Abstracts for Algebra Seminar

A density function for an algebraic invariant is a measurable function on $\mathbb{R}$ that evaluates the invariant on a real scale. This approach encapsulates richer information about the invariant without requiring additional data. In this talk, we introduce and develop density functions $f_{I_n}$ corresponding to filtrations $\left\{I_n\right\}_{n \in \mathbb{N}}$ of graded ideals within a standard graded domain $A$. This is joint work with Suprajo Das and Sudeshna Roy.

There are many conjectures from the representation theory of finite-dimensional algebras that have been transplanted into commutative algebra, and this process has significantly enriched both fields. A notable example is the Auslander–Reiten Conjecture, which asserts that a finitely generated module M over a finite-dimensional algebra A is projective if the Ext groups $Ext^n(M,M)$ and $Ext^n(M,A)$ vanish for all positive integers n.

The theory of perfectoid towers, introduced by Ishiro-Nakazato-Shimomoto, provides an axiomatic approach to perfectoid theory in commutative algebra through tower-theoretic approximations. In contrast, Bhatt and Scholze introduced prisms as a "deperfection" of perfectoid rings. Our main result shows that a "gradual perfection" of a prism becomes a perfectoid tower. As a consequence, we prove that any p-torsion-free p-adically complete delta-ring that is reduced modulo p admits a perfectoid tower.