For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fa...

In this paper, we generalize Izumi’s result on uniqueness of realization of finite C-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C -tensor categories by applying Popa’s classification theorem of strongly amenable subfactors of type III1.

Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of characteristic p > 0. We assume that G is defined and split over the prime field k0. In general, for any positive integer r, and any affine fc-group scheme H defined over k0, the r-th infinitesimal subgroup scheme Hr of H is defined to be the (scheme-theoretic) kernel of the r-th power of the Froben...

Consider the abelian category Ck of commutative group schemes of finite type over a field k. By results of Serre and Oort, Ck has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of Ck by the full subcategory Fk of finite k-g...

We consider the Kudla-Millson lift from elliptic modular forms of weight (p + q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p, q). We study the L-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit ...

Let A and B be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence α between A and B such that α is induced by exact functors. We present a nice correspondence between indecomposable modules over A and B. As a consequence, we have the following: (1) If A is a self-injective algebra, then so is B; (2) If A and B are finite dimensional algebras over an algeb...

In mathematics curves are defined as the images of continuous real functions defined on closed intervals and these continuous functions are called parameterizations of the corresponding curves. If only simple curves of finite lengths are considered, then parameterizations can be restricted to the injective continuous functions or even to the continuous length-normalized parameterizations. In ad...

For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fa...