2160:
587:
2234:
gave a splitting of algebraic K-theory tensored with the rationals; the summands are now called motivic cohomology (with rational coefficients). Beilinson and
Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology. Most but not all of their conjectures
380:
596:: motivic cohomology, motivic cohomology with compact support, Borel-Moore motivic homology (as above), and motivic homology with compact support. These theories have many of the formal properties of the corresponding theories in topology. For example, the
2257:
Finally, Voevodsky (building on his work with Suslin) defined the four types of motivic homology and motivic cohomology in 2000, along with the derived category of motives. Related categories were also defined by
Hanamura and Levine.
1399:
1588:
213:
1068:
751:
1967:
1819:
1700:
One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category. To describe this, first note that there are
1213:
361:
2134:
at an integer point is equal to the rank of a suitable motivic cohomology group. This is one of the central problems of number theory, incorporating earlier conjectures by
Deligne and Beilinson. The
582:{\displaystyle \cdots \rightarrow H_{2i+1}(X-Z,\mathbf {Z} (i))\rightarrow H_{2i}(Z,\mathbf {Z} (i))\rightarrow H_{2i}(X,\mathbf {Z} (i))\rightarrow H_{2i}(X-Z,\mathbf {Z} (i))\rightarrow 0.}
1077:
with the rationals. For arbitrary schemes of finite type over a field (not necessarily smooth), there is an analogous spectral sequence from motivic homology to G-theory (the K-theory of
1439:
A frequent goal in algebraic geometry or number theory is to compute motivic cohomology, whereas Ă©tale cohomology is often easier to understand. For example, if the base field
903:
The four versions of motivic homology and cohomology can be defined with coefficients in any abelian group. The theories with different coefficients are related by the
1243:. Since Milnor K-theory of a field is defined explicitly by generators and relations, this is a useful description of one piece of the motivic cohomology of
1285:
1477:
114:
2238:
Bloch's definition of higher Chow groups (1986) was the first integral (as opposed to rational) definition of motivic homology for schemes over a field
2795:
2261:
The work of
Elmanto and Morrow has extended the construction of motivic cohomology to arbitrary quasi-compact, quasi-separated schemes over a field.
954:
646:
2770:
2277:
Bloch, Algebraic cycles and higher K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and
Proposition 4.2.9.
2212:
The first clear sign of a possible generalization from Chow groups to a more general motivic cohomology theory for algebraic varieties was
1846:
1730:
2138:
is a special case. More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of
2043:
374:, there is a long exact localization sequence for motivic homology groups, ending with the localization sequence for Chow groups:
2085:
a subfield of the complex numbers, a candidate for the abelian category of mixed motives has been defined by Nori. If a category
1135:
2750:
282:
2556:
2512:
2139:
2135:
1451:, says that many motivic cohomology groups are in fact isomorphic to Ă©tale cohomology groups. This is a consequence of the
2177:
945:
63:
2589:
2199:
2678:
1452:
2181:
2548:
2504:
2015:
in the category of mixed motives. This is far from known. Concretely, Beilinson's conjecture would imply the
904:
231:
2366:
Nori, Lectures at TIFR; Huber and MĂŒller-Stach, On the relation between Nori motives and
Kontsevich periods.
2790:
2785:
2246:
is a natural generalization of the definition of Chow groups, involving algebraic cycles on the product of
2663:
2544:
1662:
35:
2581:
2242:(and hence motivic cohomology, in the case of smooth schemes). The definition of higher Chow groups of
2404:
1455:. Namely, the Beilinson-Lichtenbaum conjecture (Voevodsky's theorem) says that for a smooth scheme
90:
2011:. In particular, the conjecture would imply that motivic cohomology groups can be identified with
2692:
2431:
2170:
2391:
2127:
1630:
2697:
2376:
Elmanto, Elden; Morrow, Matthew (2023). "Motivic cohomology of equicharacteristic schemes".
