Contributions to the Stable Derived Categories of Gorenstein Rings

Source abstract

The stable derived category $D_{sg}(R)$ of a Gorenstein ring $R$ is the Verdier quotient of the bounded derived category $D^b(\mathsf{mod}\, R)$ by the thick subcategory of perfect complexes, and was introduced by Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation-theoretic and Koszul duality aspects. In Part I, we complete (over $\mathbb{C}$) the classification of homogeneous complete intersection isolated singularities $R$ for which $D^{\mathbb{Z}}_{sg}(R)$ (respectively, $D^b(\mathsf{coh}\, X)$ for $X = \mathsf{proj}\, R$) contains a tilting object. We then use results of Buchweitz-Iyama-Yamaura to describe the indecomposable objects in $D_{sg}^{\mathbb{Z}}(R_Y)$ for $R_Y$ the coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position, and classify their Betti tables. In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq D_{sg}^{\mathbb{Z}}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case $D^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong D_{sg}^{\mathbb{Z}}(R)$. We then relate the heart to modules over the Koszul dual $R^!$. This extends the BGG correspondence to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein, applying e.g. to elliptic normal curves and del Pezzo surfaces of degree $\geq 4$. Finally, we relate our results to conjectures of Bondal and Minamoto on the graded coherence of AS-regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimensions $\geq 4$.