arXiv subjects
Publications and source records attributed to Dmitry Badulin.
We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type algebras over a field. We also show that ind-pro completion inherits properties of the base ring such as normality, regularity, local equidimensionality, etc.
We prove embeddings of adelic groups on an excellent scheme of special type and a flat quasicoherent sheaf on it. For a normal excellent scheme of special type we establish the equality $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})=\mathbb{A}_{I\setminus0}(X,\mathcal{F})$ in the case $I\cap J=I\setminus0$. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes equals the group of global sections of this sheaf. Using this result, we deduce a theorem on intersections of adelic groups for normal projective surfaces. We also compute cohomology groups of a curtailed adelic complex and, as a consequence, show that on a three-dimensional regular projective variety over a countable field the intersection $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})$ equals $\mathbb{A}_{I\cap J}(X,\mathcal{F})$ for any $I,J\subset\{0,1,2,3\}$ and any locally free sheaf $\mathcal{F}$ on $X$.
A graph $G$ on $n$ vertices of diameter $D$ is called $H$-palindromic if $\alpha(G,k) = \alpha(G,D-k)$ for all $k=0, 1, \dots, \left \lfloor{\frac{D}{2}}\right \rfloor$, where $\alpha(G,k)$ is the number of unordered pairs of vertices at distance $k$. Quantities $\alpha(G,k)$ form coefficients of the Hosoya polynomial. In 1999, Caporossi, Dobrynin, Gutman and Hansen showed that there are exactly five $H$-palindromic trees of even diameter and conjectured that there are no such trees of odd diameter. We prove this conjecture for bipartite graphs. An infinite family of $H$-palindromic trees of diameter $6$ is also constructed.