The derived category of a locally complete intersection ring

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small.Comment: 14 page

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Compares out-of-pocket costs of maternity care under consumer-driven health plans (CDHP) to a traditional health insurance plan. Explores related factors including prenatal care coverage and unpredictability of costs for delivery and hospital stays

The Derived Category of a Locally Complete Intersection Ring

Let R be a commutative noetherian ring. A well-known theorem in commutative algebra states that R is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known ``localization problem for regular local rings. The main result of this thesis is establishing an analogous characterization for when R is locally a complete intersection. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a nontrivial perfect complex in the derived category using finitely many cones and retracts. This answers a question of Dwyer, Greenlees and Iyengar posed in 2006 and yields a completely triangulated category characterization of locally complete intersection rings. Moreover, this work gives a new proof that a complete intersection localizes. Advisors: Luchezar L. Avramov and Mark E. Walke

A partial converse ghost lemma for the derived category of a commutative noetherian ring

In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$ and complexes of $R$-modules with finitely generated homology $M$ and $N$, we show $N$ is in the thick subcategory generated by $M$ if and only if the ghost index of $N_\mathfrak{p}$ with respect to $M_\mathfrak{p}$ is finite for each prime $\mathfrak{p}$ of $R$. To do so, we establish a "converse coghost lemma" for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.Comment: 10 pages, comments welcom

Bounds on cohomological support varieties

Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$ of finitely generated $R$-modules an algebraic variety $V_R(M)$ that encodes homological properties of $M$. We give lower bounds for the dimension of $V_R(M)$ in terms of classical invariants of $R$. In particular, when $R$ is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$ has finite projective dimension, we also give an upper bound for $\dim V_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.Comment: 23 pages. Comments welcom