We show certain standard constructions of the theory of Verdier triangulated categories to be valid in the Heller triangulated framework as well; viz. Karoubi hull, exactness of adjoints, localisation. Updated 13.01.13.

We consider the Heller operator Omega in an abelian category with enough projectives. A counterexample against Omega having a right adjoint is given. I have not found a counterexample against Omega having a left adjoint, only affirmative examples. Updated 16.12.11.

Suppose given a commutative quadrangle in a Verdier triangulated category with an induced isomorphism on the horizontal cones. Suppose that the endomorphism ring of its initial or terminal corner satisfies a finiteness condition. Then it is homotopy cartesian (aka a distinguished weak square).

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle. Moreover, this counterexample generalises to Heller triangulated categories. Updated 16.03.09.

Suppose given functors A x A' -F-> B -G-> C between abelian categories, an object X in A and an object X' in A' such that certain conditions hold. We show that, E_1-terms exempt, the Grothendieck spectral sequence of the composition of F(X,-) and G evaluated at X' is isomorphic to the Grothendieck spectral sequence of the composition of F(-,X') and G evaluated at X. So instead of "resolving X' twice", we may just as well "resolve X twice". Updated 08.06.09.

Some slides dvi ps pdf.

Heller showed that triangulations (octahedra excepted) on stable categories of Frobenius categories are governed by the shift functor. We enlarge the framework so as to include octahedra. Updated 23.12.12.

A general introduction dvi ps pdf. Updated 24.01.13.

Some slides dvi ps pdf. Updated 17.02.10.

Under a combinatorial condition on a poset D, we replace diagrams of shape D that stably commute by diagrams consisting of pure monomorphisms that commute.

We calculate certain additive galois cohomology rings using twisted group rings.

Some slides (for "On representations of twisted group rings" and this item) dvi ps pdf.

We aim to describe the minimal polynomial of the canonical generator of the "p-part" of a cyclotomic extension.

We consider Gram matrices of Specht modules over the Hecke algebra of the symmetric group. H. H. Andersen remarked that such a Gram matrix is not always diagonalizable. For hook partitions, it is, as we show by construction of a suitable basis. Moreover, we give some more examples in which it is not.

Certain aspects of classical untwisted representation theory are generalized to the twisted case.

We determine the elementary divisors of Specht modules for two-row partitions (and some related ones). Moreover, for a general partition lambda at a prime p > n - lambda_1, we determine the composition factors of S_p^lambda and their distribution over the Jantzen subquotients.

Some slides dvi ps pdf.

We give a formula for the elementary divisors of the Dedekind embedding of the tensor product of two copies of a cyclotomic extension of Z_(p) into a direct product of copies of this extension, given by restriction of the rational isomorphism (being injective by Dedekind's Lemma). Similarly, we investigate the cyclic Wedderburn embedding.

We give a formula for a modular morphism between Specht lattices, the target partition arising from the start partition by a downwards shift of two boxes situated at the bottom of a column.

We aim to describe the integral group ring of the symmetric group as a subring of a product of matrix rings via the Wedderburn embedding. The key idea is that the required congruences of matrix entries, the so-called ties, result from modular morphisms between lattices over ZS

A research announcement (ERA AMS) dvi ps pdf.

Some slides dvi ps pdf.

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