such that for all $\alpha \in H^n(K, \mu_\ell^{\otimes n})$, $\lambda(\alpha)
  \leq N$. Let $N'$ be the maximum of $N_n$ for $ 1 \leq n < M$. Thus corollary follows from (\[theorem:krashen1\]).

Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. Then there is a residue homomorphism $\partial : H^n(K, \mu_\ell^{\otimes m} ) \to H^{n-1}(\kappa,
\mu_\ell^{\otimes (m-1)})$ with kernel $H^n_{{\mathrm{\acute{e}t}}}(R,\mu_\ell^{\otimes
  m})$.

Let $A$ be an integral domain with field of fractions $K$. Let $\ell$ be a prime which is a unit in $A$. We have the natural map $H^n_{\mathrm{\acute{e}t}}(A, \mu_\ell^{\otimes m}) \buildrel{\iota}\over{\to} H^n(K,
\mu_\ell^{\otimes m})$. An element $\alpha$ of $H^n(K, \mu_\ell^{\otimes m})$ is said to be [ *unramified*]{} on $A$ if $\alpha$ is in the image of $\iota$. Suppose that $A$ is a regular ring. For each height one prime ideal $P$ of $A$, we have the residue homomorphism $\partial_P : H^n(K, \mu_\ell^{\otimes m}) \to
H^{n-1}(\kappa(P), \mu_\ell^{\otimes (m-1)})$, where $\kappa(P)$ is the residue field at $P$. We have the following

([@auslander-goldman 7.4]) \[auslander-goldman\] Let $A$ be a regular two dimensional integral domain with field of fractions $K$ and $\ell$ a prime which is a unit 