An equivalence linking CM-types $A_\infty$ and $D_\infty$

TL;DR

The paper proves a triangulated equivalence between the stable categories of graded MCM modules over the $A_ fty$-type and $D_ infty$-type hypersurface singularities in a specified grading. It constructs the equivalence explicitly by analyzing indecomposable generators for the D_infty ring $R = k[x,y]/(x^2 y)$, showing the graded endomorphism ring of a generator $C_0$ is isomorphic to $k[t]$, and matching objects $C_i, D_i, E_{i,j}, F_{i,j}, G_{i,j}, H_{i,j}$ with their counterparts in the A_infty model; in particular $C_i o k[y](2i)$ and $D_i o k[y](2i+1)$ and the others to generated ideals with explicit shifts. The construction provides a nontrivial equivalence beyond Knörrer periodicity in the graded setting and clarifies how stable MCM categories relate via endomorphism algebras, despite the grading not being preserved. The work also connects to combinatorial arc models on the completed ∞-gon, offering a geometric interpretation of the equivalence.

Abstract

We show that, for a specific grading, the stable categories of graded maximal Cohen-Macaulay modules over hypersurfaces of type $A_\infty$ and $D_\infty$ are equivalent.

Figures (2)

• Figure 1: The indecomposable graded maximal Cohen-Macaulay modules over $R = k[x,y]/\langle x^2y \rangle$ with the grading $|x| = 1$ and $|y| = -1$, and their corresponding matrix factorizations.
• Figure 2: The table provides the condition when $\operatorname{Hom}^{}_{\underline{\mathrm{MCM}}_{\mathbb{Z}}(R)}(\Omega X,Y)$ is 1-dimensional over $k$. In all other cases the set of morphisms in $\underline{\mathrm{MCM}}_{\mathbb{Z}}(R)$ is zero.

Theorems & Definitions (8)

• Theorem A
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of \ref{['T:main_thm']}
• Remark 3.3
• Remark 3.4