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Abstract Consider a pair of elementsfandgin a commutative ringQ. Given a matrix factorization offand another ofg, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum$$f+g$$. We will study the tensor product ofd-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen–Macaulay and Ulrich modules over hypersurface domains of a certain form. 

The work of Mann and Rafi [Geom. Topol. 27 (2023), pp. 2237–2296] gives a classification of surfaces $\mathrm{\Sigma <\#comment/>}$when $\mathrm{M}\mathrm{a}\mathrm{p}\left(\mathrm{\Sigma <\#comment/>}\right)$is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann–Rafi, we prove that $\mathrm{P}\mathrm{M}\mathrm{a}\mathrm{p}\left(\mathrm{\Sigma <\#comment/>}\right)$is globally CB if and only if $\mathrm{\Sigma <\#comment/>}$is the Loch Ness monster surface, and locally CB or CB generated if and only if $\mathrm{\Sigma <\#comment/>}$has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures. 

We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible sheaves whose th tensor power is trivial. 

Finite $$F$$-representation type is an important notion in characteristic-$$p$$commutative algebra, but explicit examples of varieties with or without thisproperty are few. We prove that a large class of homogeneous coordinate ringsin positive characteristic will fail to have finite $$F$$-representation type. Todo so, we prove a connection between differential operators on the homogeneouscoordinate ring of $$X$$ and the existence of global sections of a twist of$$(\mathrm{Sym}^m \Omega_X)^\vee$$. By results of Takagi and Takahashi, thisallows us to rule out FFRT for coordinate rings of varieties with$$(\mathrm{Sym}^m \Omega_X)^\vee$$ not ``positive''. By using results positivityand semistability conditions for the (co)tangent sheaves, we show that severalclasses of varieties fail to have finite $$F$$-representation type, includingabelian varieties, most Calabi--Yau varieties, and complete intersections ofgeneral type. Our work also provides examples of the structure of the ring ofdifferential operators for non-$$F$$-pure varieties, which to this point havelargely been unexplored.