2725:
2648:
2599:
2566:
2522:
2492:
2462:
893:
27:
2046:
and Murre's conjectures on Chow motives, would imply the existence of an abelian category
8:
1988:
1444:
921:
836:
219:
67:
2737:
2729:
2018:
1276:
2715:
2652:
2626:
2606:
2573:
2538:
2530:
2377:
2221:
2217:
2093:) with the expected properties exists (notably that the Betti realization functor from
937:
847:
613:
593:
43:
2042:
Conversely, a variant of the
Beilinson-Soulé conjecture, together with Grothendieck's
1394:{\displaystyle H^{i}(X,\mathbf {Z} /m(j))\rightarrow H_{et}^{i}(X,\mathbf {Z} /m(j)),}
225:
This problem was resolved by generalizing Chow groups to a bigraded family of groups,
2585:
2552:
2508:
2450:
2445:
1583:{\displaystyle H^{i}(X,\mathbf {Z} /m(j))\rightarrow H_{et}^{i}(X,\mathbf {Z} /m(j))}
933:
23:
2756:
2656:
1093:
Motivic cohomology provides a rich invariant already for fields. (Note that a field
2636:
2478:
2440:
2106:
1996:
925:
897:
871:
828:
637:
208:{\displaystyle CH_{i}(Z)\rightarrow CH_{i}(X)\rightarrow CH_{i}(X-Z)\rightarrow 0,}
2644:
2640:
2595:
2562:
2518:
2488:
2458:
2313:
Levine, K-theory and motivic cohomology of schemes I, eq. (2.9) and
Theorem 14.7.
2231:
2126:
be a smooth projective variety over a number field. The Bloch-Kato conjecture on
1236:
820:
1447:(with finite coefficients). A powerful result proved by Voevodsky, known as the
1433:
2534:
2213:
2143:
1078:
1074:
102:
2667:
2483:
1063:{\displaystyle E_{2}^{pq}=H^{p}(X,\mathbf {Z} (-q/2))\Rightarrow K_{-p-q}(X).}
93:
in topology, but some things are missing. For example, for a closed subscheme
2779:
2764:
2454:
2422:
2295:
Mazza, Voevodsky, Weibel, Lecture Notes on
Motivic Cohomology, Example 13.11.
1690:
1082:
929:
832:
625:
236:
47:
2250:
with affine space which meet a set of hyperplanes (viewed as the faces of a
1724:, such that the motive of projective space is a direct sum of Tate motives:
746:{\displaystyle H^{i}(X,\mathbf {Z} (j))\cong H_{2n-i}(X,\mathbf {Z} (n-j)).}
2322:
Mazza, Voevodsky, Weibel, Lecture Notes on
Motivic Cohomology, Theorem 5.1.
2304:
Mazza, Voevodsky, Weibel, Lecture Notes on Motivic Cohomology, Theorem 4.1.
1832:
denotes the shift or "translation functor" in the triangulated category DM(
900:, and the shift means that this sheaf is viewed as a complex in degree 1.
2230:
of vector bundles. In the early 1980s, Beilinson and Soulé observed that
2055:
1101:), for which motivic cohomology is defined.) Although motivic cohomology
2348:
Jannsen, Motivic sheaves and filtrations on Chow groups, Conjecture 4.1.
2286:
Voevodsky, Triangulated categories of motives over a field, section 2.2.
2499:
Jannsen, Uwe (1994), "Motivic sheaves and filtrations on Chow groups",
2184: in this section. Unsourced material may be challenged and removed.
42:
of algebraic cycles as a special case. Some of the deepest problems in
31:
2765:
Comparison of the Categories of Motives defined by Voevodsky and Nori
2012:
74:
39:
2159:
1962:{\displaystyle H^{i}(X,R(j))\cong {\text{Hom}}_{DM(k;R)}(M(X),R(j))}
2469:
Hanamura, Masaki (1999), "Mixed motives and algebraic cycles III",
2382:
1814:{\displaystyle M(\mathbf {P} _{k}^{n})\cong \oplus _{j=0}^{n}R(j),}
850:, but this gives the same motivic cohomology groups.) For example,
2720:
2631:
2251:
592:
In fact, this is one of a family of four theories constructed by
81:, because they give strong information about all subvarieties of
1840:). In these terms, motivic cohomology (for example) is given by
1121:
is far from understood in general, there is a description when
2357:
Hanamura, Mixed motives and algebraic cycles III, Theorem 3.4.
