4 INTRODUCTION R E L A T I O N S H I P C H A R T F O R T I G H T C L O S U R E A N D IT S A P P L I C A T I O N S IMPROVED HOMOLOGICAL CONJECTURES, VANISHING THEORE M FOR MAP S O F T O R ARITHMETIC MACAULAYFICATIONS X BIG COHEN-MACAULAY ALGEBRAS, R+, AN D SOLID CLOSUR E X VANISHING THEOREM S ON HOMOLOGY : GRAUERT- RIEMENSCHNEIDER, KODAIRA, ETC . \ PHANTOM HOMOLOGY 1 / TIGH T \ I CLOSUR E J i DIRECT SUMMAND S OF REGULA R RING S AND INVARIANT THEOR Y DIFFERENTIAL OPERATORS AN D STRONGLY F-REGULAR RING S UNIFORM ARTIN-REES THEOREMS BRIANgON-SKODA THEOREMS, INTEGRA L CLOSURES, AN D RATIONAL SINGULARITIES \ CLASSIFICATION O F SINGULARITIES: F-PURE , F-RATIONAL , AND F-REGULAR RING S X
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