What is the m-adic completion of the local ring Z_(p) (integers localized at the prime (p))?
AThe real numbers R
BThe p-adic integers Z_p
CThe field Q_p of p-adic numbers
DThe polynomial ring Z[x]
The m-adic completion of Z_(p) with respect to its maximal ideal (p) is the ring of p-adic integers Z_p = lim Z/p^nZ. This is a complete local ring with maximal ideal (p) and residue field F_p. The fraction field of Z_p is Q_p, the p-adic numbers.
Question 2 True / False
Hensel's lemma allows lifting a factorization of a polynomial modulo m to a factorization over the complete local ring.
TTrue
FFalse
Answer: True
In its multiplicative form, Hensel's lemma states: if R-hat is a complete local ring with residue field k, and f in R-hat[x] is monic with f mod m = g_0 * h_0 where gcd(g_0, h_0) = 1 in k[x], then f = g * h in R-hat[x] with g, h lifting g_0, h_0. This fails for non-complete rings in general.
Question 3 Short Answer
Is the natural map R → R-hat (from a Noetherian local ring to its completion) always injective?
Think about your answer, then reveal below.
Model answer: Yes, by the Krull intersection theorem: the intersection of m^n over all n is zero in a Noetherian local ring whose maximal ideal contains no idempotents (in particular in a local domain).
The kernel of R → R-hat is the intersection of all powers of m. The Krull intersection theorem states that in a Noetherian ring, this intersection is killed by an element of the form 1 - a with a in I. For a local ring with I = m, the element 1 - a is a unit, so the intersection is zero. Thus R embeds in R-hat.
Question 4 True / False
The completion of the polynomial ring k[x] at the ideal (x) is the formal power series ring k[[x]].
TTrue
FFalse
Answer: True
The quotients k[x]/(x^n) are polynomial rings truncated at degree n. The inverse limit of these is exactly k[[x]], the ring of formal power series. Elements of k[[x]] are formal infinite sums a_0 + a_1 x + a_2 x^2 + ..., which is precisely what completion with respect to (x) produces.
Question 5 Short Answer
Why does completion preserve the Noetherian property, and why is this important?
Think about your answer, then reveal below.
Model answer: If R is a Noetherian local ring, its completion R-hat is also Noetherian (and local). This follows from the fact that R-hat/m-hat^n ≅ R/m^n and a careful analysis of ideal generation. It is important because it allows Noetherian techniques (primary decomposition, dimension theory) to be applied in the completed setting.
The key technical result is that faithful flatness of R → R-hat carries Noetherianness upward. Since R-hat is Noetherian, ideals of R-hat are finitely generated, and many questions about R can be reduced to the complete case where Hensel's lemma and the Cohen structure theorem provide extra tools.