# Proper connective differential graded algebras and their geometric realizations

## TL;DR

This paper proves that proper connective DG-algebras can be geometrically realized via smooth projective schemes with full exceptional collections, linking algebraic and geometric structures and analyzing their properties.

## Contribution

It establishes a geometric realization for proper connective DG-algebras and explores their algebraic and motivic properties, including quasi-isomorphism to finite-dimensional DG-algebras.

## Key findings

- Proper connective DG-algebras admit geometric realizations as smooth projective schemes.
- Such DG-algebras are quasi-isomorphic to finite-dimensional DG-algebras.
- The noncommutative Chow motive of these DG-algebras is computed in the smooth case.

## Abstract

We prove that every proper connective DG-algebra $A$ admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. As a corollary we obtain that $A$ is quasi-isomorphic to a finite dimensional DG-algebra and in the smooth case we compute the noncommutative Chow motive of $A$. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective DG-algebras.