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I have some question (see below) concerning an argument (red question mark) in following example from Bosch's "Algebraic Geometry and Commutative Algebra". Here the excerpt:

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We consider the ring identification $A = K[t_1,t_2]/(t_2^2-t_1^3) \cong \{\sum_{i \in \mathbb{N}}c_i t^i; c_1 =0\}$ and it's maximal ideal $m=(\bar{t_1},\bar{t_2})= (t^2,t^3) \subset A$.

I don't understand why $t^2$ defines a system of parameters of the local (localized by $m$) ring $A_m$? Here the expression "system of parameters" refers to the property introduced in Prop. 2.4.18: The system of parameters is the minimal set generating $m$ in $A_m$. That's not clear to me how to see that $m$ in $A_m$ is generated by the single element $t^2$.

user26857
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user267839
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  • I do not think the definition of a system of parameters in your post agrees with the definition widely used. $m$ is not principal, but $\sqrt{t_1A} = m$. – Youngsu Sep 13 '18 at 18:50
  • You missunderstood what the textbook says: Prop. 2.4.18 refers to *regular systems of parameters*. – user26857 Sep 13 '18 at 20:32
  • @Youngsu: Ah sure, you mean that in sence of $t^3 \simeq t_2 \in \sqrt{t_1A}$ just because $(t^2)^3= (t^3)^2$ and as user26857 correctly pointed out my confusion between usual sys of pararameters and the regular s o p. Thank you. – user267839 Sep 14 '18 at 12:31

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