Higher-dimensional module factorizations and complete intersections
Xiao-Wu Chen
TL;DR
This work generalizes Eisenbud's hypersurface matrix factorizations to higher dimensions by introducing $n$-dimensional matrix factorizations (commutative cubes with edges in $\mathbf{MF}(S; \omega_i)$) associated to a regular sequence $(\omega_1,\dots,\omega_n)$. The central mechanism is the total-cokernel functor $\mathrm{TCok}$, which realizes every maximal Cohen–Macaulay module over the complete intersection $R=S/{(\omega_1,\dots,\omega_n)}$ as the total cokernel of an $n$-dimensional factorization, yielding a triangle equivalence $\underline{\mathbf{MF}}^{\bf 0}(S; \omega_1,\dots,\omega_n) \simeq \underline{\mathrm{MCM}}(R)$. The authors extend this framework to noncommutative settings via regular sequences of type $(\sigma_1,\dots,\sigma_n;\xi_{ij})$ in $A$, establishing triangle equivalences between $\underline{\mathbf{GF}}^{\bf 0}(A; \omega_1,\dots,\omega_n)$ and $B-\underline{\mathrm{GProj}}$, and, under noetherian hypotheses, between $\underline{\mathbf{MF}}^{\bf 0}$ and $B-\underline{\mathrm{Gproj}}^{<\infty}$. The results are augmented by a thorough development of the exact structure on factorization categories, two-dimensional and multi-dimensional extensions, and culminate with applications to commutative and quantum complete intersections, connecting to HMF theory and twisted matrix rings.
Abstract
We introduce higher-dimensional module factorizations associated to a regular sequence. They include higher-dimensional matrix factorizations, which are commutative cubes consisting of free modules with edges being classical matrix factorizations. We characterize the stable category of maximal Cohen-Macaulay modules over a complete intersection via higher-dimensional matrix factorizations over the corresponding regular local ring. The result generalizes to noncommutative rings, including quantum complete intersections.