We give an example of a direct summand of a serial module that does not admit an indecomposable decomposition.

We describe the main ideas of Ray Heitmann’s proof of the Direct Summand Conjecture in dimension 3 for a ring of mixed characteristic [1]. In the first section we describe the main methods which are used and prove the necessary lemmas. In the second section we prove the main result of Heitmann’s paper. Finally, in the third section we give a proof of the Canonical Element Conjecture using this ...

In this paper, we address the problem of deciding absolute irreducibility of multivariate polynomials. Our work has been motivated by a recent work due to Gao et. al. [1, 2, 3] where they have considered the problem for bivariate polynomials by studying the integral decomposability of polygons in the sense of Minkowski sum. We have generalized their result to polynomials containing arbitrary nu...

We prove that for n ≥ 1 and q > 0 the (oriented) cobordism group of fold maps of (oriented) (n + q)-dimensional manifolds into R contains the direct sum of ⌊(q + 1)/2⌋ copies of the (n − 1)th stable homotopy group of spheres as a direct summand. We also prove that for k ≥ 1 and q = 2k − 1 the cobordism group of fold maps of unoriented (n+ q)-dimensional manifolds into R also contains the nth st...

We generalize this theorem when the motive M(X) ∈ CM(F ; Λ) is replaced by a direct summand (M(X), p) associated with a projector p ∈ EndCM(F ;Λ)(M(X)). The proof given by N. Karpenko in [2] cannot be used in the case where M(X) is replaced by a direct summand because of the use on the multiplicity ([1], §75) as the multiplicity of a projector in the category CM(F ; Λ) is not always equal to 1 ...