Knowledge (XXG)

Noncommutative projective geometry

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247: 109: 130: 25: 289:. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring. 382: 50: 322: 303: 21: 359: 260: 308: 298: 351: 277:
is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of
347: 282: 376: 335: 38: 269:
is the quotient category of the category of finitely generated graded modules over
242:{\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})} 358:
Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry".
364: 133: 53: 241: 103: 37:The quantum plane, the most basic example, is the 324:Modules over regular algebras and quantum planes 281:in this sense is equivalent to the category of 104:{\displaystyle k\langle x,y\rangle /(yx-qxy)} 8: 169: 137: 69: 57: 273:by the subcategory of torsion modules. If 363: 265:By definition, the Proj of a graded ring 230: 220: 207: 194: 184: 172: 163: 144: 132: 72: 52: 338:(1992), "Geometry of quantum planes", 7: 18:noncommutative projective geometry 14: 26:noncommutative algebraic geometry 236: 177: 98: 77: 20:is a noncommutative analog of 1: 399: 258: 321:Ajitabh, Kaushal (1994), 340:Contemporary Mathematics 119:quantum polynomial ring 243: 105: 285:on the usual Proj of 244: 121:is the quotient ring: 106: 131: 117:More generally, the 51: 22:projective geometry 383:Fields of geometry 304:Calabi–Yau algebra 239: 101: 24:in the setting of 261:Proj construction 255:Proj construction 41:of the free ring: 390: 369: 367: 354: 331: 329: 309:Sklyanin algebra 299:Elliptic algebra 283:coherent sheaves 248: 246: 245: 240: 235: 234: 225: 224: 215: 214: 199: 198: 189: 188: 176: 168: 167: 149: 148: 110: 108: 107: 102: 76: 16:In mathematics, 398: 397: 393: 392: 391: 389: 388: 387: 373: 372: 357: 334: 327: 320: 317: 295: 263: 257: 226: 216: 203: 190: 180: 159: 140: 129: 128: 49: 48: 34: 12: 11: 5: 396: 394: 386: 385: 375: 374: 371: 370: 355: 336:Artin, Michael 332: 330:(Ph.D. thesis) 316: 313: 312: 311: 306: 301: 294: 291: 256: 253: 252: 251: 250: 249: 238: 233: 229: 223: 219: 213: 210: 206: 202: 197: 193: 187: 183: 179: 175: 171: 166: 162: 158: 155: 152: 147: 143: 139: 136: 123: 122: 114: 113: 112: 111: 100: 97: 94: 91: 88: 85: 82: 79: 75: 71: 68: 65: 62: 59: 56: 43: 42: 33: 30: 13: 10: 9: 6: 4: 3: 2: 395: 384: 381: 380: 378: 366: 361: 356: 353: 349: 345: 341: 337: 333: 326: 325: 319: 318: 314: 310: 307: 305: 302: 300: 297: 296: 292: 290: 288: 284: 280: 276: 272: 268: 262: 254: 231: 227: 221: 217: 211: 208: 204: 200: 195: 191: 185: 181: 173: 164: 160: 156: 153: 150: 145: 141: 134: 127: 126: 125: 124: 120: 116: 115: 95: 92: 89: 86: 83: 80: 73: 66: 63: 60: 54: 47: 46: 45: 44: 40: 39:quotient ring 36: 35: 31: 29: 27: 23: 19: 343: 339: 323: 286: 278: 274: 270: 266: 264: 118: 17: 15: 315:References 259:See also: 365:1403.3065 201:− 170:⟩ 154:… 138:⟨ 87:− 70:⟩ 58:⟨ 377:Category 346:: 1–15, 293:See also 32:Examples 352:1144023 350:  360:arXiv 328:(PDF) 344:124 379:: 348:MR 342:, 28:. 368:. 362:: 287:R 279:R 275:R 271:R 267:R 237:) 232:i 228:x 222:j 218:x 212:j 209:i 205:q 196:j 192:x 186:i 182:x 178:( 174:/ 165:n 161:x 157:, 151:, 146:1 142:x 135:k 99:) 96:y 93:x 90:q 84:x 81:y 78:( 74:/ 67:y 64:, 61:x 55:k

Index

projective geometry
noncommutative algebraic geometry
quotient ring
Proj construction
coherent sheaves
Elliptic algebra
Calabi–Yau algebra
Sklyanin algebra
Modules over regular algebras and quantum planes
Artin, Michael
MR
1144023
arXiv
1403.3065
Category
Fields of geometry

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