1987:
are the rational numbers, a modern version of a conjecture by
1443:
is the complex numbers, then Ă©tale cohomology coincides with
1436:
from the Chow ring of a smooth variety to Ă©tale cohomology.
16:
Invariant of algebraic varieties and of more general schemes
2712:
On the relation between Nori motives and Kontsevich periods
2576:(2000), "Triangulated categories of motives over a field",
2070:). More would be needed in order to identify Ext groups in
1208:{\displaystyle K_{j}^{M}(k)\cong H^{j}(k,\mathbf {Z} (j)),}
1991:
predicts that the subcategory of compact objects in DM(k;
2130:
predicts that the order of vanishing of an L-function of
910:
356:{\displaystyle CH_{i}(X)\cong H_{2i}(X,\mathbf {Z} (i)).}
1073:
As in topology, the spectral sequence degenerates after
2112:
1995:) is equivalent to the bounded derived category of an
2668:"WATCH: Motivic Cohomology: past, present and future"
1849:
1733:
1480:
1288:
1138:
957:
649:
383:
285:
276:)), with the usual Chow group being the special case
117:
73:. A key goal of algebraic geometry is to compute the
2528:
2109:), then it must be equivalent to Nori's category.
1961:
1813:
1582:
1393:
1207:
1062:
745:
581:
355:
207:
53:
2709:
846:. (Some properties are easier to prove using the
2777:
2578:Cycles, Transfers, and Motivic Homology Theories
50:are attempts to understand motivic cohomology.
1088:
2710:Huber, Annette; MĂŒller-Stach, Stefan (2011),
2375:
1267:be a positive integer which is invertible in
2739:K-theory and motivic cohomology of schemes I
2039:< 0, which is known only in a few cases.
1271:. Then there is a natural homomorphism (the
218:whereas in topology this would be part of a
1250:
2796:Topological methods of algebraic geometry
2719:
2630:
2605:
2572:
2482:
2444:
2381:
2200:Learn how and when to remove this message
2468:
2117:
1661:determines two objects in DM called the
1432:th roots of unity. This generalizes the
2771:p-adic motivic cohomology in arithmetic
2498:
1416:) on the right means the Ă©tale sheaf (ÎŒ
915:
2778:
2662:
2331:Voevodsky, On motivic cohomology with
1612:
911:Relations to other cohomology theories
89:have some of the formal properties of
2425:(1986), "Algebraic cycles and higher
2421:
227:(BorelâMoore) motivic homology groups
2609:(2011), "On motivic cohomology with
2182:adding citations to reliable sources
2153:
2540:Lecture Notes on Motivic Cohomology
2113:Applications to Arithmetic Geometry
946:Atiyah-Hirzebruch spectral sequence
13:
2735:
896:) denotes the sheaf of invertible
620:of finite type over a field. When
14:
2807:
2703:
2216:'s definition and development of
1467:a positive integer invertible in
2748:
2679:International Mathematical Union
2158:
2136:BirchâSwinnerton-Dyer conjecture
1742:
1556:
1501:
1453:norm residue isomorphism theorem
1449:Beilinson-Lichtenbaum conjecture
1364:
1309:
1259:be a smooth scheme over a field
1186:
999:
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670:
557:
509:
467:
425:
334:
2369:
2360:
2169:needs additional citations for
944:over a field, analogous to the
831:with coefficients in a certain
54:Motivic homology and cohomology
2351:
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2316:
2307:
2298:
2289:
2280:
2271:
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1947:
1944:
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1878:
1872:
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1190:
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1048:
1026:
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989:
756:In particular, the Chow group
737:
734:
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683:
680:
674:
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567:
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429:
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162:
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2549:American Mathematical Society
2505:American Mathematical Society
2471:Mathematical Research Letters
2415:
2254:) in the expected dimension.
