We determine the $T$-equivariant integral cohomology of $F_4/T$ combinatorially by the Goresky, Kottwitz, and MacPherson (GKM) theory, where $T$ is a maximal torus of the exceptional Lie group $F_4$ and acts on $F_4/T$ by the left multiplication.

We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories, respectively. We study the properties of these homological dimensions and codimensions and unify some important properties possessed by some known homological dimensions. Then we apply the obtained properties to special subcategories and in particular to module categories. Finally we propose some open questions and conjectures which are closely related to the Auslander–Reiten conjecture and the strong Nakayama conjecture.

Arakelov–Green functions defined on metrized graphs have an important role in relating arithmetical problems on algebraic curves to graph-theoretical problems. In this paper, we clarify the combinatorial interpretation of certain Arakelov–Green functions by using electric circuit theory. The formulas we give clearly show that such functions are piecewisely defined, and each piece is a linear or quadratic function on each pair of edges of metrized graphs. These formulas lead to an efficient algorithm for explicit computation of Arakelov–Green functions.

A detailed study of the characters of ${\mathfrak{S}}_{\infty }\left(T\right)$, the wreath product of compact group $T$ with the infinite symmetric group ${\mathfrak{S}}_{\infty }$, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product ${\mathfrak{S}}_n\left(T\right)$ as $n\to \infty$ and its connection with characters of ${\mathfrak{S}}_{\infty }\left(T\right)$. This paper takes a dual approach to these problems. We study harmonic functions on $\mathbb{Y}\left(\widehat{T}\right)$, the branching graph of the inductive system of ${\mathfrak{S}}_n\left(T\right)$’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of ${\mathfrak{S}}_{\infty }\left(T\right)$, which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on $\mathbb{Y}\left(\widehat{T}\right)$ and Martin integral expressions for harmonic functions.

Soit $X$ un schéma torique projective lisse sur $Spec\left(\mathbb{Z}\right)$. Si $\overline{D}$ est un fibré en droites équivariant sur $X$, muni d’une métrique continue et invariante par l’action du tore compact de $X\left(\mathbb{C}\right)$, nous montrons que son volume arithmétique s’exprime en fonction de la transformée de Fenchel–Legendre associée à sa métrique. Lorsque $\overline{D}$ est en plus admissible, nous montrons que le fait que $\overline{D}$ soit arithmétiquement ample, nef, ou gros est caractérisé par des objets issus de la géométrie convexe.

Let $X$ be a smooth projective toric scheme over $Spec\left(\mathbb{Z}\right)$. Let $\overline{D}$ be an equivariant line bundle on $X$, endowed with a continuous Hermitian metric which is invariant by the action of the compact torus of $X\left(\mathbb{C}\right)$. We show that its arithmetic volume is given in terms of the Legendre–Fenchel transform associated to the metric. When $\overline{D}$ is supposed admissible, we characterize when it is arithmetically ample, nef, or big in terms of combinatorial date.

We extend $SL\left(2\right)$-orbit theorems for the degeneration of mixed Hodge structures to a situation in which we do not assume the polarizability of graded quotients. We also obtain analogous results on Deligne systems.

Let $B$, $B$ be square irreducible matrices with entries in $\{0,1\}$. We will show that if the one-sided topological Markov shifts $(X_A,{\sigma }_A)$ and $(X_B,{\sigma }_B)$ are continuously orbit equivalent, then the two-sided topological Markov shifts $({\bar{X}}_A,{\bar{\sigma }}_A)$ and $({\bar{X}}_B,{\bar{\sigma }}_B)$ are flow equivalent, and hence $det(id-A)=det(id-B)$. As a result, the one-sided topological Markov shifts $(X_A,{\sigma }_A)$ and $(X_B,{\sigma }_B)$ are continuously orbit equivalent if and only if the Cuntz–Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ are isomorphic and $det(id-A)=det(id-B)$.

We study the homological behavior of modules over local rings modulo exact zero-divisors. We obtain new results which are in some sense “opposite” to those known for modules over local rings modulo regular elements.