what is a spectrum of an Algebra

Question

In this lectures Fields the author defines

An $\underline{\text { infinitesimally thickened point }}$ $$ \mathbb{D}:=\operatorname{Spec}(A) $$ is represented by a commutative algebra $A \in \mathbb{R A l g}$ which as a real vector space is a direct sum $$ A \simeq_{\mathbb{R}}\langle 1\rangle \oplus V $$ of the 1 -dimensional space $\langle 1\rangle=\mathbb{R}$ of multiples of 1 with a finite dimensional vector space $V$ that is a nilpotent ideal in that for each element $a \in V$ there exists a natural number $n > \in \mathbb{N}$ such that $$ a^{n+1}=0 $$

My question is what is $Spec(A)$?

score 1 · Answer 1 · answered Nov 14 '21 at 19:10

1

${\rm Spec}(A)$, read "the spectrum of $A$", is the set of all prime ideals of $A$.

answered Nov 14 '21 at 19:10

Bonnaduck

4,178

$A$ is an algebra not a ring. – amilton moreira Nov 14 '21 at 19:22
If $A$ is an $\Bbb R$-algebra, then it is also a ring - see here. – Dietrich Burde Nov 14 '21 at 19:27
@Dietrich Burde so you are saying that $Spec(A)$ is the spectrum of $A$ when we treat $A$ as a ring? – amilton moreira Nov 14 '21 at 19:30
The notation ${\rm Spec}(A)$ refers to the set of prime ideals in a ring $A$, yes. – Dietrich Burde Nov 14 '21 at 19:31

what is a spectrum of an Algebra

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