Cohomological Blow Ups of Graded Artinian Gorenstein Algebras Along Surjective Maps

TL;DR

This paper introduces the cohomological blow up (BUG) for graded Artinian Gorenstein algebras, exploring its properties, classification, and relation to geometric blow ups, while preserving key algebraic features.

Contribution

It defines the BUG operation for AG algebras, analyzes its properties, and classifies BUGs that are complete intersections, connecting algebraic and geometric concepts.

Findings

BUG is a connected sum operation.

BUG preserves the strong Lefschetz property.

Standard graded compressed algebras are rarely BUGs.

Abstract

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.