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A representation of\mathfrak{gl}(V)=V \otimes V^{\ast}is a linear map\mu \colon \mathfrak{gl}(V) \otimes M \rightarrow Msatisfying a certain identity. By currying, giving a linear map\muis equivalent to giving a linear mapa \colon V \otimes M \rightarrow V \otimes M, and one can translate the condition for\muto be a representation into a condition ona. This alternate formulation does not use the dual ofVand makes sense for any objectVin a tensor category\mathcal{C}. We call such objects representations of thecurried general linear algebraonV. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore. 

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the canonical ribbon conjecture of Bayer and Eisenbud holds over a field of characteristic $$0$$ or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.