527:
7290:, implies the equivalence of pure motives with respect to homological and numerical equivalence. (In particular the former category of motives would not depend on the choice of the Weil cohomology theory). Jannsen (1992) proved the following unconditional result: the category of (pure) motives over a field is abelian and semisimple if and only if the chosen equivalence relation is numerical equivalence.
8295:
2711:
3874:
3051:
310:
2413:
6307:
4611:
will guarantee that every pair of cycles has an equivalent pair in general position that we can intersect. The Chow groups are defined using rational equivalence, but other equivalences are possible, and each defines a different sort of motive. Examples of equivalences, from strongest to weakest, are
796:
4912:
There are different definitions due to
Hanamura, Levine and Voevodsky. They are known to be equivalent in most cases and we will give Voevodsky's definition below. The category contains Chow motives as a full subcategory and gives the "right" motivic cohomology. However, Voevodsky also shows that
1065:
From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than
2129:
1875:
3483:
2570:
4082:
3628:
2867:
522:{\displaystyle \left(M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p},\operatorname {comp} _{\mathrm {DR} ,B},\operatorname {comp} _{\mathbb {A} ^{f},B},\operatorname {comp} _{\operatorname {cris} p,\mathrm {DR} },W,F_{\infty },F,\phi ,\phi _{p}\right)}
4555:
5219:
whose objects are smooth varieties and morphisms are given by smooth correspondences. The only non-trivial part of this "definition" is the fact that we need to describe compositions. These are given by a push-pull formula from the theory of Chow rings.
2833:
4231:
2251:
7509:
6496:
preserve this structure. Then one may ask when two given objects are isomorphic, and ask for a "particularly nice" representative in each isomorphism class. The classification of algebraic varieties, i.e. application of this idea in the case of
3331:
7057:
2542:
6127:
1540:
7196:
The standard conjectures are commonly considered to be very hard and are open in the general case. Grothendieck, with
Bombieri, showed the depth of the motivic approach by producing a conditional (very short and elegant) proof of the
5584:
4317:
622:
705:
6614:
theory will have such properties. There are different Weil cohomology theories, they apply in different situations and have values in different categories, and reflect different structural aspects of the variety in question:
7719:(the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions). The motivic Galois group has the surrounding representation theory. (What it is not, is a
4400:
2030:
3196:
874:
694:
1728:
3342:
2706:{\displaystyle \operatorname {Ob} \left(\operatorname {Chow} ^{\operatorname {eff} }(k)\right):=\{(X,\alpha )\mid (\alpha :X\vdash X)\in \operatorname {Corr} (k){\mbox{ such that }}\alpha \circ \alpha =\alpha \}.}
6413:, depending on the coefficients used in the construction of the category of Motives. These are fundamental building blocks in the category of motives because they form the "other part" besides Abelian varieties.
3869:{\displaystyle f_{1}\otimes f_{2}:(X_{1},\alpha _{1})\otimes (X_{2},\alpha _{2})\vdash (Y_{1},\beta _{1})\otimes (Y_{2},\beta _{2}),\qquad f_{1}\otimes f_{2}:=\pi _{1}^{*}\gamma _{1}\cdot \pi _{2}^{*}\gamma _{2}}
3963:
3046:{\displaystyle h:{\begin{cases}\operatorname {SmProj} (k)&\longrightarrow \operatorname {Chow^{eff}} (k)\\X&\longmapsto :=(X,\Delta _{X})\\f&\longmapsto :=\Gamma _{f}\subset X\times Y\end{cases}}}
6760:
5648:
1379:
5889:
7410:
1720:
1666:
7172:
6825:
5839:
6528:
is an attempt to find a universal way to linearize algebraic varieties, i.e. motives are supposed to provide a cohomology theory that embodies all these particular cohomologies. For example, the
4802:
coincides with the one predicted by algebraic K-theory, and contains the category of Chow motives in a suitable sense (and other properties). The existence of such a category was conjectured by
6458:
4797:
4738:
4411:
1990:
5265:
4968:
2167:
996:
7678:-vector spaces. It can be shown that the former category is a Tannakian category. Assuming the equivalence of homological and numerical equivalence, i.e. the above standard conjecture
5360:
1916:
7193:
were first formulated in terms of the interplay of algebraic cycles and Weil cohomology theories. The category of pure motives provides a categorical framework for these conjectures.
4593:
2461:
2199:
1948:
1285:
1160:
1112:
4154:
933:
6079:
2717:
2408:{\displaystyle \alpha +\beta :=(\alpha ,\beta )\in A^{*}(X\times X)\oplus A^{*}(Y\times Y)\hookrightarrow A^{*}\left(\left(X\coprod Y\right)\times \left(X\coprod Y\right)\right).}
5217:
5103:
5006:
4165:
8275:
5938:
5045:
1615:
1410:
7137:
5677:
1058:
can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of
7099:
6382:
6351:
6042:
7426:
8320:
7676:
7616:
7594:
7345:
6851:
6650:
6501:, is very difficult due to the highly non-linear structure of the objects. The relaxed question of studying varieties up to birational isomorphism has led to the field of
4869:
3211:
1018:
172:
6924:
5177:
5740:
1250:
1220:
963:
5297:
2022:
1579:
6302:{\displaystyle \operatorname {Hom} _{\mathcal {DM}}((A,n),(B,m))=\lim _{k\geq -n,-m}\operatorname {Hom} _{{\mathcal {DM}}_{gm}^{\operatorname {eff} }}(A(k+n),B(k+m))}
2469:
268:
230:
120:
1194:
6111:
4685:
894:
6411:
6004:
5971:
5446:
1430:
825:
294:
7323:
6478:
140:
5386:
4256:
791:{\displaystyle \operatorname {comp} _{\mathrm {DR} ,B},\operatorname {comp} _{\mathbb {A} ^{f},B},\operatorname {comp} _{\operatorname {cris} p,\mathrm {DR} }}
4743:
taking values on all varieties (not just smooth projective ones as it was the case with pure motives). This should be such that motivic cohomology defined by
541:
6864:
is an attempt to find a universal theory which embodies all these particular cohomologies and their structures and provides a framework for "equations" like
8197:
4632:
The literature occasionally calls every type of pure motive a Chow motive, in which case a motive with respect to algebraic equivalence would be called a
2124:{\displaystyle F:{\begin{cases}\operatorname {SmProj} (k)\longrightarrow \operatorname {Corr} (k)\\X\longmapsto X\\f\longmapsto \Gamma _{f}\end{cases}}}
5457:
4329:
8299:
7287:
7190:
3092:
830:
1870:{\displaystyle \beta \circ \alpha :=\pi _{XZ*}\left(\pi _{XY}^{*}(\alpha )\cdot \pi _{YZ}^{*}(\beta )\right)\in \operatorname {Corr} ^{r+s}(X,Z),}
636:
3478:{\displaystyle \pi _{X}:(X\times Y)\times (X\times Y)\to X\times X,\quad {\text{and}}\quad \pi _{Y}:(X\times Y)\times (X\times Y)\to Y\times Y.}
8186:
8092:
8071:
8045:
8017:
7968:
7938:
7861:
4077:{\displaystyle L:=(\mathbb {P} ^{1},\lambda ),\qquad \lambda :=pt\times \mathbb {P} ^{1}\in A^{1}(\mathbb {P} ^{1}\times \mathbb {P} ^{1})}
6547:. So, the motive of the curve should contain the genus information. Of course, the genus is a rather coarse invariant, so the motive of
6424:
1167:
175:
4607:
In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing a suitable
1020:-variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained in a motive.
1028:
The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including
6044:
Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives
6689:
6524:
There are several important cohomology theories, which reflect different structural aspects of varieties. The (partly conjectural)
5596:
1062:
where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.
1305:
5847:
7570:
this functor is shown to be an equivalence of categories. Notice that fields are 0-dimensional. Motives of this kind are called
7353:
1671:
1620:
7142:
6773:
6322:
There are several elementary examples of motives which are readily accessible. One of them being the Tate motives, denoted
5756:
7596:-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite
5392:
this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms
4608:
1067:
8325:
7523:), which takes values in semi-simple representations. (The latter part is automatic in the case of the Hodge analogue).
