# Finite-dimensional differential graded algebras and their geometric realizations

## TL;DR

This paper establishes a correspondence between finite-dimensional differential graded algebras with separable semisimple parts and certain categories of perfect complexes on smooth projective schemes, providing a geometric realization framework.

## Contribution

It proves an equivalence between categories of perfect modules over such algebras and subcategories of perfect complexes on smooth projective schemes with semi-exceptional collections.

## Key findings

- Category of perfect modules is equivalent to a subcategory of perfect complexes on a smooth projective scheme.
- Provides a characterization of these categories via idempotent completeness and classical generators.
- Connects algebraic structures with geometric realizations through semi-exceptional collections.

## Abstract

We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable semi-exceptional collection. Moreover, we also show that it gives a characterization of such categories assuming that a subcategory is idempotent complete and has a classical generator.