Trace map and regularity of finite extensions of a DVR (Q344122)

In this paper, \(R\) denotes a discrete valuation ring with residue field \(k_R\) and maximal ideal \(m_R\). The algebraic closure of \(k_R\) is denoted by \(\overline{k_R}\). Let \(A\) be a finite and flat \(R\)-algebra. There is a well-defined trace function from \(\mathrm{Hom}_R(A,A)\) in \(R\) which extends the usual trace of matrices and commute with arbitrary extensions of scalars. For \(a\) in \(A\), the author denotes by \(\mathrm{tr}_{A/R}(a)\) the trace of the morphism \(\begin{cases} A\to R\\ x\mapsto ax \end{cases}\) and by \(\tilde{\mathrm{tr}}_{A/R}\) the map \(\begin{cases} A\to \mathrm{Hom}_R(A,R)\\ a \mapsto \mathrm{tr}_{A/R}(a\cdot -)\end{cases}\). The cokernel of \(\tilde{\mathrm{tr}}_{A/R}\) is denoted by \({\mathcal Q}_{A/R}\) (i.e. the quotient of \(\mathrm{Hom}_R(A,R)\) by the image of \(\tilde{\mathrm{tr}}_{A/R}\)). The lenght of \({\mathcal Q}_{A/R}\) (i.e. the size of a maximal chain of submodules) is denoted by \(f^{A/R}\).NEWLINENEWLINEThe aim of this paper is to prove the inequality \(f^{A/R}\geq \mathrm{rk}(A)-|\mathrm{Spec}(A\otimes_R \overline{k_R})|\), and that equality holds if and only if one of the following equivalent conditions holds:NEWLINENEWLINE{(i)} \(A/R\) is tame with separable residue fields and \(A\) is regular,NEWLINENEWLINE{(ii)} \(A/R\) is tame with separable residue fields and \(m_R{\mathcal Q}_{A/R}=0\).NEWLINENEWLINEHere \(\mathrm{rk}(A)\) is the rank of \(A\) and, as usual, \(|\mathrm{Spec}(A\otimes_R \overline{k_R})|\) is the number of prime ideals of \(A\otimes_R \overline{k_R}\). There are several ways of defining flat \(R\)-modules, for example: \(A\) is flat if for every finitely generated ideal \(I\) of \(R\), the induced morphism from \(I \otimes_R A\) in \(R \otimes_R A\) is injective. For a prime ideal \(p\) of \(A\), we let \(k(p)=A/p\), and for a prime ideal \(q\) of \(R\), \(k(q)\) denotes the residue field of the localization \(R_q\). Given a prime ideal \(p\) of \(A\) lying over \(q\), the ramification index of \(p\) in the extension \(A/R\) is \({e(p,A/R)=\frac{\mathrm{dim}_{k(q)} \left(A_p\otimes k_R\right)} {[k(p):k(q)]}}\). \(A/R\) is said to be tame if, for every prime ideal \(q\) of \(R\) and every ideal \(p\) of \(A\) lying over \(q\), \(e(p,A/R)\) is coprime with the characteristic of \(k(q)\). \(A/R\) is said to have separable residue fields if, for every maximal ideal \(p\) of \(A\), the finite extension \(k(p)/k(q)\) is separable.