6670:
4906:
4749:
4550:{\displaystyle f:(X,p,m)\to (Y,q,n),\quad f\in \operatorname {Corr} ^{n-m}(X,Y){\mbox{ such that }}f\circ p=f=q\circ f,}
5680:
8315:
8037:
4693:
4653:
7625:
is to extend the above equivalence to higher-dimensional varieties. In order to do this, the technical machinery of
1956:
5389:
7630:
5231:
8220:
6488:
A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a
4937:
2137:
968:
6539:
which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first
5313:
1886:
7963:, Proceedings of Symposia in Pure Mathematics, vol. 55, Providence, R.I.: American Mathematical Society,
4566:
2434:
2172:
1921:
1258:
1133:
1085:
2828:{\displaystyle \operatorname {Mor} ((X,\alpha ),(Y,\beta )):=\{f:X\vdash Y|f\circ \alpha =f=\beta \circ f\}.}
7768:
7758:
6683:
6596:
3942:
3083:
31:
7708:
6610:
has the same structure in any reasonable cohomology theory with good formal properties; in particular, any
4226:{\displaystyle \mathbf {1} \cong \left(\mathbb {P} ^{1},\mathbb {P} ^{1}\times \operatorname {pt} \right).}
4110:
899:
6676:
6489:
6047:
1079:
1044:
936:
187:
74:
54:
7728:
7178:
6653:
5590:
5186:
5076:
4973:
4886:
8281:
8241:
5894:
5015:
1588:
1387:
7504:{\displaystyle H:M(k)_{\mathbb {Q} _{\ell }}\to \operatorname {Rep} _{\ell }(\operatorname {Gal} (k))}
7108:
5653:
3950:
8125:
7806:
7694:
7512:
7073:
6356:
6325:
3326:{\displaystyle (,\alpha )\otimes (,\beta ):=(X\times Y,\pi _{X}^{*}\alpha \cdot \pi _{Y}^{*}\beta ),}
2242:
533:
6009:
2882:
2045:
7786:
7732:
7276:
comes from a similar decomposition of, say, de-Rham cohomology of smooth projective varieties, see
7052:{\displaystyle H^{n}(X,m):=H^{n}(X,\mathbb {Z} (m)):=\operatorname {Hom} _{DM}(X,\mathbb {Z} (m)),}
6659:
6600:
6585:
6529:
6502:
6498:
4803:
1037:
62:
7659:
7599:
7577:
7328:
6834:
6633:
4826:
1918:
notice that the composition of degree 0 correspondences is degree 0. Hence we define morphisms of
1001:
8212:
8168:
8141:
8027:
7909:
7888:
7796:
7763:
7753:
7626:
6915:
6910:
6770:) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology
6577:
4898:
1033:
628:
145:
66:
38:
5679:-homotopies of varieties while the second will give the category of geometric mixed motives the
5140:
2537:{\displaystyle \operatorname {Chow} ^{\operatorname {eff} }(k):=Split(\operatorname {Corr} (k))}
6901:
Beginning with
Grothendieck, people have tried to precisely define this theory for many years.
5689:
2838:
Composition is the above defined composition of correspondences, and the identity morphism of (
1229:
1199:
941:
8182:
8088:
8067:
8041:
8013:
7964:
7956:
7934:
7872:
7840:
7547:→ non-empty finite sets with a (continuous) transitive action of the absolute Galois group of
6421:
The motive of a curve can be explicitly understood with relative ease: their Chow ring is just
5308:
5270:
5063:
3202:
2220:
1995:
1548:
186:, however, such a triple contains almost no information outside the context of Grothendieck's
78:
58:
235:
197:
87:
8133:
7716:
7634:
7423:
is equivalent to: the so-called Tate realization, i.e. ℓ-adic cohomology, is a full functor
7294:
7198:
6581:
6540:
4818:
2428:
1535:{\displaystyle \operatorname {Corr} ^{r}(k)(X,Y):=\bigoplus _{i}A^{d_{i}+r}(X_{i}\times Y),}
1173:
1029:
70:
8055:
7978:
7948:
7884:
7852:
6084:
5686:
Also, note that this category has a tensor structure given by the product of varieties, so
4658:
879:
8051:
7974:
7944:
7880:
7848:
7748:
7724:
7698:
7686:
is an exact faithful tensor-functor. Applying the
Tannakian formalism, one concludes that
7642:
7638:
7420:
7413:
6387:
6121:
an integer representing the twist by the Tate motive. The hom-groups are then the colimit
5980:
5947:
5398:
5070:
1082:
of pure motives often proceeds in three steps. Below we describe the case of Chow motives
50:
4312:{\displaystyle \operatorname {Chow} (k):=\operatorname {Chow} ^{\operatorname {eff} }(k)}
1255:
It will be useful to describe correspondences of arbitrary degree, although morphisms in
804:
273:
84:
In the formulation of
Grothendieck for smooth projective varieties, a motive is a triple
8129:
7810:
617:{\displaystyle M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p}}
8110:
8082:
8031:
7308:
6620:
6510:
6463:
5056:
4596:
297:
125:
5365:
5362:
of bounded complexes of smooth correspondences. Here smooth varieties will be denoted
8309:
8172:
8154:
Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970)
8145:
7567:
8216:
8181:, Annals of Mathematics Studies, Princeton, New Jersey: Princeton University Press,
1066:
rational equivalence. The admissible equivalences are given by the definition of an
17:
7901:
7720:
7712:
7633:, but a purely algebraic theory) is used. Its purpose is to shed light on both the
7516:
7348:
7277:
6666:
6592:
6514:
4902:
3953:. The effect is that motives become triples instead of pairs. The Lefschetz motive
7817:
7825:
5579:{\displaystyle {\xrightarrow {j_{U}'+j_{V}'}}\oplus {\xrightarrow {j_{U}-j_{V}}}}
7736:
6767:
4238:
2548:
In other words, effective Chow motives are pairs of smooth projective varieties
7933:, Panoramas et Synthèses, vol. 17, Paris: Société Mathématique de France,
7416:). Here pure motive means pure motive with respect to homological equivalence.
4395:{\displaystyle (X\in \operatorname {SmProj} (k),p:X\vdash X,n\in \mathbb {Z} )}
3949:) a formal inverse (with respect to the tensor product) of a motive called the
6876:
directly gives all the information about the several Weil cohomology theories
6509:
an object of more linear nature, i.e. an object amenable to the techniques of
3191:{\displaystyle (,\alpha )\oplus (,\beta ):=\left(\left,\alpha +\beta \right),}
1059:
7876:
7844:
7715:. Again speaking in rough terms, the Hodge and Tate conjectures are types of
869:{\displaystyle \operatorname {Gal} ({\overline {\mathbb {Q} }},\mathbb {Q} )}
304:. In that article, a motive is a "system of realisations" – that is, a tuple
8294:
8162:
4560:
and the composition of morphisms comes from composition of correspondences.
1223:
689:{\displaystyle \mathbb {Q} ,\mathbb {Q} ,\mathbb {A} ^{f},\mathbb {Q} _{p},}
8237:
8176:
8066:. Mathematical surveys and monographs, 57. American Mathematical Society.
7790:
6914:
itself had been invented before the creation of mixed motives by means of
6682:
All these cohomology theories share common properties, e.g. existence of
6624:
6493:
1880:
where the dot denotes the product in the Chow ring (i.e., intersection).
191:
8137:
7202:
4893:
The current state of the theory is that we do have a suitable category
4810:
183:
7735:, it predicts the image of the Galois group, or, more accurately, its
6563:
has a corresponding motive , so the simplest examples of motives are:
4913:(with integral coefficients) it does not admit a motivic t-structure.
7801:
6505:. Another way to handle the question is to attach to a given variety
4622:
Smash-nilpotence equivalence (sometimes called
Voevodsky equivalence)
7305:
mapping any pure motive with rational coefficients (over a subfield
5815:
5540:
5481:
7931:
Une introduction aux motifs (motifs purs, motifs mixtes, périodes)
7914:
6619:
Betti cohomology is defined for varieties over (subfields of) the
6533:
1585:. Correspondences are often denoted using the "⊢"-notation, e.g.,
998:. This data is modeled on the cohomologies of a smooth projective
7656:(pure motives using numerical equivalence) to finite-dimensional
8040:, vol. 2, Providence, R.I.: American Mathematical Society,
7531:
To motivate the (conjectural) motivic Galois group, fix a field
302:
Le Groupe
Fondamental de la Droite Projective Moins Trois Points
1295:
be smooth projective varieties and consider a decomposition of
801:
between the obvious base changes of these modules, filtrations
7707:
The motivic Galois group is to the theory of motives what the
7298:
6755:{\displaystyle H^{*}(X)\cong H^{*}(X\times \mathbb {A} ^{1}),}
5643:{\displaystyle {\mathcal {DM}}_{\text{gm}}^{\text{eff}}(k,A).}
5228:
Typical examples of prime correspondences come from the graph
77:. Philosophically, a "motif" is the "cohomology essence" of a
6918:. The above category provides a neat way to (re)define it by
5750:
Using the triangulated structure we can construct a triangle
1374:{\displaystyle X=\coprod _{i}X_{i},\qquad d_{i}:=\dim X_{i}.}
8192:(Voevodsky's definition of mixed motives. Highly technical).
