Let [Formula: see text] be an text]-finite, Krull–Schmidt and text]-linear text]-exangulated category with a commutative artinian ring. In this note, we prove that has Auslander–Reiten–Serre duality if only Auslander–Reiten text]-exangles. Moreover, also give equivalent condition for the existence of Serre (which is special type duality). Finally, assume further duality. We exploit bijection tr...

Functorially finite subcategories in module categories over Artin algebras have been introduced by Auslander and Smalø [2] to provide a convenient setting for existence of relative Auslander-Reiten sequences in subcategories. Given a functorially finite subcategory, it is generally not well understood how to compute approximations, and the end terms of relative Auslander-Reiten sequences. In th...

We study the possible weights of an irreducible 2-dimensional mod p representation of Gal(F/F ) which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the ...

Let M be a closed oriented d-dimensionnal manifold and let LM be the space of free loops on M . In this paper, we give a geometrical interpretation of the loop-coproduct and we study it’s compatibility with the Serre spectral sequence associated to the fibration ΩM → LM ev(0) → M . Then, we show that the spectral sequence associated to the free loop fibration LN → LX → LM of some Serre fibratio...