D = C1 ** C2The tensor product is a ZZ/d-graded factorization $D$ whose $i$th component is the direct sum of $C1_j \otimes C2_k$ over all $i = j+k \mod d$. The differential on $C1_j \otimes C2_k$ is the differential $dd^{C1} \otimes id_{C2} + t^j id_{C1} \otimes dd^{C2}$, where $t$ is some primitive $d$th root of unity (assuming both factorizations have period $d$).
The use of a primitive $d$th root of unity adds some subtlety to this construction, since for $d > 2$ the user may need to adjoin a root of unity using the adjoinRoot command. In general, if $C1$ is a factorizations of $f$ and $C2$ is a factorization of $g$, then the tensor product $C1 \otimes C2$ is a factorization of the sum $f + g$. As the next example illustrates, this allows one to always construct matrix factorizations by taking the tensor product of "trivial" factorizations.
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Let us see an example of a longer factorization and tensor products of these objects. In the following example, we use the notation $t$ to denote a primitive $3$rd root of unity:
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If one of the arguments is a module, it is considered as a ZZ/d-graded factorization concentrated in homological degree 0.
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If one of the arguments is a complex, the function will automatically Fold the complex and then take the tensor product. Since we are tensoring with a complex, the potential will remain unchanged.
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Because the tensor product can be regarded as the totalization of a bifactorization, each term of the tensor product comes with pairs of indices, labelling the summands.
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Note in the above how the ZZ/d-graded tensor product differs from the standard tensor product of complexes. Since the indices are taken modulo the period, we see that both $\{ 0, 0 \}$ and $\{ 1, 1 \}$ are indices in degree $0$ (and likewise the indices for $D12$ are taken modulo $3$).
The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:1402:0.