7902:"A guided tour through the garden of noncommutative motives"
7618:-vector spaces together with an action of the Galois group.
6230:
6227:
6140:
6137:
6057:
6054:
5884:{\displaystyle \mathbb {P} ^{1}\to \operatorname {Spec} (k)}
5606:
5603:
5344:
5341:
5338:
5335:
5332:
5204:
5201:
5198:
5195:
5192:
5024:
5021:
4985:
4982:
4979:
4809:
Instead of constructing such a category, it was proposed by
4236:
The tensor inverse of the
Lefschetz motive is known as the
3039:
2419:
Second step: category of pure effective Chow motives, Chow(
2117:
7405:{\displaystyle H:M(k)_{\mathbb {Q} }\to HS_{\mathbb {Q} }}
5125:. Then, we can take the set of prime correspondences from
57:
in the 1960s to unify the vast array of similarly behaved
8152:
Kleiman, Steven L. (1972), "Motives", in Oort, F. (ed.),
4625:
Homological equivalence (in the sense of Weil cohomology)
3488:
The tensor product of morphisms may also be defined. Let
2564:, and morphisms are of a certain type of correspondence:
1715:{\displaystyle \beta \in \operatorname {Corr} ^{s}(Y,Z),}
1661:{\displaystyle \alpha \in \operatorname {Corr} ^{r}(X,Y)}
1122:
First step: category of (degree 0) correspondences, Corr(
7814:(technical introduction with comparatively short proofs)
7511:(pure motives up to homological equivalence, continuous
5944:. Taking the iterative tensor product lets us construct
7167:{\displaystyle \mathbb {P} ^{1}\to \operatorname {pt} }
6665:(over any field of characteristic ≠ l) has a canonical
5012:
are separated schemes of finite type. We will also let
6820:{\displaystyle H_{\text{Betti}}^{*}(X,\mathbb {Z} /n)}
6576:
These 'equations' hold in many situations, namely for
6517:. This "linearization" goes usually under the name of
6480:, hence Jacobians embed into the category of motives.
5650:
Note that the first class of morphisms are localizing
4511:
4250:. Then we define the category of pure Chow motives by
2676:
8244:
7662:
7602:
7580:
7429:
7356:
7331:
7311:
7297:, may be neatly reformulated using motives: it holds
7145:
7111:
7076:
6927:
6872:
In particular, calculating the motive of any variety
6837:
6776:
6692:
6636:
6466:
6427:
6390:
6359:
6328:
6130:
6087:
6050:
6012:
5983:
5950:
5897:
5850:
5834:{\displaystyle \mathbb {L} \to \to {\xrightarrow {}}}
5759:
5692:
5656:
5599:
5460:
5401:
5368:
5316:
5273:
5234:
5189:
5143:
5079:
5018:
4976:
4940:
4829:
4752:
4696:
4661:
4569:
4414:
4332:
4259:
4168:
4113:
3966:
3631:
3345:
3214:
3095:
2870:
2720:
2573:
2472:
2437:
2254:
2175:
2140:
2033:
1998:
1959:
1924:
1889:
1731:
1674:
1623:
1591:
1551:
1433:
1390:
1308:
1261:
1232:
1202:
1176:
1136:
1088:
1004:
971:
944:
902:
882:
833:
807:
708:
639:
544:
313:
276:
238:
200:
148:
128:
90:
8111:"Motives, numerical equivalence and semi-simplicity"
8008:Huber, Annette; Müller-Stach, Stefan (2017-03-20),
5008:as the category of quasi-projective varieties over
4897:. Already this category is useful in applications.
8269:
7670:
7610:
7588:
7503:
7404:
7339:
7317:
7166:
7131:
7093:
7051:
6845:
6819:
6754:
6644:
6472:
6453:{\displaystyle \mathbb {Z} \oplus {\text{Pic}}(C)}
6452:
6405:
6376:
6345:
6301:
6105:
6073:
6036:
5998:
5965:
5932:
5883:
5833:
5734:
5671:
5642:
5578:
5440:
5380:
5354:
5291:
5259:
5211:
5171:
5097:
5039:
5000:
4962:
4863:
4792:{\displaystyle \operatorname {Ext} _{MM}^{*}(1,?)}
4791:
4732:
4679:
4587:
4549:
4394:
4311:
4225:
4148:
4076:
3868:
3477:
3325:
3190:
3086:. The direct sum of effective motives is given by
3045:
2827:
2705:
2536:
2455:
2407:
2193:
2161:
2123:
2016:
1984:
1942:
1910:
1869:
1714:
1660:
1609:
1573:
1534:
1404:
1373:
1279:
1244:
1214:
1188:
1154:
1106:
1012:
990:
957:
927:
888:
868:
819:
790:
688:
616:
521:
288:
262:
224:
166:
134:
114:
8081:Friedlander, Eric M.; Grayson, Daniel R. (2005).
7212:, which states the existence of algebraic cycles
6623:, it has the advantage of being defined over the
4405:such that morphisms are given by correspondences
8178:Cycles, transfers, and motivic homology theories
6192:
4733:{\displaystyle \operatorname {Var} (k)\to MM(k)}
3933:Third step: category of pure Chow motives, Chow(
3071:, is a functor. The motive is often called the
2427:The transition to motives is made by taking the
1287:are correspondences of degree 0. In detail, let
8277:: Arithmetic spin structures on elliptic curves
7896:(high-level introduction to motives in French).
1196:, which can be associated with their graphs in
8156:, Groningen: Wolters-Noordhoff, pp. 53–82
7205:), assuming the standard conjectures to hold.
4883:would then be accomplished by a (conjectural)
1985:{\displaystyle \Gamma _{f}\subseteq X\times Y}
699:respectively, various comparison isomorphisms
1953:The following association is a functor (here
296:. A more object-focused approach is taken by
8:
7833:Notices of the American Mathematical Society
7527:Tannakian formalism and motivic Galois group
7139:which in Voevodsky's setting is the complex
5260:{\displaystyle \Gamma _{f}\subset X\times Y}
2819:
2769:
2697:
2615:
1162:are simply smooth projective varieties over
8158:(adequate equivalence relations on cycles).
7286:, stating the concordance of numerical and
6857:-adic cohomology with finite coefficients.
