What is the meaning of $\mathbb C[t]_t$ in the theory of local ring?

Question

I was reading local ring, multiplicative closed set, localization etc..where it was written that $(f(t)/t^m) \in \mathbb C[t]_t$ for some $m \in \mathbb N$ and $f(t)\in \mathbb C[t].$ Here what is the meaning of $\mathbb C[t]_t$ ?

Please someone help..

Thank you..

Answer 1

The four typical meanings I would see that notation given are:

• The completion of $\mathbb{C}[t]$ at the prime ideal $(t)$ — that is, the power series ring $\mathbb{C}[[t]]$.
• The local ring of $\mathbb{C}[t]$ at the prime ideal $(t)$ — that is, the ring of all rational functions whose denominator is not divisible by $t$.
• The ring obtained by inverting $t$ — that is, $\mathbb{C}[t, t^{-1}]$. It is the ring of all polynomials in $t$ and $t^{-1}$, or equivalently the ring of all rational functions whose denominator is a power of $t$.
• The quotient ring $\mathbb{C}[t] / (t)$.

The last is, I think, by far the least common. The third is the next least common — but given the context you saw it in, it's the only one that makes sense.

Since you are talking about the basics, note:

• The second bullet above is obtained by inverting the multiplicatively closed subset of all polynomials that aren't in the ideal $(t)$
• The third bullet above is obtained by inverting the multiplicatively closed set generated by $t$ — that is, the set of powers of $t$.