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Proper connective differential graded algebras and their geometric realizations

Theo Raedschelders, Greg Stevenson

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.

Proper connective differential graded algebras and their geometric realizations

Abstract

We prove that every proper connective DG-algebra admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. As a corollary we obtain that is quasi-isomorphic to a finite dimensional DG-algebra and in the smooth case we compute the noncommutative Chow motive of . 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.

Paper Structure

This paper contains 15 sections, 26 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and conventions
  3. Geometric realizations (following Orlov)
  4. An $A_{\infty}$-version of the radical filtration
  5. Existence of realizations
  6. An isomorphism of noncommutative Chow motives
  7. Recollections and the geometric proof
  8. A direct proof
  9. Examples
  10. Smoothness, regularity and t-structures
  11. Smoothness implies regularity
  12. From regularity to smoothness
  13. Regularity of the dual
  14. Through the looking glass
  15. Smoothness of $\mathsf{H}^0$

Key Result

Lemma 3.5

If $(\Lambda,m_*)$ is a minimal finite-dimensional $A_{\infty}$-algebra which is connective (i.e. $\Lambda^i=0$ for $i>0$), then:

Theorems & Definitions (71)

  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Lemma 3.7
  • proof
  • Definition 3.8
  • ...and 61 more