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-fun...

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory E(n), or Morava E-theory En. For nice spaces the E2-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between ExtUBP∗BP (BP∗,M) and ExtUEn∗En (En∗, En∗⊗BP∗ M) ...

In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist’s point of view. In 1983 Bousfield published a paper about the cate...

A numerical method for a class of forward-backward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the Four Step Scheme (Douglas-Ma-Protter, 1996) but with a Hermite-spectral method to approximate the solution to the decoupled quasilinear PDE on the whole space. A rigorous synthetic error analysis is carried out for a fully discretized scheme, ...

We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal of morphisms with certain properties, and that if has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study v...

LET p : E-t B be a fibration of pointed spaces with fiber F. Let A be any abelian group, and suppose that the base B is connected. Our main result is: THEOREM. The mod A Eilenberg-Moore spectral sequence of p cotverges strongly to H,(F, A) ifand oni,v if z,(B) acts nilpotently on H,(F, A) for each i 2 0. This statement has to be explained. First of all, the theorem refers to the general " Eilen...

In the mid 1970’s Mark Mahowald constructed a new infinite family of elements in the 2–component of the stable homotopy groups of spheres, ηj ∈ π 2j (S)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in §3 below), ηj is detected by h1hj ∈ Ext2,∗ A (Z/2,Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2...

The Kervaire invariant one problem has been one of the most fundamental and challenging problems in topology [6,8,9, 18, 19,231. Of course, the pivotal work was [S], which translated the original geometric problem [18, 191 into the problem of the stable homotopy groups of the sphere: Is It; E Ext;“” (2/2,;2/2) a permanent cycle in the Adams spectral sequence of the sphere? The traditional belie...