5973:. If we have an effective geometric motive
4963:{\displaystyle A=\mathbb {Q} ,\mathbb {Z} }
2162:{\displaystyle \operatorname {SmProj} (k),}
991:{\displaystyle M_{\operatorname {cris} ,p}}
6853:with finite coefficients is isomorphic to
6656:, it is a differential-geometric invariant
4817:having the properties one expects for the
1047:. The general hope is that equations like
27:Structure for unifying cohomology theories
8321:Topological methods of algebraic geometry
8256:
8246:
8245:
8243:
7913:
7800:
7664:
7663:
7661:
7604:
7603:
7601:
7582:
7581:
7579:
7471:
7456:
7452:
7451:
7449:
7428:
7396:
7395:
7394:
7378:
7377:
7376:
7355:
7333:
7332:
7330:
7310:
7152:
7148:
7147:
7144:
7113:
7112:
7110:
7078:
7077:
7075:
7021:
7020:
6999:
6976:
6975:
6960:
6932:
6926:
6839:
6838:
6836:
6806:
6802:
6801:
6786:
6781:
6775:
6740:
6736:
6735:
6719:
6697:
6691:
6638:
6637:
6635:
6465:
6436:
6429:
6428:
6426:
6389:
6361:
6360:
6358:
6330:
6329:
6327:
6243:
6235:
6226:
6225:
6223:
6195:
6136:
6135:
6129:
6086:
6062:
6053:
6052:
6049:
6011:
5982:
5949:
5914:
5913:
5896:
5857:
5853:
5852:
5849:
5810:
5777:
5773:
5772:
5761:
5760:
5758:
5691:
5663:
5659:
5658:
5655:
5616:
5611:
5602:
5601:
5598:
5558:
5545:
5535:
5502:
5486:
5476:
5459:
5417:
5413:
5412:
5400:
5367:
5355:{\displaystyle K^{b}({\mathcal {SmCor}})}
5331:
5330:
5321:
5315:
5272:
5239:
5233:
5191:
5190:
5188:
5183:. Then, we can form an additive category
5148:
5142:
5078:
5029:
5020:
5019:
5017:
4990:
4978:
4977:
4975:
4956:
4955:
4948:
4947:
4939:
4834:
4828:
4765:
4757:
4751:
4695:
4660:
4568:
4510:
4480:
4413:
4385:
4384:
4331:
4282:
4258:
4203:
4199:
4198:
4188:
4184:
4183:
4169:
4167:
4135:
4123:
4119:
4118:
4112:
4065:
4061:
4060:
4050:
4046:
4045:
4035:
4022:
4018:
4017:
3982:
3978:
3977:
3965:
3860:
3850:
3845:
3832:
3822:
3817:
3804:
3791:
3774:
3761:
3742:
3729:
3710:
3697:
3678:
3665:
3649:
3636:
3630:
3418:
3408:
3350:
3344:
3308:
3303:
3287:
3282:
3213:
3094:
3018:
2981:
2919:
2905:
2877:
2869:
2787:
2719:
2675:
2589:
2572:
2477:
2471:
2436:
2345:
2317:
2289:
2253:
2174:
2139:
2108:
2040:
2032:
1997:
1964:
1958:
1923:
1911:{\displaystyle \operatorname {Corr} (k),}
1888:
1834:
1807:
1799:
1777:
1769:
1748:
1730:
1685:
1673:
1634:
1622:
1590:
1556:
1550:
1514:
1493:
1488:
1478:
1438:
1432:
1398:
1397:
1389:
1362:
1343:
1329:
1319:
1307:
1260:
1231:
1201:
1175:
1170:. They generalize morphisms of varieties
1135:
1087:
1006:
1005:
1003:
976:
970:
949:
943:
914:
910:
909:
907:
901:
881:
859:
858:
846:
845:
843:
832:
806:
778:
765:
744:
740:
739:
737:
714:
713:
707:
677:
673:
672:
662:
658:
657:
649:
648:
641:
640:
638:
602:
587:
583:
582:
580:
563:
562:
549:
543:
508:
483:
460:
447:
426:
422:
421:
419:
396:
395:
376:
361:
357:
356:
354:
337:
336:
323:
312:
275:
237:
199:
147:
127:
89:
7201:(which are proven by different means by
6117:an effective geometric mixed motive and
5047:be the subcategory of smooth varieties.
4909:uses these motives as a key ingredient.
4687:, together with a contravariant functor
4634:Chow motive modulo algebraic equivalence
4588:{\displaystyle \operatorname {Chow} (k)}
2456:{\displaystyle \operatorname {Corr} (k)}
2194:{\displaystyle \operatorname {Corr} (k)}
1943:{\displaystyle \operatorname {Corr} (k)}
1280:{\displaystyle \operatorname {Corr} (k)}
1155:{\displaystyle \operatorname {Corr} (k)}
1107:{\displaystyle \operatorname {Chow} (k)}
8002:. (detailed exposition of Chow motives)
7871:(in French) (198): 11, 333–349 (1992),
6630:de Rham cohomology (for varieties over
1883:Returning to constructing the category
1581:denotes the Chow-cycles of codimension
270:is given by a correspondence of degree
7645:theory. Fix a Weil cohomology theory
7256:decomposes in graded pieces of weight
7704:, known as the motivic Galois group.
7558:to the (finite) set of embeddings of
6555:The search for a universal cohomology
5051:Smooth varieties with correspondences
4149:{\displaystyle =\mathbf {1} \oplus L}
2245:. The sum of morphisms is defined by
1412:, then the correspondences of degree
928:{\displaystyle M_{\mathbb {A} ^{f}},}
7:
8198:"Realization of Voevodsky's motives"
7105:-th tensor power of the Tate object
6074:{\displaystyle {\mathcal {DM}}_{gm}}
3558:) be morphisms of motives. Then let
8033:Lecture notes on motivic cohomology
5109:and surjective over a component of
3205:of effective motives is defined by
7224:inducing the canonical projectors
5236:
5212:{\displaystyle {\mathcal {SmCor}}}
5098:{\displaystyle W\subset X\times Y}
5001:{\displaystyle {\mathcal {Var}}/k}
3063: := denotes the diagonal of
3015:
2978:
2926:
2923:
2920:
2916:
2912:
2909:
2906:
2105:
1961:
782:
779:
718:
715:
567:
564:
484:
464:
461:
400:
397:
341:
338:
25:
8270:{\displaystyle \mathbb {Q} (1/4)}
7792:A DG guide to Voevodsky's motives
7693:is equivalent to the category of
7252:) implies that every pure motive
5933:{\displaystyle A(1)=\mathbb {L} }
5040:{\displaystyle {\mathcal {Sm}}/k}
1610:{\displaystyle \alpha :X\vdash Y}
1405:{\displaystyle r\in \mathbb {Z} }
8293:
7132:{\displaystyle \mathbb {Z} (1),}
6591:, the number of points over any
6460:for any smooth projective curve
5672:{\displaystyle \mathbb {A} ^{1}}
5303:Localizing the homotopy category
4170:
4136:
3073:motive associated to the variety
1950:to be degree 0 correspondences.
1722:their composition is defined by
142:is a smooth projective variety,
8175:; Friedlander, Eric M. (2000),
7955:Jannsen, Uwe; Kleiman, Steven;
7641:, the outstanding questions in
7094:{\displaystyle \mathbb {Z} (m)}
6551:is more than just this number.
6484:Explanation for non-specialists
6377:{\displaystyle \mathbb {Z} (n)}
6346:{\displaystyle \mathbb {Q} (n)}
5593:of effective geometric motives
4469:
4000:
3786:
3413:
3407:
1338:
53:usage) is a theory proposed by
8282:What are "Fractional Motives"?
8264:
8250:
8163:Motives — Grothendieck’s Dream
7498:
7495:
7489:
7480:
7464:
7446:
7439:
7384:
7373:
7366:
7347:) to its Hodge structure is a
7185:Conjectures related to motives
7181:in the triangulated category.
