A new proof of Serre's homological characterization of regular local rings (Q2520555)

Hereafter, \(A\) will denote a commutative Noetherian local ring with maximal ideal \(\mathfrak{m}\) and residue field \(k\); remember that the \textbf{global dimension} of \(A\) \(\text{gldim}(A)\) is defined as the common value of the \textbf{global injective dimension} \(\text{glidim}(A)\) and the \textbf{global projective dimension} \(\text{glpdim}(A)\), where \[ \text{glidim}(A):=\sup_{M}\{\text{idim}(M)\}, \;\text{glpdim}(A):=\sup_{M}\{\text{pdim}(M)\}, \] \(M\) runs through all the possible \(A\)-modules, and \(\text{idim}(M)\) (respectively, \(\text{pdim}(M)\)) denotes the injective (respectively, projective) dimension of \(M\). Finally, recall that \(A\) is said to be \textbf{regular} provided \(\dim_k \left(\mathfrak{m}/\mathfrak{m}^2\right)=\dim (A)\). The below result is the classical homological characterization of regular local rings obtained by \textit{J.-P. Serre} [in: Proc. internat. Sympos. algebraic number theory, Tokyo \& Nikko Sept. 1955, 175--189 (1956; Zbl 0073.26004)]. Theorem (Serre) The following statements are equivalent: {\parindent=0.7cm \begin{itemize}\item[--] \(A\) is regular. \item[--] \(\text{gldim}(A)=\dim (A)\). \item[--] \(\text{gldim}(A)<\infty\). \end{itemize}} Among the three implications that one needs to prove, the hardest one is to show that \((3)\) implies \((1)\); Serre's original proof of this step works essentially as follows: First of all, one shows that, if \(\text{gldim}(A)<\infty\), then \(\text{gldim}(A)\leq\dim (A)\) using the Auslander-Buchsbaum formula (roughly, the interaction between regular sequences and projective dimension), for this reason often Serre's Theorem is stated as the Auslander-Buchsbaum-Serre Theorem in several textbooks. By means of the Koszul complex, one then shows that \(\dim_k \left(\mathfrak{m}/\mathfrak{m}^2\right)\leq\text{gldim}(A)\). The chain of inequalities \[ \text{gldim}(A)\leq\dim (A)\leq\dim_k \left(\mathfrak{m}/\mathfrak{m}^2\right)\leq\text{gldim}(A) \] implies the result. In the paper under review, the authors produce a new proof of the implication \((3)\) implies \((1)\) which doesn't use a Koszul complex; for the reader's benefit, we give below a rough overview of this proof. One shows that, if \(\text{gldim}(A)<\infty\), then \(\mathfrak{m}\) is not an associated prime of \(A\). Since \(\mathfrak{m}\) is not an associated prime of \(A\), the Prime Avoidance Lemma ensures the existence of an element \(f\in\mathfrak{m}-\mathfrak{m}^2\) which is not a zerodivisor of \(A\). Given this element, one then shows that \[ \text{gldim}(A)=\text{gldim}(A/(f))+1; \] this equality is definitely the core of the proof, and it is based on the non-trivial fact that the derived tensor product \(k\otimes_A^{\mathbb{L}} A/(f)\) has the same homology, as complex of \(A/(f)\)-modules, to the complex \(k\otimes k[1]\). In other words, these two complexes are isomorphic in the derived category of \(A/(f)\)-modules. Finally, using all the above facts, one concludes by doing an increasing induction on \(\text{gldim}(A).\)