Knowledge (XXG)

Arithmetic and geometric Frobenius

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in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a
282: 152: 275: 214: 316: 301: 311: 268: 306: 32: 172: 164: 21: 248: 105: 210: 59: 252: 25: 224: 220: 206: 198: 176: 205:, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) , vol. 13, Berlin, New York: 295: 156: 148: 125: 180: 168: 86: 43: 240: 141: 17: 184: 90: 74: 256: 163:. The reason for a careful terminology is that the 85:-th powers. In some important cases, for example 179:of the geometric Frobenius. As in the case of a 93:. Otherwise φ is an endomorphism but not a ring 276: 8: 283: 269: 203:Étale cohomology and the Weil conjecture 108:construction to φ. This gives a mapping 7: 237: 235: 46:. Namely, the mapping φ that takes 255:. You can help Knowledge (XXG) by 14: 136:this is not the identity, unless 239: 1: 333: 234: 317:Algebraic geometry stubs 302:Mathematical terminology 312:Algebraic number theory 104:arises by applying the 69:The image of φ is then 251:–related article is a 173:transport of structure 165:Frobenius automorphism 128:. Even in cases where 22:Frobenius endomorphism 147:Mappings created by 197:Freitag, Eberhard; 161:geometric Frobenius 102:geometric Frobenius 100:The terminology of 307:Algebraic geometry 249:algebraic geometry 106:spectrum of a ring 24:is defined in any 264: 263: 216:978-3-540-12175-6 60:ring endomorphism 324: 285: 278: 271: 243: 236: 227: 199:Kiehl, Reinhardt 171:, or defined by 26:commutative ring 332: 331: 327: 326: 325: 323: 322: 321: 292: 291: 290: 289: 232: 217: 207:Springer-Verlag 196: 193: 177:inverse mapping 175:, is often the 12: 11: 5: 330: 328: 320: 319: 314: 309: 304: 294: 293: 288: 287: 280: 273: 265: 262: 261: 244: 230: 229: 215: 192: 189: 151:with φ*, i.e. 126:affine schemes 122: 121: 81:consisting of 33:characteristic 13: 10: 9: 6: 4: 3: 2: 329: 318: 315: 313: 310: 308: 305: 303: 300: 299: 297: 286: 281: 279: 274: 272: 267: 266: 260: 258: 254: 250: 245: 242: 238: 233: 226: 222: 218: 212: 208: 204: 200: 195: 194: 190: 188: 186: 182: 178: 174: 170: 169:Galois groups 166: 162: 159:to be called 158: 157:scheme theory 154: 150: 149:fibre product 145: 143: 139: 135: 131: 127: 119: 115: 111: 110: 109: 107: 103: 98: 96: 92: 88: 87:finite fields 84: 80: 76: 72: 67: 65: 61: 57: 53: 49: 45: 41: 37: 34: 30: 27: 23: 19: 257:expanding it 246: 231: 202: 187:may appear. 181:cyclic group 160: 153:base changes 146: 137: 133: 129: 123: 117: 113: 101: 99: 95:automorphism 94: 82: 78: 70: 68: 63: 55: 51: 47: 44:prime number 39: 35: 28: 15: 228:, p. 5 142:prime field 18:mathematics 296:Categories 191:References 185:minus sign 155:, tend in 91:surjective 116:) → Spec( 112:φ*: Spec( 31:that has 201:(1988), 38:, where 225:0926276 140:is the 89:, φ is 75:subring 223:  213:  73:, the 20:, the 247:This 58:is a 42:is a 253:stub 211:ISBN 167:in 144:. 124:of 77:of 62:of 54:to 50:in 16:In 298:: 221:MR 219:, 209:, 132:= 97:. 66:. 284:e 277:t 270:v 259:. 138:R 134:R 130:R 120:) 118:R 114:R 83:p 79:R 71:R 64:R 56:r 52:R 48:r 40:p 36:p 29:R

Index

mathematics
Frobenius endomorphism
commutative ring
characteristic
prime number
ring endomorphism
subring
finite fields
surjective
spectrum of a ring
affine schemes
prime field
fibre product
base changes
scheme theory
Frobenius automorphism
Galois groups
transport of structure
inverse mapping
cyclic group
minus sign
Kiehl, Reinhardt
Springer-Verlag
ISBN
978-3-540-12175-6
MR
0926276
Stub icon
algebraic geometry
stub

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