Support varieties - an ideal approach

Homological theory of $k$-idempotent ideals in dualizing varieties

In this work, we develop the theory of $k$-idempotent ideals in the setting of dualizing varieties. Several results given previously in \cite{APG} by M. Auslander, M. I. Platzeck, and G. Todorov are extended to this context. Given an ideal $\mathcal{I}$ (which is the trace of a projective module), we construct a canonical recollement which is the analog to a well-known recollement in categories of modules over artin algebras. Moreover, we study the homological properties of the categories involved in such a recollement. Consequently, we find conditions on the ideal $\mathcal{I}$ to obtain quas…

Ideals in BIT speciale varieties

Ideals in BIT speciale varieties are characterized. In particular, it is proved that, for any finitary BIT speciale variety, there is a finite set of ideal terms determining ideals. Several ideal term sets of this kind are given. For the variety of groups one of these sets consists of the terms $y_1y_2$, $xyx^{-1}$, $x^{-1}y^{-1}x$ (and $y$ which can be ignored), while for rings it consists of the terms $y_1+y_2$, $-y$, $xy$, $yx$. For each of the following varieties -- groups with multiple operators, semi-loops, and divisible involutory groupoids -- one of the term sets found in this paper al…

Prime spectra of abelian 2-categories and categorifications of Richardson varieties

We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…

Support varieties - an axiomatic approach

We provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category, where the tensor product is not necessarily symmetric. This is illustrated by examples, taken in particular from representation theory of finite dimensional algebras.

图示代数几何：从理想和代数簇到量子算子

Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi

本文提出图示代数几何（GAG），扩展了图示线性代数程序，证明其在代数结构的（co）span语义下是通用且完整的，展示了GAG在多项式约束网络中的重写系统应用及对量子计算中qudit ZH算子的扩展。