254:, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
250:
in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the
38:
207:. In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any
184:
119:
490:
302:
115:
413:
27:
371:
251:
195:. This means there is not enough room for a regular function to do anything interesting outside of
88:
46:
54:
42:
222:) to be the common fraction field of any ring of regular functions on any open affine subset of
382:
58:
475:
73:
424:
169:
142:
484:
231:
34:
247:
227:
416:
of this field extension. All finite transcendence degree field extensions of
188:
226:. Alternatively, one can define the function field in this case to be the
430:, that is, dimension 1, it follows that any two non-constant functions
322:) and whose restriction maps are induced from the restriction maps of
199:, and consequently the behavior of the rational functions on
172:
whose value is the fraction field of the global sections of
420:
correspond to the rational function field of some variety.
203:
should determine the behavior of the rational functions on
187:
but not affine, then any non-empty affine open set will be
476:
https://www.e-periodica.ch/cntmng?pid=ens-001:1979:25::101
246:is no longer integral. Then it is possible to have
359:is defined, it is possible to study properties of
157:will be a localization of the global sections of
329:by the universal property of localization. Then
257:The correct solution is to proceed as follows:
8:
153:is affine, the ring of regular functions on
49:, such a sheaf associates to each open set
283:) that are not zero divisors in any stalk
103:In the simplest cases, the definition of
336:is the sheaf associated to the presheaf
39:function field of an algebraic variety
7:
145:of the ring of regular functions on
463:Kleiman, S., "Misconceptions about
297:be the presheaf whose sections on
61:on that open set; in other words,
14:
474:25 (1979), 203–206, available at
272:be the set of all elements in Γ(
442:satisfy a polynomial equation
1:
423:In the particular case of an
72:) is the set of fractions of
18:sheaf of rational functions
507:
393:we have a field extension
389:, then over each open set
370:. This is the subject of
33:is the generalization to
242:The trouble starts when
120:affine algebraic variety
110:is straightforward. If
87:does not always give a
412:will be equal to the
363:which depend only on
126:is an open subset of
91:for a general scheme
414:transcendence degree
80:. Despite its name,
408:. The dimension of
372:birational geometry
252:total quotient ring
161:, and consequently
47:algebraic varieties
261:For each open set
59:rational functions
43:algebraic geometry
383:algebraic variety
74:regular functions
45:. In the case of
37:of the notion of
498:
506:
505:
501:
500:
499:
497:
496:
495:
481:
480:
468:
460:
425:algebraic curve
398:
368:
357:
350:
341:
334:
327:
320:
309:
295:
288:
281:
270:
240:
216:
166:
135:
108:
101:
85:
66:
24:
12:
11:
5:
504:
502:
494:
493:
483:
482:
479:
478:
472:Enseign. Math.
466:
459:
456:
396:
366:
355:
349:
348:Further issues
346:
345:
344:
339:
332:
325:
318:
307:
293:
286:
279:
268:
239:
236:
214:
170:constant sheaf
164:
143:fraction field
141:) will be the
133:
106:
100:
97:
83:
64:
22:
13:
10:
9:
6:
4:
3:
2:
503:
492:
491:Scheme theory
489:
488:
486:
477:
473:
469:
462:
461:
457:
455:
453:
449:
445:
441:
437:
433:
429:
426:
421:
419:
415:
411:
407:
403:
399:
392:
388:
385:over a field
384:
380:
375:
373:
369:
362:
358:
347:
342:
335:
328:
321:
314:
310:
304:
303:localizations
300:
296:
289:
282:
275:
271:
264:
260:
259:
258:
255:
253:
249:
248:zero divisors
245:
237:
235:
233:
232:generic point
229:
225:
221:
217:
210:
206:
202:
198:
194:
190:
186:
182:
177:
175:
171:
167:
160:
156:
152:
148:
144:
140:
136:
129:
125:
121:
117:
113:
109:
98:
96:
94:
90:
86:
79:
75:
71:
67:
60:
56:
52:
48:
44:
41:in classical
40:
36:
35:scheme theory
32:
29:
25:
19:
471:
464:
458:Bibliography
451:
447:
443:
439:
435:
431:
427:
422:
417:
409:
405:
401:
394:
390:
386:
378:
376:
364:
360:
353:
351:
337:
330:
323:
316:
312:
305:
298:
291:
284:
277:
273:
266:
262:
256:
243:
241:
238:General case
223:
219:
212:
208:
204:
200:
196:
192:
180:
178:
173:
168:will be the
162:
158:
154:
150:
146:
138:
131:
127:
123:
111:
104:
102:
99:Simple cases
92:
81:
77:
69:
62:
50:
30:
20:
17:
15:
116:irreducible
228:local ring
149:. Because
122:, and if
485:Category
185:integral
454:) = 0.
230:of the
130:, then
114:is an (
57:of all
381:is an
290:. Let
265:, let
28:scheme
404:) of
352:Once
189:dense
89:field
26:of a
434:and
301:are
55:ring
53:the
16:The
470:",
438:on
377:If
287:X,x
191:in
183:is
179:If
76:on
487::
374:.
315:,
311:Γ(
276:,
234:.
211:,
176:.
118:)
95:.
467:X
465:K
452:G
450:,
448:F
446:(
444:P
440:C
436:G
432:F
428:C
418:k
410:U
406:k
402:U
400:(
397:X
395:K
391:U
387:k
379:X
367:X
365:K
361:X
356:X
354:K
343:.
340:X
338:K
333:X
331:K
326:X
324:O
319:X
317:O
313:U
308:U
306:S
299:U
294:X
292:K
285:O
280:X
278:O
274:U
269:U
267:S
263:U
244:X
224:X
220:U
218:(
215:X
213:K
209:U
205:X
201:U
197:U
193:X
181:X
174:X
165:X
163:K
159:X
155:U
151:X
147:U
139:U
137:(
134:X
132:K
128:X
124:U
112:X
107:X
105:K
93:X
84:X
82:K
78:U
70:U
68:(
65:X
63:K
51:U
31:X
23:X
21:K
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.