The Hilbert-Kunz functions of two-dimensional rings of type ADE (Q330170)

Let \(k\) be an algebraically closed field of prime characteristic \(p\), let \(F\in k[x,y,z]\) be a homogeneous polynomial (where \(k[x,y,z]\) is \(\mathbb{Z}\)-graded, but not necessarily in the standard way), and set \(R:=k[x,y,z]/(F).\) The object of study of the paper under review is the Hilbert-Kunz function of \(R,\) which in this case is given, for any integer \(e\geq 0,\) by NEWLINE\[NEWLINE HK(R,p^e):=\dim_k \left(k[x,y,z]/(F,x^{p^e},y^{p^e},z^{p^e})\right). NEWLINE\]NEWLINE For instance, a classical result of Kunz says that, when \(F=x^{n+1}-yz\) (\(n\geq 0\) an integer), the Hilbert-Kunz function of \(R\) is exactly NEWLINE\[NEWLINE HK(R,p^e)=\left(2-\frac{1}{n+1}\right)p^{2e}-r+\frac{r^2}{n+1}, NEWLINE\]NEWLINE where \(p^e\equiv r\pmod{n+1}\), and \(r\) is chosen to be the smallest non-negative representative; notice that, in this case, \(R\) is a ring of type \(A_n.\)NEWLINENEWLINEIn the paper under review, the author computes explicitly the Hilbert-Kunz functions of the below rings: NEWLINE\[NEWLINE\begin{aligned} D_n&:=k[x,y,z]/(x^2+y^{n-1}+yz^2),\\ E_6&:=k[x,y,z]/(x^2+y^3+z^4),\\ E_7&:=k[x,y,z]/(x^2+y^3+yz^3),\\ E_8&:=k[x,y,z]/(x^2+y^3+z^5). \end{aligned}NEWLINE\]NEWLINE The explicit formulas are located in Theorem 5.5, Theorem 5.3, Theorem 5.4 and Theorem 5.8 respectively; instead of writing down here these formulas, we prefer to briefly outline what is roughly the strategy to obtain them.NEWLINENEWLINEFirstly, one has to study syzygy modules of the form \(\text{Syz}_R (x^{p^e},y^{p^e},z^{p^e}),\) where \(R\) is one of the above \(DE\) rings. Since \(R\) is, in this case, a graded ring of finite Cohen-Macaulay type, and all \(\text{Syz}_R (x^{p^e},y^{p^e},z^{p^e})\) are maximal Cohen-Macaulay (hereafter, MCM for short) modules, the isomorphism class of \(\text{Syz}_R (x^{p^e},y^{p^e},z^{p^e})\), for \(e\gg 0,\) depends only on the residue class of \(p^e\) modulo a certain invariant, and this invariant is essentially given for the number of isomorphism classes of indecomposable (non-free) MCM modules; this list is obtained using \textit{D. Eisenbud}'s theory of matrix factorizations (see [Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)] and [\textit{G. J. Leuschke} and \textit{R. Wiegand}, Cohen-Macaulay representations. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1252.13001)], Chapter 7, for details about matrix factorizations).NEWLINENEWLINESecondly, by embedding \(R\) into a standard \(\mathbb{Z}\)-graded ring of type \(A_n,\) one can use Brenner-Trivedi (see [\textit{H. Brenner}, Math. Ann. 334, No. 1, 91--110 (2006; Zbl 1098.13017)] and [\textit{V. Trivedi}, J. Algebra 284, No. 2, 627--644 (2005; Zbl 1094.14024)]) geometric interpretation of Hilbert-Kunz functions and the theory of vector bundles to show that all \(\text{Syz}_R (x^{p^e},y^{p^e},z^{p^e})\) (again, for \(e\gg 0\)) are indecomposable.NEWLINENEWLINEFinally, one calculates the Hilbert series of these indecomposable objects, which is likely the most technical part of the whole argument.