POINCARÉ DUALITY ALGEBRAS AND RELATED DUAL SYSTEMS
CONTENTS INTRODUCTION PART I POINCAR´E DUALITY QUOTIENTS I 1 POINCAR´E DUALITY, GORENSTEIN ALGEBRAS, AND IRREDUCIBLE IDEALS I 2 PROPERTIES OF IRREDUCIBLE IDEALS I 3 THE ANCESTOR IDEALS I 4 FUNDAMENTAL CLASSES I 5 POINCAR´E DUALITY QUOTIENTS OF F V I 6 COUNTING POINCAR´E DUALITY QUOTIENTS UP TO ISOMORPHISM PART II MACAULAY’S DUAL SYSTEMS AND FROBENIUS POWERS II 1 DIVIDED POWER ALGEBRAS AND OPERATIONS II 2 MACAULAY’S DUAL PRINCIPAL SYSTEMS II 3 AN ILLUSTRATIVE EXAMPLE II 4 RELATION TO THE CLASSICAL FORM PROBLEM II 5 THE K L PARADIGM: A COMPUTATIONAL TOOL II 6 FROBENIUS POWERS PART III POINCAR´E DUALITY AND THE STEENROD ALGEBRA III 1 P?-UNSTABLE POINCAR´E DUALITY QUOTIENTS OF FQ V III 2 THE P?-DOUBLE ANNIHILATOR THEOREM III 3 WU CLASSES III 4 P?-INDECOMPOSABLES: HIT PROBLEMS III 5 A DUAL INTERPRETATION OF THE P?-DOUBLE ANNIHILATOR THEOREM III 6 STEENROD OPERATIONS AND FROBENIUS POWERS III 7 EXAMPLES PART IV DICKSON, SYMMETRIC, AND OTHER COINVARIANTS IV 1 DICKSON COINVARIANTS IV 2 WU CLASSES OF ALGEBRAS OF COINVARIANTS IV 3 THE MACAULAY DUAL OF DICKSON COINVARIANTS MOD 2 IV 4 SYMMETRIC COINVARIANTS PART V THE HIT PROBLEM MOD V 1 POWERS OF DICKSON POLYNOMIALS IN 2 VARIABLES V 2 A ?-INDECOMPOSABLE ELEMENTS IN F2 X, Y V 3 POWERS OF DICKSON POLYNOMIALS IN 3 VARIABLES V 4 POWERS OF DICKSON POLYNOMIALS IN MANY VARIABLES V 5 POWERS OF MOD 2 STIEFEL–WHITNEY CLASSES IN 3 VARIABLES PART VI MACAULAY’S INVERSE SYSTEMS AND APPLICATIONS VI 1 MACAULAY’S INVERSE PRINCIPAL SYSTEMS VI 2 CATALECTICANT MATRICES AND ANCESTOR IDEALS VI 3 REGULAR IDEALS VI 4 LYING OVER FOR IRREDUCIBLE IDEALS VI 5 CHANGE OF RINGS AND INVERSE POLYNOMIALS VI 6 INVERSE SYSTEMS AND THE STEENROD ALGEBRA VI 7 CHANGE OF RINGS AND WU CLASSES VI 8 THE HIT PROBLEM FOR THE DICKSON AND OTHER ALGEBRAS REFERENCES NOTATION