Flat Unramified extension of DVR

by iarnab_kundu, Oct 23, 2017, 9:46 AM

A flat extension of rings is one where $B$ is flat as an $A$ module. An unramified extension of Noetherian local rings $A\to B$ is one where the maximal ideal $m$ of $A$ generates the maximal ideal of $B$ , that is $m\cdot B$ is the maximal ideal of $B$ ; and the residue field extension $\kappa(A)\to\kappa(B)$ is a finite separable extension.

Lemma: If $A\to B$ is an unramified extension of Noetherian local rings. Suppose $A\to C$ is any ring extension. Then $C\to C\otimes B$ is an \etale map.

Lemma: Let $A$ , $B$ be Noetherian local rings with maximal ideal $m$ and $n$ respectively. If $A\to B$ is a flat unramified extension of local rings. Then $B/n\to B\otimes A/m$ is an isomorphism of rings, where the map is given by $\bar{b}\mapsto b\otimes \bar{1}$ .
Proof: Consider the exact sequence $0\to m\to A\to A/m\to 0$ . Given that $B$ is flat, tensoring with $B$ produces the sequence $0\to n\to B\to B\otimes A/m\to 0$ . We have used the fact that $m\otimes B\to mB=n$ is an isomorphism, where the map is $x\otimes t\mapsto xt$ .

Lemma: Flat module over an integral domain is torsion free.

Lemma: (Krull's intersection theorem) Let $A$ be a Noetherian local ring, with a proper ideal $I$ . Then $\cap_{n=1}^{\infty}I^n=0$ .

Lemma: Let $A$ , $B$ be Noetherian local rings with maximal ideals $m$ and $n$ respectively. Suppose $A$ is a dvr, and $A \to B$ is a flat, unramified extension. Then $B$ is a integral domain.
Proof: At the outset, assume that $B$ is not a field. Let $t\in m$ be a coordinate of $A$ . Then by the unramifiedness we have that $u=\phi(t)$ generates $n$ . If $u=0$ then the maximal ideal $n=0$ implying that $B$ is a field; and therefore we may assume $u\neq 0$ . We shall first prove that $u$ is not nilpotent. Suppose on the contrary $u$ is nilpotent. Choose the smallest $n>1$ such that $u^n=0$ . But $B$ being flat over an integral domain, is in fact torsion free. However, $u^n=\phi(t)^{n-1}\phi(t)=0$ . Thus, $\phi(t^{n-1})=0$ , which contradicts the minimality of $n$ .Thus we have proved that $u$ is not nilpotent.
Now we shall prove that $B$ is an integral domain. Suppose there are non-zero zero-divisors, $x$ and $y$ . By Krull's intersection theorem we have that there is a positive integer $p$ such that $x\in n^p\setminus n^{p+1}$ . Hence we may factor $x=u^{p}r$ where $r\notin n$ and is therefore a unit. Similarly there is a positive integer $q$ and a unit $s$ such that $y=u^qs$ . Then $xy=u^{p+q}st=0$ , which implies that $u$ is nilpotent. Thus we are done.

Corrollary: With $A,B$ be Noetherian local rings as above. If $A$ is a dvr, and $A\to B$ is a flat unramified extension of rings; then $B$ is also a dvr.
Proof: We shall follow through the above proof. We have already shown that $B$ is an integral domain, and that $u$ generates the maximal ideal $n$ . From the proof it is also clear that any element $x\in n$ can be written as $x=u^ps$ where $p$ is a positive integer and $s$ is a unit. We may now show that $B$ is a UFD, and therefore $n$ being principal is of height $1$ . Thus the result follows.

This post has been edited 5 times. Last edited by iarnab_kundu, Oct 24, 2017, 10:31 AM

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Here is a restatement of proof which makes it easier for regular local rings:

Since the ring extension is flat, t is a non-zero divisor on B. By Krull's Principal Ideal theorem, (krull) dimB = ht(n) >= 1. By unramifiedness, (chevalley dimension) s(B) =< 1, therefore by dimension theorem for noetherian local rings s(B) = dimB = 1. It is easy to see that dim(n/n^{2}) = 1 (as tensor product commutes with direct sum!), hence we are done by 4.20 from http://www.math.tifr.res.in/~publ/pamphlets/homological.pdf

In general, because of flatness , a regular sequence will get carried to a regular sequence. It is known that depth =< dim, i.e., dim A =< depth =< dimB. As the ring is unramified, dim A >= chevalley dimension >= dimB. Therefore dim(B) = dim(A) = dim(n/n^{2}) (as tensor product commutes with direct sum!), hence we are done by 4.20 from http://www.math.tifr.res.in/~publ/pamphlets/homological.pdf

by Shravu, Nov 5, 2017, 11:33 AM

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Cracking the Nut Open

Flat Unramified extension of DVR

by iarnab_kundu, Oct 23, 2017, 9:46 AM

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