7158:
7123:
7117:
7088:
7082:
7043:
7040:
7034:
7031:
7025:
7011:
6989:
6986:
6980:
6966:
6950:
6938:
6814:
6792:
6746:
6725:
6709:
6703:
6673:of the (absolute) Galois group
6627:and is a topological invariant
6447:
6441:
6400:
6394:
6371:
6365:
6340:
6334:
6296:
6293:
6281:
6272:
6260:
6254:
6185:
6182:
6170:
6164:
6152:
6149:
6100:
6088:
6037:{\displaystyle M\otimes A(k).}
6028:
6022:
5993:
5987:
5960:
5954:
5927:
5918:
5907:
5901:
5878:
5872:
5863:
5825:
5816:
5807:
5804:
5798:
5789:
5786:
5783:
5768:
5765:
5729:
5717:
5711:
5705:
5699:
5693:
5634:
5622:
5573:
5567:
5532:
5526:
5520:
5514:
5473:
5461:
5435:
5429:
5426:
5423:
5402:
5375:
5369:
5349:
5327:
5283:
5166:
5154:
4858:
4855:
4849:
4840:
4813:to first construct a category
4786:
4774:
4727:
4721:
4712:
4709:
4703:
4674:
4668:
4609:equivalence relation on cycles
4582:
4576:
4507:
4495:
4463:
4445:
4442:
4439:
4421:
4389:
4354:
4348:
4333:
4306:
4300:
4297:
4291:
4272:
4266:
4129:
4114:
4104:)), then the elegant equation
4071:
4041:
3994:
3973:
3780:
3754:
3748:
3722:
3716:
3690:
3684:
3658:
3460:
3457:
3445:
3439:
3427:
3392:
3389:
3377:
3371:
3359:
3317:
3263:
3257:
3248:
3242:
3239:
3233:
3224:
3218:
3215:
3138:
3129:
3123:
3120:
3114:
3105:
3099:
3096:
3008:
3002:
2999:
2987:
2968:
2962:
2956:
2953:
2941:
2935:
2902:
2897:
2891:
2788:
2763:
2760:
2748:
2742:
2730:
2727:
2672:
2666:
2654:
2636:
2630:
2618:
2604:
2598:
2531:
2528:
2522:
2513:
2492:
2486:
2450:
2444:
2338:
2335:
2323:
2307:
2295:
2279:
2267:
2188:
2182:
2153:
2147:
2101:
2088:
2078:
2072:
2063:
2060:
2054:
2008:
1937:
1931:
1902:
1896:
1861:
1849:
1819:
1813:
1789:
1783:
1706:
1694:
1655:
1643:
1568:
1562:
1526:
1507:
1468:
1456:
1453:
1447:
1274:
1268:
1180:
1149:
1143:
1101:
1095:
863:
840:
257:
239:
219:
201:
109:
91:
1:
8205:Journal of Algebraic Geometry
7562:into an algebraic closure of
6606:The general idea is that one
4970:be our coefficient ring. Set
1068:adequate equivalence relation
7789:; Vologodsky, Vadim (2007),
7671:{\displaystyle \mathbb {Q} }
7611:{\displaystyle \mathbb {Q} }
7589:{\displaystyle \mathbb {Q} }
7539:finite separable extensions
7340:{\displaystyle \mathbb {C} }
6846:{\displaystyle \mathbb {C} }
6645:{\displaystyle \mathbb {C} }
4864:{\displaystyle D^{b}(MM(k))}
1013:{\displaystyle \mathbb {Q} }
850:
8300:Motive (algebraic geometry)
8038:Clay Mathematics Monographs
7860:Serre, Jean-Pierre (1991),
7856:(motives-for-dummies text).
7248:) (for any Weil cohomology
7210:Künneth standard conjecture
6669:action, i.e. has values in
5267:of a morphism of varieties
5224:Examples of correspondences
1299:into connected components:
167:{\displaystyle p:X\vdash X}
8342:
8030:; Weibel, Charles (2006),
7818:Motives over Finite Fields
7723:; however in terms of the
7649:. It gives a functor from
5307:From here we can form the
5179:. Its elements are called
5172:{\displaystyle C_{A}(X,Y)}
4323:A motive is then a triple
3941:To proceed to motives, we
1074:Definition of pure motives
29:
7900:Tabauda, Goncalo (2011),
7535:and consider the functor
6081:as the category of pairs
5746:Inverting the Tate motive
5735:{\displaystyle \otimes =}
4926:Here we will fix a field
4599:pseudo-abelian category.
1245:{\displaystyle X\times Y}
1215:{\displaystyle X\times Y}
958:{\displaystyle \phi _{p}}
190:of pure motives, where a
8010:Periods and Nori Motives
7993:The standard conjectures
6684:Mayer-Vietoris sequences
5292:{\displaystyle f:X\to Y}
4088:If we define the motive
3608:) be representatives of
2017:{\displaystyle f:X\to Y}
1574:{\displaystyle A^{k}(X)}
937:"Frobenius" automorphism
8196:Huber, Annette (2000).
7826:"What is ... a motive?"
7759:Presheaf with transfers
7288:homological equivalence
6597:multiplicative notation
6559:Each algebraic variety
6532:of a smooth projective
5844:from the canonical map
5681:Mayer–Vietoris sequence
4917:Geometric Mixed Motives
4654:abelian tensor category
4644:For a fixed base field
3084:pseudo-abelian category
2429:pseudo-abelian envelope
1222:, to fixed dimensional
263:{\displaystyle (Y,q,n)}
225:{\displaystyle (X,p,m)}
115:{\displaystyle (X,p,m)}
32:Motive (disambiguation)
8298:Quotations related to
8271:
7769:L-functions of motives
7729:Galois representations
7672:
7629:theory (going back to
7612:
7590:
7505:
7406:
7341:
7319:
7168:
7133:
7095:
7053:
6847:
6821:
6756:
6686:, homotopy invariance
6677:crystalline cohomology
6646:
6474:
6454:
6407:
6378:
6347:
6303:
6107:
6075:
6038:
6000:
5967:
5934:
5885:
5835:
5736:
5673:
5644:
5580:
5442:
5382:
5356:
5293:
5261:
5213:
5181:finite correspondences
5173:
5099:
5041:
5002:
4964:
4905:-winning proof of the
4865:
4793:
4734:
4681:
4603:Other types of motives
4589:
4551:
4396:
4313:
4227:
4150:
4078:
3870:
3479:
3327:
3192:
3047:
2829:
2707:
2538:
2457:
2409:
2195:
2163:
2125:
2018:
1986:
1944:
1912:
1871:
1716:
1662:
1611:
1575:
1536:
1406:
1375:
1281:
1246:
1216:
1190:
1189:{\displaystyle X\to Y}
1156:
1108:
1045:crystalline cohomology
1014:
992:
959:
929:
890:
870:
821:
792:
690:
618:
523:
290:
264:
226:
168:
136:
116:
75:crystalline cohomology
55:Alexander Grothendieck
8272:
8109:Jannsen, Uwe (1992),
8062:Levine, Marc (1998).
7824:Mazur, Barry (2004),
7673:
7631:Tannaka–Krein duality
7621:The objective of the
7613:
7591:
7506:
7407:
7342:
7320:
7169:
7134:
7096:
7054:
6848:
6822:
6757:
6654:mixed Hodge structure
6647:
6475:
6455:
6408:
6379:
6348:
6304:
6108:
6106:{\displaystyle (M,n)}
6076:
6039:
6001:
5968:
5935:
5886:
5836:
5737:
5674:
5645:
5591:triangulated category
5589:then we can form the
5581:
5443:
5383:
5357:
5294:
5262:
5214:
5174:
5133:and construct a free
5105:which is finite over
5100:
5042:
5003:
4965:
4866:
4794:
4735:
4682:
4680:{\displaystyle MM(k)}
4628:Numerical equivalence
4619:Algebraic equivalence
4590:
4552:
4513: such that
4397:
4314:
4228:
4151:
4079:
3929:are the projections.