1685:); the two are isomorphic if
1275:) from motivic cohomology to
905:universal coefficient theorem
2641:10.4007/annals.2011.174.1.11
2446:10.1016/0001-8708(86)90081-2
243:of finite type over a field
239:). Namely, for every scheme
7:
2686:
2545:Clay Mathematics Monographs
2339:coefficients, Theorem 6.17.
2078:) with motivic cohomology.
1635:derived category of motives
1089:Relation to Milnor K-theory
936:from motivic cohomology to
255:, we have an abelian group
10:
2812:
2582:Princeton University Press
2149:
1675:compactly supported motive
1593:is an isomorphism for all
862:(0) is the constant sheaf
2484:10.4310/MRL.1999.v6.n1.a5
2220:(1973), generalizing the
1601:and is injective for all
1097:determines a scheme Spec(
932:, and Levine, there is a
870:(1) is isomorphic in the
229:(which were first called
2264:
940:for every smooth scheme
768:cycles is isomorphic to
2693:Presheaf with transfers
2432:Advances in Mathematics
2146:on motivic cohomology.
1625:, Voevodsky defined an
1251:Map to Ă©tale cohomology
795:The motivic cohomology
366:For a closed subscheme
2399:Cite journal requires
2235:have now been proved.
1983:When the coefficients
1963:
1815:
1584:
1395:
1209:
1064:
811:)) of a smooth scheme
747:
583:
357:
209:
2619:Annals of Mathematics
2128:values of L-functions
2118:Values of L-functions
1964:
1816:
1641:with coefficients in
1631:triangulated category
1621:and commutative ring
1585:
1396:
1210:
1065:
748:
584:
358:
210:
107:localization sequence
85:. The Chow groups of
2584:, pp. 188â238,
2507:, pp. 245â302,
2503:, Providence, R.I.:
2178:improve this article
2054:) as the heart of a
2044:standard conjectures
1976:of finite type over
1847:
1731:
1478:
1286:
1136:
955:
916:Relation to K-theory
894:multiplicative group
647:
381:
283:
115:
105:of Chow groups, the
91:BorelâMoore homology
26:and of more general
2791:Homotopical algebra
2786:Cohomology theories
2730:2011arXiv1105.0865H
2607:Voevodsky, Vladimir
2574:Voevodsky, Vladimir
2531:Voevodsky, Vladimir
2003:), the category of
1783:
1756:
1720:) for all integers
1613:Relation to motives
1545:
1445:singular cohomology
1353:
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975:
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220:long exact sequence
24:algebraic varieties
22:is an invariant of
2698:AÂč homotopy theory
2222:Grothendieck group
2218:algebraic K-theory
2105:-vector spaces is
1959:
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938:algebraic K-theory
907:, as in topology.
848:Nisnevich topology
743:
598:motivic cohomology
579:
353:
232:higher Chow groups
205:
44:algebraic geometry
30:. It is a type of
20:Motivic cohomology
2769:WiesĆawa NizioĆ,
2666:(July 12, 2022).
