Additive group actions on affine T-varieties of complexity one in arbitrary characteristic

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these additive group actions in the toric situation.Comment: 31 page

Gamma-invariant ideals in Iwasawa algebras

Let kG be the completed group algebra of a uniform pro-p group G with coefficients in a field k of characteristic p. We study right ideals I in kG that are invariant under the action of another uniform pro-p group Gamma. We prove that if I is non-zero then an irreducible component of the characteristic support of kG/I must be contained in a certain finite union of rational linear subspaces of Spec gr kG. The minimal codimension of these subspaces gives a lower bound on the homological height of I in terms of the action of a certain Lie algebra on G/G^p. If we take Gamma to be G acting on itself by conjugation, then Gamma-invariant right ideals of kG are precisely the two-sided ideals of kG, and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when G is open in SL_n(\Zp) this lower bound equals 2n - 2. This gives a significant improvement of the results of Ardakov, Wei and Zhang on reflexive ideals in Iwasawa algebras

Calabi-Yau Frobenius algebras

We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include (centrally extended) preprojective algebras of (generalized) Dynkin quivers, and group algebras of classical periodic groups. We use this theory to compute (for the first time) the Hochschild cohomology of many algebras related to quivers, and to simplify the description of known results. Furthermore, we compute the maps on cohomology from extended Dynkin preprojective algebras to the Dynkin ones, which relates our CY property (for Frobenius algebras) to that of Ginzburg (for algebras of finite Hochschild dimension).Comment: 39 pages; v3 has several corrections and some reorganizatio