3871:
3480:
3328:
3193:
3048:
2830:
2708:
2678: such that
2539:
2458:
2410:
2196:
2164:
2126:
2019:
1992:denotes the graph of
1987:
1945:
1913:
1872:
1717:
1663:
1612:
1576:
1537:
1407:
1376:
1282:
1247:
1217:
1191:
1157:
1109:
1015:
993:
960:
930:
891:
889:{\displaystyle \phi }
871:
822:
793:
691:
619:
524:
291:
265:
227:
169:
137:
117:
8242:
8103:Reference Literature
8084:Handbook of K-Theory
7986:Tannakian categories
7929:André, Yves (2004),
7787:Beilinson, Alexander
7660:
7623:motivic Galois group
7600:
7578:
7427:
7354:
7329:
7309:
7191:standard conjectures
7143:
7109:
7074:
6925:
6835:
6827:of a smooth variety
6774:
6690:
6634:
6601:local zeta-functions
6464:
6425:
6406:{\displaystyle A(n)}
6388:
6357:
6326:
6128:
6085:
6048:
6010:
5999:{\displaystyle M(k)}
5981:
5966:{\displaystyle A(k)}
5948:
5895:
5848:
5757:
5690:
5654:
5597:
5458:
5441:{\displaystyle \to }
5399:
5366:
5314:
5271:
5232:
5187:
5141:
5115:prime correspondence
5077:
5016:
4974:
4938:
4827:
4750:
4694:
4659:
4616:Rational equivalence
4567:
4412:
4330:
4257:
4166:
4111:
3964:
3629:
3343:
3212:
3093:
2868:
2718:
2571:
2470:
2435:
2252:
2243:preadditive category
2173:
2138:
2031:
1996:
1957:
1922:
1887:
1729:
1672:
1621:
1589:
1549:
1431:
1388:
1306:
1259:
1230:
1200:
1174:
1166:. The morphisms are
1134:
1086:
1002:
969:
942:
900:
880:
831:
805:
706:
637:
542:
311:
274:
236:
198:
146:
126:
88:
30:For other uses, see
18:Motive (mathematics)
8326:Homological algebra
8169:Voevodsky, Vladimir
8130:1992InMat.107..447J
8028:Voevodsky, Vladimir
7906:Journal of K-theory
7811:2006math......4004B
7174:shifted by –2, and
6791:
6503:birational geometry
6499:algebraic varieties
6313:Examples of motives
6248:
5828:
5621:
5564:
5511:
5510:
5494:
4804:Alexander Beilinson
4770:
4094:trivial Tate motive
3855:
3827:
3313:
3292:
2846:) is defined to be
2556:correspondences α:
1812:
1782:
820:{\displaystyle W,F}
289:{\displaystyle n-m}
63:singular cohomology
59:cohomology theories
8316:Algebraic geometry
8267:
8138:10.1007/BF01231898
7957:Serre, Jean-Pierre
7764:Mixed Hodge module
7754:Motivic cohomology
7709:Mumford–Tate group
7668:
7627:Tannakian category
7608:
7586:
7519:of the base field
7501:
7402:
7337:
7315:
7272:. The terminology
7164:
7129:
7091:
7049:
6916:algebraic K-theory
6911:Motivic cohomology
6905:Motivic cohomology
6843:
6817:
6777:
6752:
6642:
6578:de Rham cohomology
6470:
6450:
6403:
6374:
6343:
6299:
6224:
6218:
6103:
6071:
6034:
5996:
5963:
5930:
5881:
5831:
5732:
5669:
5640:
5600:
5576:
5498:
5482:
5438:
5378:
5352:
5289:
5257:
5209:
5169:
5095:
5037:
4998:
4960:
4930:of characteristic
4899:Vladimir Voevodsky
4861:
4789:
4753:
4730:
4677:
4648:, the category of
4585:
4547:
4515:
4392:
4309:
4223:
4146:
4074:
3866:
3841:
3813:
3475:
3323:
3299:
3278:
3188:
3078:As intended, Chow(
3043:
3038:
2825:
2703:
2680:
2534:
2453:
2405:
2191:
2159:
2121:
2116:
2014:
1982:
1940:
1908:
1867:
1795:
1765:
1712:
1658:
1607:
1571:
1532:
1483:
1402:
1371:
1324:
1277:
1242:
1212:
1186:
1152:
1104:
1034:de Rham cohomology
1010:
988:
955:
925:
886:
866:
817:
788:
686:
614:
519:
286:
260:
222:
164:
132:
112:
67:de Rham cohomology
39:algebraic geometry
8232:Future directions
8188:978-0-691-04814-7
8118:Inventiones Math.
8094:978-3-540-23019-9
8073:978-0-8218-0785-9
8047:978-0-8218-3847-1
8019:978-3-319-50925-9
8000:Classical motives
7970:978-0-8218-1636-3
7940:978-2-85629-164-1
7839:(10): 1214–1216,
7318:{\displaystyle k}
7303:Hodge realization
7208:For example, the
7070:are integers and
6862:theory of motives
6784:
6526:theory of motives
6473:{\displaystyle C}
6439:
6417:Motives of curves
6191:
5829:
5619:
5614:
5565:
5512:
5309:homotopy category
5073:closed subscheme
4907:Milnor conjecture
4652:is a conjectural
4514:
3411:
2861:The association,
2679:
2201:has direct sums (
1474:
1315:
853:
174:is an idempotent
135:{\displaystyle X}
16:(Redirected from
8333:
8297:
8276:
8274:
8273:
8268:
8260:
8249:
8227:
8225:
8219:. Archived from
8202:
8191:
8161:Milne, James S.
8157:
8148:
8115:
8098:
8077:
8058:
8022:
7981:
7951:
7918:
7917:
7895:
7893:
7887:, archived from
7866:
7855:
7830:
7813:
7804:
7795:, p. 4004,
7733:étale cohomology
7717:invariant theory
7677:
7675:
7674:
7669:
7667:
7635:Hodge conjecture
7617:
7615:
7614:
7609:
7607:
7595:
7593:
7592:
7587:
7585:
7515:of the absolute
7510:
7508:
7507:
7502:
7476:
7475:
7463:
7462:
7461:
7460:
7455:
7414:Hodge structures
7411:
7409:
7408:
7403:
7401:
7400:
7399:
7383:
7382:
7381:
7346:
7344:
7343:
7338:
7336:
7324:
7322:
7321:
7316:
7295:Hodge conjecture
7199:Weil conjectures
7177:means the usual
7173:
7171:
7170:
7165:
7157:
7156:
7151:
7138:
7136:
7135:
7130:
7116:
7100:
7098:
7097:
7092:
7081:
7058:
7056:
7055:
7050:
7024:
7007:
7006:
6979:
6965:
6964:
6937:
6936:
6852:
6850:
6849:
6844:
6842:
6826:
6824:
6823:
6818:
6810:
6805:
6790:
6785:
6782:
6761:
6759:
6758:
6753:
6745:
6744:
6739:
6724:
6723:
6702:
6701:
6663:-adic cohomology
6651:
6649:
6648:
6643:
6641:
6589:-adic cohomology
6582:Betti cohomology
6541:Betti cohomology
6513:, for example a
6479:
6477:
6476:
6471:
6459:
6457:
6456:
6451:
6440:
6437:
6432:
6412:
6410:
6409:
6404:
6383:
6381:
6380:
6375:
6364:
6352:
6350:
6349:
6344:
6333:
6308:
6306:
6305:
6300:
6250:
6249:
6247:
6242:
6234:
6233:
6217:
6145:
6144:
6143:
6120:
6116:
6112:
6110:
6109:
6104:
6080:
6078:
6077:
6072:
6070:
6069:
6061:
6060:
6043:
6041:
6040:
6035:
6005:
6003:
6002:
5997:
5976:
5972:
5970:
5969:
5964:
5940:and call it the
5939:
5937:
5936:
5931:
5917:
5890:
5888:
5887:
5882:
5862:
5861:
5856:
5840:
5838:
5837:
5832:
5830:
5811:
5782:
5781:
5776:
5764:
5741:
5739:
5738:
5733:
5678:
5676:
5675:
5670:
5668:
5667:
5662:
5649:
5647:
5646:
5641:
5620:
5617:
5615:
5612:
5610:
5609:
5585:
5583:
5582:
5577:
5566:
5563:
5562:
5550:
5549:
5536:
5513:
5506:
5490:
5477:
5447:
5445:
5444:
5439:
5422:
5421:
5416:
5387:
5385:
5384:
5381:{\displaystyle }
5379:
5361:
5359:
5358:
5353:
5348:
5347:
5326:
5325:
5298:
5296:
5295:
5290:
5266:
5264:
5263:
5258:
5244:
5243:
5218:
5216:
5215:
5210:
5208:
5207:
5178:
5176:
5175:
5170:
5153:
5152:
5136:
5132:
5128:
5124:
5120:
5112:
5108:
5104:
5102:
5101:
5096:
5068:
5061:
5046:
5044:
5043:
5038:
5033:
