Hochschild homology of rings of algebraic integers
In 1970, \textit{D. Quillen} showed how the (now so-called) André-Quillen cotangent complex of a commutative algebra can be used to study its Hochschild homology [see Appl. Categorical Algebra, Proc. Symp. Pure Math. 17, 65-87 (1970; Zbl 0234.18010)]. The main purpose of the paper under review is to remind that the André-Quillen homology theory is a very useful tool to study Hochschild homology of commutative algebras. The author chooses as an example the computation of Hochschild homology of Dedekind extensions. More concretely, if \(R\) is a Dedekind domain with fraction field \(K\), \(L\) is a finite separable field extension of \(K\), \(S\) is the integral closure of \(R\) in \(L\), and all the residue fields of \(R\) are perfect, the author proves that the Hochschild homology \[ HH_n(S/R)= \begin{cases} S,\quad & \text{if }n=0,\\ \Omega_{S/R},\quad & \text{if }n\text{ is odd},\\ 0,\quad &\text{if } n>0\text{ is even}.\end{cases}. \]
scientific article; zbMATH DE number 1538410