358 INDEX desingularization graph, 81, 97, 101, 101, 102, 104 determinant, 74 diagonal map, 58, 66–68, 152, 165, 274, 322, 323–329 diagonalizable, 69, 79, 335, 338 diagonalization, 4, 265 Dickson’s Lemma, 15, 16, 28, 47 Dieterich, Ernst, 268, 276, 280, 284, 285, 298 Dieudonn´ e, Jean, 31 Ding, Songqing, 43, 192, 195, 196, 199, 200 Diophantine monoid, see Krull monoid direct image, 98 direct limit, 164, 165, 167, 168 direct-sum cancellation, 3, 27 direct-sum decomposition, 1, 7, 10, 13–15, 19, 27, 73, 75, 106, 164, 180, 181, 189, 224, 230, 231, 263, 278, 298, 306 discrete valuation ring, DVR, 24, 46, 48, 56, 59, 126, 142, 143, 146, 249–251, 254, 257, 272, 303, 327, 330, 338 divisor, 100, 105 divisor class group Cl(R), 14, 23–26, 99, 289, 306, 307 divisor homomorphism, 14, 14, 16, 26, 27 divisorial ideal, 24, 26, 307 double branched cover, 128, 128–142, 145, 146, 160, 234, 245, 297, 299 double sharp 5 , 138, 138–139 Drozd, Yuriy, 32, 38, 50, 241, 244, 245, 248, 263 Drozd-Ro˘ ıter conditions, 29, 32–39, 41–44, 48–52, 54, 56, 59, 60, 170, 172, 174, 283, 299 necessity of, 42–45 sufficiency of, 45–50, 52 Du Val singularity, see Kleinian singularity Du Val, Patrick, 87, 97, 100 dual, 223, 234 duality, canonical, 177, 222, 250, 261, 281 e(R), e(M), e(I, M), 313 EGA, 310 eigenvalue, 339 Eisenbud, David, 119, 126, 268, 270, 287, 293, 295 Elkik, Ren´ ee, 169 embedding dimension, 100 End, 203 endomorphism ring, 1–3, 5, 11, 32, 44, 48, 55, 56, 59–61, 65, 67–70, 72, 83, 111, 177, 181, 196, 198, 203, 207, 225, 237, 239, 261, 262, 264, 302–304 nc-local, 1, 3–6, 10, 134, 203, 218, 223, 225, 226, 238, 272, 273 enveloping algebra, 323 equivalence of matrix factorizations, 125, 126, 132, 134 Esnault, el` ene, 117, 118 essentially of finite type, 22, 66, 116, 164, 173, 176, 319, 321–326 ´ etale covering, 117 ´ etale fundamental group πet(X), 1 117–119 ´ etale homomorphism, 117, 164, 164, 167, 168, 173, 321–328 Euclid, 84 Euclidean diagram, see extended ADE Coxeter-Dynkin diagram Euler characteristic, 98 evaluation homomorphism, 198 Evans, E. Graham, 3, 7, 311 excellent ring, 98, 118, 163, 167, 167, 169, 171, 173, 242, 268, 276, 279, 280, 304 exceptional fiber, 99–105 expanded subsemigroup, 14, 16, 22, 23 Ext-minimal FID hull, 183 Ext-minimal MCM approximation, 183, 190, 192 extended ADE Coxeter-Dynkin diagram, 91–97, 232, 236–238, see also ADE Coxeter-Dynkin diagram extended module, 17, 18, 23, 24, 26, 43, 164, 169, 305, 306
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