Realizability and the Avrunin–Scott theorem for higher-order support varieties

We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin–Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.

Operations on Semigraphs

In this paper the structural equivalence of union, intersection ring sum and decomposition of semigraphs are explored by using the various types of isomorphisms such as isomorphism, ev-isomorphism, a-isomorphism and e-isomorphism for Ge, Ga and Gca. We establish various types of binary operations in semigraphs.

On Rational Equivalence in Tropical Geometry

This article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Asymptotic prime divisors over complete intersection rings

Let A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that $$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$ is a finite set. Moreover, we prove that there exist i0, n0 ⩾ 0 such that for all i ⩾ i0 and n ⩾ n0, we have $$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$ We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity cxA(M, N/InN) is constant for all sufficiently large n.