362 INDEX Koszul complex, 65, 73, 74, 228, 230, 292, 296 Koszul relations, 289 Kronecker, Leopold, 31 Kr¨ oning, Heike, 159, 160 KRS, see Krull-Remak-Schmidt Theorem Krull Intersection Theorem, 77, 115, 219 Krull monoid, 14, 16, 16, 21–23 Krull, Wolfgang, 1 Krull-Remak-Schmidt Theorem, 1–7, 10, 19, 27, 47, 48, 114, 131, 137, 210, 212, 298 failure of, 3, 13, 19, 22 for Artinian pairs, 59 for skew group rings, 70, 72 in an additive category, 4 over Artinian pairs, 32 over Artinian rings, 5, 27 over complete local rings, 7, 9, 13, 15, 82, 83 over Henselian local rings, 6 R (M), 313 left minimal FID hull, 183 Lemme d’Acyclicit´ e, 206, 215 length sequence, 269, 270 Leray spectral sequence, 98 Leuschke, Graham J., 109, 114 Levin, Gerson, 200 Levy, Lawrence S., 17 lifting Drozd-Ro˘ ıter conditions, 43 factorizations, 317 field extensions, 170, 171 homomorphisms, 7, 8 idempotents, 5, 6, 72, 317, 318 modules from Artinian pairs, 43 simple roots, 145, 319 lifting number, 7, 8, 9 lifting property, 210, 211–215, 217–219, 223, 226–228, 238 of FID hulls, 179, 180 of MCM approximations, 179, 180 Lipman, Joseph, 119 R (M), 195 local cohomology, 99, 106, 183, 201 local duality, 202 local fundamental group, 117 local-global theorem, 57, 66 locally finite, AR quiver is, 228, 279, 282 locally free in codimension one, 264 on the punctured spectrum, 114, 116, 121, 169, 219, 220, 222, 223, 331, 333, 334 locally free sheaf, 105, 106 Loewy length, 195, 196 Luckas, Melissa, 56 lying over, 326, 337 Maclaurin series, 134 mapping cone, 127 Maranda’s Theorem, 272 Maranda, Jean-Marie, 272 Martsinkovsky, Alex, 184, 189, 192, 194 Maschke’s Theorem, 79 Matlis duality, 223 matrix decompositions, 38, 49, 50, 172 matrix factorization, 123, 123–141, 151–159, 234–238, 245, 253, 255, 256, 258–260 maximally generated, 315 McKay correspondence, 71–77, 81, 91–97, 229–233 geometric, 97–106 McKay quiver, 61, 73, 73–75, see also McKay-Gabriel quiver McKay, John, 81, 91, 104 McKay-Gabriel quiver, 71, 75, 76, 77, 81, 91–97, 104, 229, 230 MCM approximation, 175, 179, 179–202, 208, 243 Ext-minimal, 183, 190, 192 minimal, 180, 180, 182–184, 186, 188–192, 196, 200, 231, 234, 278, 282 right minimal, 180, 180, 182, 183 MCM module, 29, 57 over the skew group ring, 135 mess, 34, 36 minimal FID hull, 186, 189 uniqueness, 183
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