Let $(R,\mathfrak m)$ be an unramified regular local ring of mixed characteristic $(0,p)$ and dimension $d$ and let $I\subseteq R$ be an ideal. We prove that $depth(R/I)\geq 3$ implies $cd(I)\leq d-3$, and if $R$ is essentially of finite type over a DVR, then $depth(R/I)\geq 4$ implies $cd(I)\leq d-4$. More generally, $H_I^j(R)$ is a $\mathbb{Q}$-vector space whenever $j>d-depth(R/I)$, thus vanishing of local cohomology in this range is determined completely by the characteristic zero fiber.
Let $Q$ be a Gorenstein local ring, let $I$ be a proper ideal of $Q$, and set $R=Q/I$. We investigate when finite injective dimension of the conormal module $I/I^2$ forces $I$ to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When $Q$ is re...
Let $A$ and $X$ be connected based CW complexes with $H_1(A;\mathbb Q)\neq 0$, and let $m\geq 1$ be the least degree with $\widetilde H_m(X;\mathbb Q)\neq0$. We prove that the Bousfield--Kan rational completion map of $A\vee X$ induces a map on $H_{2m}(-;\mathbb Q)$ whose cokernel has dimension at least $2^{\aleph_0}$....
Let $G=\operatorname{Gr}(k,V)$ be a complex Grassmannian in its Pl$\"u$cker embedding, and let $Y_E$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics using the kernel and span of a conic. These define a flag variety $B=\operatorname{Fl}(k-2,k+2;V)$ and a relative...
For $r>2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field o...
Let $K(n)$ be the generalized Kronecker quiver, and let $(T(n),\Omega)$ be the universal covering of $K(n)$, where $n\geq 3$ and $\Omega$ is a bipartite orientation. We introduce three different types of representations for quiver $(T(n),\Omega)$ by $K(n)$: maps are injective in a representation maps are all surjective...
Let $X$ be a smooth scheme of dimension $d$ over a field $F$. We study the abelian-group structure of the higher Chow groups $CH^{d+i}(X,j)$. For a prime $l$ different from the characteristic of $F$, we prove divisibility and torsion-freeness results when $i\ge$ the $l$-cohomological dimension of $F$. If $X$ is smooth...
Toshiro Hiranouchi, Rin Sugiyama· 0 citations
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