We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C = R, this recovers the standard projective dimension. We show that three natural definitions of finite Cprojective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results th...

The shadow V ′ of a self-dual vertex operator superalgebra V is defined as the direct sum of the irreducible modules of its even vertex operator subalgebra V(0) not contained in V = V(0)⊕V(1). We describe the self-dual “very nice” unitary rational vertex operator superalgebras V of rank c whose shadows have the largest possible minimal weights c 8 or c 8 −1. The results are analogous to and imp...

I review the proposal of Berenstein–Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, a...

We show, in full generality, that Lusztig’s a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of...

The existence and construction of self-dual codes in a permutation module of a finite group for the semisimple case are described from two aspects, one is from the point of view of the composition factors which are self-dual modules, the other one is from the point of view of the Galois group of the coefficient field.

The existence and construction of self-dual codes in a permutation module of a finite group for the semisimple case are described from two aspects, one is from the point of view of the composition factors which are self-dual modules, the other one is from the point of view of the Galois group of the coefficient field.

in this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.