5028:
5027:
5011:
5007:
5005:
5004:
4999:
4994:
4989:
4988:
4969:
4967:
4966:
4961:
4959:
4951:
4933:
4929:
4870:
4868:
4867:
4862:
4839:
4838:
4819:derived category
4798:
4796:
4795:
4790:
4769:
4764:
4739:
4737:
4736:
4731:
4686:
4684:
4683:
4678:
4594:
4592:
4591:
4586:
4556:
4554:
4553:
4548:
4516:
4512:
4491:
4490:
4401:
4399:
4398:
4393:
4388:
4318:
4316:
4315:
4310:
4287:
4286:
4232:
4230:
4229:
4224:
4219:
4215:
4208:
4207:
4202:
4193:
4192:
4187:
4173:
4155:
4153:
4152:
4147:
4139:
4128:
4127:
4122:
4100: := h(Spec(
4083:
4081:
4080:
4075:
4070:
4069:
4064:
4055:
4054:
4049:
4040:
4039:
4027:
4026:
4021:
3987:
3986:
3981:
3951:Lefschetz motive
3875:
3873:
3872:
3867:
3865:
3864:
3854:
3849:
3837:
3836:
3826:
3821:
3809:
3808:
3796:
3795:
3779:
3778:
3766:
3765:
3747:
3746:
3734:
3733:
3715:
3714:
3702:
3701:
3683:
3682:
3670:
3669:
3654:
3653:
3641:
3640:
3484:
3482:
3481:
3476:
3423:
3422:
3412:
3409:
3355:
3354:
3332:
3330:
3329:
3324:
3312:
3307:
3291:
3286:
3197:
3195:
3194:
3189:
3184:
3180:
3167:
3163:
3052:
3050:
3049:
3044:
3042:
3041:
3023:
3022:
2986:
2985:
2931:
2930:
2929:
2834:
2832:
2831:
2826:
2791:
2712:
2710:
2709:
2704:
2681:
2677:
2611:
2607:
2594:
2593:
2543:
2541:
2540:
2535:
2482:
2481:
2462:
2460:
2459:
2454:
2414:
2412:
2411:
2406:
2401:
2397:
2396:
2392:
2374:
2370:
2350:
2349:
2322:
2321:
2294:
2293:
2240:
2218:
2200:
2198:
2197:
2192:
2168:
2166:
2165:
2160:
2130:
2128:
2127:
2122:
2120:
2119:
2113:
2112:
2023:
2021:
2020:
2015:
1991:
1989:
1988:
1983:
1969:
1968:
1949:
1947:
1946:
1941:
1917:
1915:
1914:
1909:
1876:
1874:
1873:
1868:
1845:
1844:
1826:
1822:
1811:
1806:
1781:
1776:
1759:
1758:
1721:
1719:
1718:
1713:
1690:
1689:
1667:
1665:
1664:
1659:
1639:
1638:
1616:
1614:
1613:
1608:
1580:
1578:
1577:
1572:
1561:
1560:
1541:
1539:
1538:
1533:
1519:
1518:
1506:
1505:
1498:
1497:
1482:
1443:
1442:
1411:
1409:
1408:
1403:
1401:
1380:
1378:
1377:
1372:
1367:
1366:
1348:
1347:
1334:
1333:
1323:
1286:
1284:
1283:
1278:
1251:
1249:
1248:
1243:
1221:
1219:
1218:
1213:
1195:
1193:
1192:
1187:
1161:
1159:
1158:
1153:
1113:
1111:
1110:
1105:
1041:-adic cohomology
1030:Betti cohomology
1019:
1017:
1016:
1011:
1009:
997:
995:
994:
989:
987:
986:
964:
962:
961:
956:
954:
953:
934:
932:
931:
926:
921:
920:
919:
918:
913:
895:
893:
892:
887:
875:
873:
872:
867:
862:
854:
849:
844:
826:
824:
823:
818:
797:
795:
794:
789:
787:
786:
785:
757:
756:
749:
748:
743:
729:
728:
721:
695:
693:
692:
687:
682:
681:
676:
667:
666:
661:
652:
644:
623:
621:
620:
615:
613:
612:
594:
593:
592:
591:
586:
572:
571:
570:
554:
553:
528:
526:
525:
520:
518:
514:
513:
512:
488:
487:
469:
468:
467:
439:
438:
431:
430:
425:
411:
410:
403:
387:
386:
368:
367:
366:
365:
360:
346:
345:
344:
328:
327:
295:
293:
292:
287:
269:
267:
266:
261:
231:
229:
228:
223:
173:
171:
170:
165:
141:
139:
138:
133:
121:
119:
118:
113:
71:etale cohomology
21:
8341:
8340:
8336:
8335:
8334:
8332:
8331:
8330:
8306:
8305:
8290:
8240:
8239:
8234:
8223:
8200:
8195:
8189:
8167:
8151:
8113:
8108:
8105:
8095:
8080:
8074:
8061:
8048:
8025:
8020:
8007:
7971:
7959:, eds. (1994),
7954:
7941:
7928:
7925:
7899:
7891:
7864:
7859:
7828:
7823:
7785:
7782:
7780:Survey Articles
7777:
7749:Ring of periods
7745:
7725:Tate conjecture
7699:algebraic group
7695:representations
7691:
7658:
7657:
7654:
7643:algebraic cycle
7639:Tate conjecture
7598:
7597:
7576:
7575:
7529:
7513:representations
7467:
7450:
7445:
7425:
7424:
7421:Tate conjecture
7419:Similarly, the
7390:
7372:
7352:
7351:
7327:
7326:
7307:
7306:
7269:
7187:
7146:
7141:
7140:
7107:
7106:
7072:
7071:
6995:
6956:
6928:
6923:
6922:
6907:
6893:
6882:
6833:
6832:
6772:
6771:
6762:the product of
6734:
6715:
6693:
6688:
6687:
6671:representations
6652:) comes with a
6632:
6631:
6621:complex numbers
6612:Weil cohomology
6557:
6486:
6462:
6461:
6423:
6422:
6419:
6386:
6385:
6355:
6354:
6324:
6323:
6320:
6315:
6219:
6131:
6126:
6125:
6118:
6114:
6083:
6082:
6051:
6046:
6045:
6008:
6007:
5979:
5978:
5974:
5946:
5945:
5893:
5892:
5851:
5846:
5845:
5771:
5755:
5754:
5748:
5688:
5687:
5657:
5652:
5651:
5595:
5594:
5554:
5541:
5456:
5455:
5411:
5397:
5396:
5364:
5363:
5317:
5312:
5311:
5305:
5269:
5268:
5235:
5230:
5229:
5226:
5185:
5184:
5144:
5139:
5138:
5134:
5130:
5126:
5122:
5118:
5110:
5106:
5075:
5074:
5066:
5059:
5053:
5014:
5013:
5009:
4972:
4971:
4936:
4935:
4931:
4927:
4924:
4919:
4830:
4825:
4824:
4748:
4747:
4692:
4691:
4657:
4656:
4642:
4605:
4565:
4564:
4476:
4410:
4409:
4328:
4327:
4278:
4255:
4254:
4197:
4182:
4181:
4177:
4164:
4163:
4117:
4109:
4108:
4059:
4044:
4031:
4016:
3976:
3962:
3961:
3939:
3927:
3920:
3914:
3907:
3900:
3893:
3885:
3856:
3828:
3800:
3787:
3770:
3757:
3738:
3725:
3706:
3693:
3674:
3661:
3645:
3632:
3627:
3626:
3620:
3613:
3607:
3600:
3589:
3582:
3575:
3564:
3557:
3550:
3543:
3536:
3529:
3522:
3515:
3508:
3501:
3494:
3414:
3346:
3341:
3340:
3210:
3209:
3153:
3149:
3148:
3144:
3091:
3090:
3062:
3037:
3036:
3014:
2997:
2991:
2990:
2977:
2951:
2945:
2944:
2915:
2900:
2878:
2866:
2865:
2716:
2715:
2585:
2584:
2580:
2569:
2568:
2473:
2468:
2467:
2433:
2432:
2425:
2382:
2378:
2360:
2356:
2355:
2351:
2341:
2313:
2285:
2250:
2249:
2224:
2221:tensor products
2202:
2171:
2170:
2136:
2135:
2115:
2114:
2104:
2095:
2094:
2082:
2081:
2041:
2029:
2028:
1994:
1993:
1960:
1955:
1954:
1920:
1919:
1885:
1884:
1830:
1764:
1760:
1744:
1727:
1726:
1681:
1670:
1669:
1630:
1619:
1618:
1587:
1586:
1552:
1547:
1546:
1510:
1489:
1484:
1434:
1429:
1428:
1386:
1385:
1358:
1339:
1325:
1304:
1303:
1257:
1256:
1228:
1227:
1198:
1197:
1172:
1171:
1168:correspondences
1132:
1131:
1130:The objects of
1128:
1084:
1083:
1076:
1026:
1000:
999:
972:
967:
966:
945:
940:
939:
908:
903:
898:
897:
878:
877:
829:
828:
803:
802:
761:
738:
733:
709:
704:
703:
671:
656:
635:
634:
598:
581:
576:
558:
545:
540:
539:
504:
479:
443:
420:
415:
391:
372:
355:
350:
332:
319:
318:
314:
309:
308:
272:
271:
234:
233:
196:
195:
144:
143:
124:
123:
86:
85:
35:
28:
23:
22:
15:
12:
11:
5:
8339:
8337:
8329:
8328:
8323:
8318:
8308:
8307:
8304:
8303:
8289:
8288:External links
8286:
8285:
8284:
8279:
8266:
8263:
8259:
8255:
8252:
8248:
8233:
8230:
8229:
8228:
8226:on 2017-09-26.
