Projective Resolution of Modules over the Noncommutative Algebra

We give an explicit algorithm to compute a projective resolution of a module over the noncommutative ring based on the noncommutative Gröbner bases theory. Introduction LetK be a field and Γ be a ring overK. We generally assume that Γ has a unit, ǫ : K → Γ, as well as an augmentation η : Γ → K. For graded Γ-module M and N , Ext Γ (M,N) is defined with a projective resolution of M . This Ext-functor appears in various areas. In algebraic topology, it appears as a E2-terms of the Adams spectral sequence, which is one of the most important tools to compute homotopy sets. Especially, it is given by Ext Ap(Fp,Fp)that E2-terms of the spectral sequence which converges to the stable homotopy groups of the sphere (p)Π ∗ S. Here Ap denotes the mod p Steenrod algebra and ΠS = ⊕ k lim n→∞ [S, S] (p)Π ∗ S = Π ∗ S/{elements of finite order prime to p}. The study of the stable homotopy groups of sphere has a long history, so many tools has been developed. For example, there exists a spectral sequence converging to Ext Ap(Fp,Fp). In this paper, we study an elementary algorithm to compute the Ext-functor directly from its definition, that is, to compute a projective resolution of Γ-modules. 2000 Mathematics Subject Classification. Primary 18G15; Secondary 55S10.