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We prove two kinds of $\mathbb{Z}/2$-periodic Koszul duality equivalences fortriangulated categories of matrix factorizations associated with $(-1)$-shiftedcotangents over quasi-smooth affine derived schemes. We use this result todefine $\mathbb{Z}/2$-periodic version of Donaldson-Thomas categories for localsurfaces, whose $\mathbb{C}^{\ast}$-equivariant version was introduced anddeveloped in the author's previous paper. We compare $\mathbb{Z}/2$-periodic DTcategory with the $\mathbb{C}^{\ast}$-equivariant one, and deduce wall-crossingequivalences of $\mathbb{Z}/2$-periodic DT categories from those of$\mathbb{C}^{\ast}$-equivariant DT categories.

Categorical Donaldson-Thomas theory for local surfaces: $\mathbb{Z}/2$-periodic version