Syzygy Decompositions and Projective Resolutions

We give a projective resolution of a finite dimensional 𝛫-algebra 𝛬 over its enveloping algebra 𝛬^𝑒 = 𝛬^𝑜𝑝 ⨂_𝛫𝛬. The description of this resolution is related to decompositions of the first syzygy module of 𝛬 as an 𝛬^𝑒 module. Resolutions of right 𝛬 modules 𝑀_𝛬 may be obtained by tensoring 𝑀 over 𝛬 with this bimodule resoution. We describe how to obtain such a resolution when 𝑀 is simple or when 𝑀 is given in the form of a projective presentation. Computations of 𝐸𝑥𝑡_𝛬^𝑛(𝑆_𝑣,𝑆_𝑤) for certain classes of algebras 𝛬 are made using these resolutions, and applied to obtain results on global dimension.