2558:978-0-8218-3847-1
2514:978-0-8218-1637-0
2210:
2209:
2202:
1972:for every scheme
1891:
934:spectral sequence
898:regular functions
764:) of codimension-
616:for every scheme
38:and includes the
2803:
2760:
2755:, archived from
2752:Lectures at TIFR
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2744:
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2535:Weibel, Charles
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62:be a scheme of
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2774:
2773:
2767:
2761:
2759:on 22 Sep 2016
2749:Nori, Madhav,
2746:
2736:Levine, Marc,
2733:
2705:
2704:External links
2702:
2701:
2700:
2695:
2688:
2685:
2684:
2683:
2660:
2603:
2590:
2570:
2557:
2529:Mazza, Carlo;
2526:
2513:
2496:
2466:
2439:(3): 267â304,
2423:Bloch, Spencer
2417:
2414:
2411:
2410:
2401:|journal=
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2144:height pairing
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1301:
1296:
1292:
1252:
1249:
1237:Milnor K-group
1222:
1216:
1215:
1204:
1201:
1198:
1195:
1192:
1188:
1184:
1181:
1178:
1173:
1169:
1165:
1162:
1159:
1156:
1151:
1146:
1142:
1117:)) for fields
1090:
1087:
1083:vector bundles
1081:, rather than
1071:
1070:
1059:
1056:
1053:
1050:
1045:
1042:
1039:
1036:
1032:
1028:
1025:
1022:
1019:
1015:
1011:
1008:
1005:
1001:
997:
994:
991:
986:
982:
978:
973:
970:
965:
961:
917:
914:
912:
909:
889:
882:
754:
753:
742:
739:
736:
733:
730:
727:
724:
720:
716:
713:
710:
705:
702:
699:
696:
692:
688:
685:
682:
679:
676:
672:
668:
665:
662:
657:
653:
590:
589:
578:
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572:
569:
566:
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531:
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521:
518:
515:
511:
507:
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501:
496:
493:
489:
485:
482:
479:
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473:
469:
465:
462:
459:
454:
451:
447:
443:
440:
437:
434:
431:
427:
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420:
417:
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406:
403:
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389:
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364:
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349:
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318:
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288:
259:
216:
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204:
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198:
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186:
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170:
167:
164:
161:
158:
153:
149:
145:
142:
139:
136:
133:
128:
124:
120:
103:exact sequence
101:, there is an
55:
52:
15:
9:
6:
4:
3:
2:
2808:
2797:
2794:
2792:
2789:
2787:
2784:
2783:
2781:
2772:
2768:
2766:
2762:
2758:
2754:
2753:
2747:
2741:
2740:
2734:
2731:
2727:
2722:
2717:
2713:
2708:
2707:
2699:
2696:
2694:
2691:
2690:
2680:
2676:
2669:
2665:
2661:
2658:
2654:
2650:
2646:
2642:
2638:
2633:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2601:
2597:
2593:
2591:9781400837120
2587:
2583:
2579:
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2568:
2564:
2560:
2554:
2550:
2546:
2542:
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2536:
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2524:
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2516:
2510:
2506:
2502:
2497:
2494:
2490:
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2480:
2476:
2472:
2467:
2464:
2460:
2456:
2452:
2447:
2442:
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2434:
2433:
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2424:
2420:
2419:
2406:
2393:
2384:
2379:
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2363:
2354:
2345:
2338:
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2328:
2319:
2310:
2301:
2292:
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2274:
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2262:
2259:
2255:
2253:
2249:
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2236:
2233:
2226:
2223:
2219:
2215:
2204:
2201:
2193:
2183:
2179:
2173:
2172:
2167:This section
2165:
2161:
2156:
2155:
2147:
2145:
2141:
2137:
2133:
2129:
2125:
2110:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2079:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2040:
2038:
2034:
2030:
2026:
2022:
2020:
2014:
2010:
2006:
2005:mixed motives
2002:
1998:
1994:
1990:
1986:
1981:
1979:
1975:
1950:
1941:
1935:
1932:
1926:
1920:
1909:
1906:
1903:
1897:
1894:
1884:
1875:
1869:
1866:
1863:
1855:
1851:
1843:
1842:
1841:
1839:
1835:
1831:
1827:
1808:
1802:
1799:
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1784:
1779:
1774:
1771:
1768:
1764:
1760:
1752:
1747:
1734:
1727:
1726:
1725:
1723:
1719:
1715:
1711:
1707:
1704:
1698:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1610:
1608:
1604:
1600:
1596:
1571:
1565:
1561:
1552:
1549:
1541:
1536:
1533:
1529:
1516:
1510:
1506:
1497:
1494:
1486:
1482:
1474:
1473:
1472:
1470:
1466:
1462:
1459:over a field
1458:
1454:
1450:
1446:
1442:
1437:
1435:
1431:
1426:
1420:
1415:
1411:
1407:
1388:
1379:
1373:
1369:
1360:
1357:
1349:
1344:
1341:
1337:
1324:
1318:
1314:
1305:
1302:
1294:
1290:
1282:
1281:
1280:
1278:
1274:
1270:
1266:
1262:
1258:
1248:
1246:
1242:
1238:
1234:
1230:
1225:
1221:
1202:
1193:
1182:
1179:
1171:
1167:
1163:
1157:
1149:
1144:
1140:
1132:
1131:
1130:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1086:
1084:
1080:
1076:
1057:
1051:
1043:
1040:
1037:
1034:
1030:
1017:
1013:
1009:
1006:
995:
992:
984:
980:
976:
971:
968:
963:
959:
951:
950:
949:
948:in topology:
947:
943:
939:
935:
931:
927:
923:
908:
906:
901:
899:
895:
888:
881:
877:
873:
869:
865:
861:
857:
853:
849:
845:
841:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
793:
791:
787:
783:
779:
775:
771:
767:
763:
759:
740:
731:
728:
725:
714:
711:
703:
700:
697:
694:
690:
686:
677:
666:
663:
655:
651:
643:
642:
641:
639:
636:, there is a
635:
631:
628:of dimension
627:
623:
619:
615:
611:
607:
603:
599:
595:
576:
564:
553:
550:
547:
544:
536:
533:
529:
516:
505:
502:
494:
491:
487:
474:
463:
460:
452:
449:
445:
432:
421:
418:
415:
412:
404:
401:
398:
395:
391:
384:
377:
376:
375:
373:
369:
350:
341:
330:
327:
319:
316:
312:
308:
302:
294:
290:
286:
279:
278:
277:
275:
271:
267:
262:
258:
254:
250:
247:and integers
246:
242:
238:
234:
233:
228:
223:
221:
202:
199:
190:
187:
184:
176:
172:
168:
159:
151:
147:
143:
134:
126:
122:
118:
111:
110:
109:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
69:
65:
61:
51:
49:
48:number theory
45:
41:
37:
33:
29:
25:
21:
2757:the original
2751:
2738:
2711:
2674:
2664:Levine, Marc
2622:
2618:
2614:
2610:
2577:
2539:
2500:
2474:
2470:
2436:
2430:
2426:
2392:cite journal
2371:
2362:
2353:
2344:
2336:
2332:
2327:
2318:
2309:
2300:
2291:
2282:
2273:
2260:
2256:
2247:
2243:
2239:
2237:
2224:
2211:
2196:
2190:January 2021
2187:
2176:Please help
2171:verification
2168:
2131:
2123:
2121:
2102:
2098:
2094:
2090:
2086:
2082:
2080:
2075:
2071:
2067:
2063:
2059:
2051:
2047:
2041:
2036:
2032:
2028:
2024:
2016:
2008:
2004:
2000:
1992:
1984:
1982:
1977:
1973:
1971:
1837:
1833:
1829:
1825:
1823:
1721:
1717:
1713:
1709:
1705:
1703:Tate motives
1702:
1699:
1694:
1686:
1682:
1678:
1674:
1670:
1666:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1626:
1622:
1618:
1616:
1606:
1602:
1598:
1594:
1592:
1468:
1464:
1460:
1456:
1448:
1440:
1438:
1429:
1424:
1418:
1413:
1409:
1405:
1403:
1272:
1268:
1264:
1260:
1256:
1254:
1244:
1240:
1232:
1228:
1223:
1219:
1217:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1092:
1072:
941:
919:
902:
886:
879:
875:
867:
863:
859:
855:
851:
843:
839:
824:
816:
812:
808:
804:
800:
796:
794:
789:
785:
781:
777:
773:
769:
765:
761:
757:
755:
640:isomorphism
633:
629:
621:
617:
609:
605:
601:
597:
591:
371:
370:of a scheme
367:
365:
273:
269:
265:
260:
256:
252:
248:
244:
240:
230:
226:
224:
217:
106:
98:
94:
86:
82:
78:
70:
59:
57:
19:
18:
2675:youtube.com
2625:: 401â438,
2056:t-structure
1673:), and the
1633:called the
926:Friedlander
922:Lichtenbaum
75:Chow groups
64:finite type
34:related to
2780:Categories
2429:-theory",
2416:References
2383:2309.08463
2140:regulators
2021:conjecture
2017:Beilinson-
2013:Ext groups
1428:being the
1263:, and let
920:By Bloch,
821:cohomology
32:cohomology
2721:1105.0865
2632:0805.4430
2477:: 61â82,
2455:0001-8708
1989:Beilinson
1885:≅
1765:⊕
1761:≅
1526:→
1434:cycle map
1422:), with Ό
1334:→
1273:cycle map
1231:) is the
1164:≅
1075:tensoring
1041:−
1035:−
1027:⇒
1007:−
729:−
701:−
687:≅
594:Voevodsky
574:→
548:−
526:→
484:→
442:→
416:−
388:→
385:⋯
309:≅
197:→
188:−
166:→
141:→
40:Chow ring
2687:See also
2657:15583705
2537:(2006),
2107:faithful
1712:) in DM(
1629:-linear
858:< 0,
784:)) when
2726:Bibcode
2671:(video)
2649:2811603
2600:1764202
2567:2242284
2523:1265533
2501:Motives
2493:1682709
2463:0852815
2252:simplex
2214:Quillen
2150:History
885:. Here
842:(j) on
837:sheaves
833:complex
827:in the
819:is the
600:groups
66:over a
36:motives
28:schemes
2655:
2647:
2598:
2588:
2565:
2555:
2521:
2511:
2491:
2461:
2453:
2142:and a
1824:where
1691:proper
1663:motive
1404:where
1218:where
930:Suslin
866:, and
626:smooth
2743:(PDF)
2716:arXiv
2653:S2CID
2627:arXiv
2378:arXiv
2265:Notes
2101:) to
2023:that
2019:Soulé
2007:over
1693:over
1657:over
1645:, DM(
1637:over
1609:â 1.
892:(the
815:over
632:over
237:Bloch
68:field
2586:ISBN
2553:ISBN
2509:ISBN
2451:ISSN
2405:help
2122:Let
2081:For
1681:, M(
1669:, M(
1463:and
1255:Let
614:ring
251:and
58:Let
46:and
2637:doi
2623:174
2479:doi
2441:doi
2180:by
2058:on
2027:(X,
1999:MM(
1890:Hom
1689:is
1677:of
1665:of
1239:of
1235:th
1085:).
878:to
874:of
835:of
823:of
624:is
604:(X,
235:by
97:of
77:of
2782::
2724:,
2714:,
2677:.
2673:.
2651:,
2645:MR
2643:,
2635:,
2621:,
2596:MR
2594:,
2580:,
2563:MR
2561:,
2551:,
2543:,
2533:;
2519:MR
2517:,
2489:MR
2487:,
2473:,
2459:MR
2457:,
2449:,
2437:61
2435:,
2396::
2394:}}
2390:{{
2095:MM
2087:MM
2072:MM
2066:;
2060:DM
2048:MM
1980:.
1836:;
1828:âŠ
1716:;
1697:.
1649:;
1605:â„
1597:â„
1279::
1247:.
1129::
1125:=
1109:,
928:,
924:,
803:,
792:.
758:CH
577:0.
222:.
2728::
2718::
2681:.
2639::
2629::
2615:l
2613:/
2611:Z
2481::
2475:6
2443::
2427:K
2407:)
2403:(
2386:.
2380::
2337:l
2335:/
2333:Z
2248:X
2244:X
2240:k
2228:0
2225:K
2203:)
2197:(
2192:)
2188:(
2174:.
2132:X
2124:X
2103:Q
2099:k
2097:(
2091:k
2089:(
2083:k
2076:k
2074:(
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2064:k
2062:(
2052:k
2050:(
2037:i
2033:j
2031:(
2029:Q
2025:H
2009:k
2001:k
1993:Q
1985:R
1978:k
1974:X
1957:)
1954:]
1951:i
1948:[
1945:)
1942:j
1939:(
1936:R
1933:,
1930:)
1927:X
1924:(
1921:M
1918:(
1913:)
1910:R
1907:;
1904:k
1901:(
1898:M
1895:D
1882:)
1879:)
1876:j
1873:(
1870:R
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1864:X
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1806:]
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1800:2
1797:[
1794:)
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