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
Let
 NSFPAR ID:
 10506375
 Editor(s):
 Dan Abramovich
 Publisher / Repository:
 Amer. Math. Soc.
 Date Published:
 Journal Name:
 Transactions of the American Mathematical Society
 ISSN:
 00029947
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$R$ 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 nonnegatively graded DG algebra with noetherian homology. 
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$R$ be a standard graded algebra over a field. We investigate how the singularities of$\operatorname {Spec} R$ or$\operatorname {Proj} R$ affect the$h$ vector of$R$ , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if$R$ satisfies Serre’s condition$(S_r)$ and has reasonable singularities (Du Bois on the punctured spectrum or$F$ pure), then$h_0$ , …,$h_r\geq 0$ . Furthermore the multiplicity of$R$ is at least$h_0+h_1+\dots +h_{r1}$ . We also prove that equality in many cases forces$R$ to be CohenMacaulay. The main technical tools are sharp bounds on regularity of certain$\operatorname {Ext}$ modules, which can be viewed as Kodairatype vanishing statements for Du Bois and$F$ pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be CohenMacaulay. Our results build on and extend previous work by de FernexEin, EisenbudGoto, HunekeSmith, MuraiTerai and others. 
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${\mathrm {V}}_R(M)$ that encodes homological properties of$M$ . We give lower bounds for the dimension of${\mathrm {V}}_R(M)$ 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$\dim {\mathrm {V}}_R(M)$ 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, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of$R$ . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. 
Let
$\Sigma$ be a smooth Riemannian manifold,$\Gamma \subset \Sigma$ a smooth closed oriented submanifold of codimension higher than$2$ and$T$ an integral areaminimizing current in$\Sigma$ which bounds$\Gamma$ . We prove that the set of regular points of$T$ at the boundary is dense in$\Gamma$ . Prior to our theorem the existence of any regular point was not known, except for some special choice of$\Sigma$ and$\Gamma$ . As a corollary of our theoremwe answer to a question in Almgren’s
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