[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the canonical module of the Rees algebra $S=R[It]$ is $\bigoplus\limits_{n\ge 1}I^{n-g+1}\omega_Rt^n$. But its proof is in a paper of Bruns which is unfortunately unavailable. My main question is that does this result hold without any assumption on $R$, e.g. the Cohen-Macaulayness of $R$?
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1$\begingroup$ What is $n$? The dimension of $R$? Regardless, I'm sure you can figure out the canonical module from this paper which looks at a much more general situation. Filtered rings, Filtered Blowing-Ups and normal two-dimensional singularities with "star-shaped" resolution by Tomari and Watanabe. See in particular section 3 (Corollary 3.3(ii)). This paper handles more general Rees algebras (such as symbolic Rees algebras) as well. $\endgroup$– Karl SchwedeCommented Feb 14, 2015 at 3:09
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The result of Bruns needed for determining the canonical module of the Rees algebra with respect to an ideal generated by a regular sequence is Theorem 8.8 in Bruns and Vetter, Determinantal Rings. This assumes that the ground ring is a Cohen-Macaulay normal domain.
A more general frame can be found in the paper On the divisor class group of Rees-algebras (Theorem (c)) of Herzog and Vasconcelos.
