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We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. This includes an addition-deletion and a restriction theorem, a Yoshinaga type result, and a division theorem for projective dimensions of hyperplane arrangements. These new theorems are all generalizations of classical results for free arrangements, which is the special case of projective dimension zero. To prove these results, we introduce several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, which is combinatorially determined when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is combinatorially determined.
Takuro Abe (Fri,) studied this question.