end{array}$$

Where $P(R)$$P(R)$ is the power set of $R$$R$, we treat it as a category.

For any object $A_a\in {\mathcal{C}}^{\ast }$$A_a \in \mathcal{C}^*$, we have:

$$\begin{array}{}\text{(2)}& \mathrm{sp}(A_a)=\mathrm{Spec}(A_a)=\{\lambda \in R\mid a-\lambda \text{ is not invertible}\}\end{array}$$

For any morphism $f:A_a\to B_b$$f: A_a \to B_b$ (i.e., a base point-preserving algebra homomorphism), we have:

$$\begin{array}{}\text{(3)}& \mathrm{Spec}(f(a))\subseteq \mathrm{Spec}(a)\end{array}$$

Proof of functoriality:

If $a-\lambda$$a-\lambda$ is invertible, then $f(a-\lambda )=f(a)-\lambda$$f(a-\lambda) = f(a)-\lambda$ is invertible

This shows:

$$\begin{array}{}\text{(5)}& R-\mathrm{Spec}(a)\subseteq R-\mathrm{Spec}(f(a))\end{array}$$

Taking complements:

$$\begin{array}{}\text{(6)}& \mathrm{Spec}(f(a))\subseteq \mathrm{Spec}(a)\end{array}$$

Hence if there exists two morphism $A_a\to B_b$$A_a\to B_b$ and $B_b\to A_a$$B_b\to A_a$ then $\mathrm{Spec}(a)=\mathrm{Spec}(b)$$\mathrm{Spec}(a)=\mathrm{Spec}(b)$.

Application

Let $M_{n,n}(R)$$M_{n,n}(R)$ be the matrix ring, easy to see that $A⟼PA{P}^{-1}$$A\longmapsto PAP^{-1}$ is a $R$$R$ algebra isomomorphism.

Hence

$$\begin{array}{}\text{(7)}& \mathrm{Spec}(A)=\mathrm{Spec}(PA{P}^{-1})\end{array}$$