8193:
8187:
8173:Suslin, Andrei
8165:
8159:
8149:
8104:
8101:
8100:
8099:
8093:
8078:
8072:
8059:
8046:
8026:Mazza, Carlo;
8023:
8018:
8005:
8004:
8003:
7996:
7989:
7969:
7952:
7939:
7924:
7921:
7920:
7919:
7897:
7857:
7821:
7815:
7781:
7778:
7776:
7773:
7772:
7771:
7766:
7761:
7756:
7751:
7744:
7741:
7689:
7682:, the functor
7666:
7652:
7606:
7584:
7552:
7551:
7528:
7525:
7500:
7497:
7494:
7491:
7488:
7485:
7482:
7479:
7474:
7470:
7466:
7459:
7454:
7448:
7444:
7441:
7438:
7435:
7432:
7398:
7393:
7389:
7386:
7380:
7375:
7371:
7368:
7365:
7362:
7359:
7335:
7314:
7267:
7186:
7183:
7163:
7160:
7155:
7150:
7128:
7125:
7122:
7119:
7115:
7090:
7087:
7084:
7080:
7060:
7059:
7048:
7045:
7042:
7039:
7036:
7033:
7030:
7027:
7023:
7019:
7016:
7013:
7010:
7005:
7002:
6998:
6994:
6991:
6988:
6985:
6982:
6978:
6974:
6971:
6968:
6963:
6959:
6955:
6952:
6949:
6946:
6943:
6940:
6935:
6931:
6906:
6903:
6891:
6880:
6870:
6869:
6841:
6816:
6813:
6809:
6804:
6800:
6797:
6794:
6789:
6780:
6751:
6748:
6743:
6738:
6733:
6730:
6727:
6722:
6718:
6714:
6711:
6708:
6705:
6700:
6696:
6680:
6679:
6674:
6657:
6640:
6628:
6574:
6573:
6570:
6567:
6556:
6553:
6511:linear algebra
6485:
6482:
6469:
6449:
6446:
6443:
6435:
6431:
6418:
6415:
6402:
6399:
6396:
6393:
6373:
6370:
6367:
6363:
6342:
6339:
6336:
6332:
6319:
6316:
6314:
6311:
6310:
6309:
6298:
6295:
6292:
6289:
6286:
6283:
6280:
6277:
6274:
6271:
6268:
6265:
6262:
6259:
6256:
6253:
6246:
6241:
6238:
6232:
6229:
6222:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6194:
6190:
6187:
6184:
6181:
6178:
6175:
6172:
6169:
6166:
6163:
6160:
6157:
6154:
6151:
6148:
6142:
6139:
6134:
6102:
6099:
6096:
6093:
6090:
6068:
6065:
6059:
6056:
6033:
6030:
6027:
6024:
6021:
6018:
6015:
5995:
5992:
5989:
5986:
5962:
5959:
5956:
5953:
5929:
5926:
5923:
5920:
5916:
5912:
5909:
5906:
5903:
5900:
5891:. We will set
5880:
5877:
5874:
5871:
5868:
5865:
5860:
5855:
5842:
5841:
5827:
5824:
5821:
5818:
5814:
5809:
5806:
5803:
5800:
5797:
5794:
5791:
5788:
5785:
5780:
5775:
5770:
5767:
5763:
5747:
5744:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5707:
5704:
5701:
5698:
5695:
5666:
5661:
5639:
5636:
5633:
5630:
5627:
5624:
5608:
5605:
5587:
5586:
5575:
5572:
5569:
5561:
5557:
5553:
5548:
5544:
5539:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5509:
5505:
5501:
5497:
5493:
5489:
5485:
5480:
5475:
5472:
5469:
5466:
5463:
5449:
5448:
5437:
5434:
5431:
5428:
5425:
5420:
5415:
5410:
5407:
5404:
5377:
5374:
5371:
5351:
5346:
5343:
5340:
5337:
5334:
5329:
5324:
5320:
5304:
5301:
5288:
5285:
5282:
5279:
5276:
5256:
5253:
5250:
5247:
5242:
5238:
5225:
5222:
5206:
5203:
5200:
5197:
5194:
5168:
5165:
5162:
5159:
5156:
5151:
5147:
5094:
5091:
5088:
5085:
5082:
5057:smooth variety
5052:
5049:
5036:
5032:
5026:
5023:
4997:
4993:
4987:
4984:
4981:
4958:
4954:
4950:
4946:
4943:
4923:
4920:
4918:
4915:
4873:
4872:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4837:
4833:
4800:
4799:
4788:
4785:
4782:
4779:
4776:
4773:
4768:
4763:
4760:
4756:
4741:
4740:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4676:
4673:
4670:
4667:
4664:
4641:
4638:
4630:
4629:
4626:
4623:
4620:
4617:
4604:
4601:
4584:
4581:
4578:
4575:
4572:
4558:
4557:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4509:
4506:
4503:
4500:
4497:
4494:
4489:
4486:
4483:
4479:
4475:
4472:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4403:
4402:
4391:
4387:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4321:
4320:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4285:
4281:
4277:
4274:
4271:
4268:
4265:
4262:
4234:
4233:
4222:
4218:
4214:
4211:
4206:
4201:
4196:
4191:
4186:
4180:
4176:
4172:
4157:
4156:
4145:
4142:
4138:
4134:
4131:
4126:
4121:
4116:
4086:
4085:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4034:
4030:
4025:
4020:
4015:
4012:
4009:
4006:
4003:
3999:
3996:
3993:
3990:
3985:
3980:
3975:
3972:
3969:
3938:
3931:
3925:
3918:
3912:
3905:
3898:
3891:
3883:
3878:
3877:
3863:
3859:
3853:
3848:
3844:
3840:
3835:
3831:
3825:
3820:
3816:
3812:
3807:
3803:
3799:
3794:
3790:
3785:
3782:
3777:
3773:
3769:
3764:
3760:
3756:
3753:
3750:
3745:
3741:
3737:
3732:
3728:
3724:
3721:
3718:
3713:
3709:
3705:
3700:
3696:
3692:
3689:
3686:
3681:
3677:
3673:
3668:
3664:
3660:
3657:
3652:
3648:
3644:
3639:
3635:
3618:
3611:
3605:
3598:
3587:
3580:
3573:
3562:
3555:
3548:
3541:
3534:
3527:
3520:
3513:
3506:
3499:
3492:
3486:
3485:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3421:
3417:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3353:
3349:
3334:
3333:
3322:
3319:
3316:
3311:
3306:
3302:
3298:
3295:
3290:
3285:
3281:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3203:tensor product
3199:
3198:
3187:
3183:
3179:
3176:
3173:
3170:
3166:
3162:
3159:
3156:
3152:
3147:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3058:
3055:
3054:
3040:
3035:
3032:
3029:
3026:
3021:
3017:
3013:
3010:
3007:
3004:
3001:
2998:
2996:
2993:
2992:
2989:
2984:
2980:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2950:
2947:
2946:
2943:
2940:
2937:
2934:
2928:
2925:
2922:
2918:
2914:
2911:
2908:
2904:
2901:
2899:
2896:
2893:
2890:
2887:
2884:
2883:
2881:
2876:
2873:
2836:
2835:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2790:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2713:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2610:
2606:
2603:
2600:
2597:
2592:
2588:
2583:
2579:
2576:
2546:
2545:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2480:
2476:
2452:
2449:
2446:
2443:
2440:
2424:
2417:
2416:
2415:
2404:
2400:
2395:
2391:
2388:
2385:
2381:
2377:
2373:
2369:
2366:
2363:
2359:
2354:
2348:
2344:
2340:
2337:
2334:
2331:
2328:
2325:
2320:
2316:
2312:
2309:
2306:
2303:
2300:
2297:
2292:
2288:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2190:
2187:
2184:
2181:
2178:
2158:
2155:
2152:
2149:
2146:
2143:
2132:
2131:
2118:
2111:
2107:
2103:
2100:
2097:
2096:
2093:
2090:
2087:
2084:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2046:
2044:
2039:
2036:
2013:
2010:
2007:
2004:
2001:
1981:
1978:
1975:
1972:
1967:
1963:
1939:
1936:
1933:
1930:
1927:
1907:
1904:
1901:
1898:
1895:
1892:
1878:
1877:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1843:
1840:
1837:
1833:
1829:
1825:
1821:
1818:
1815:
1810:
1805:
1802:
1798:
1794:
1791:
1788:
1785:
1780:
1775:
1772:
1768:
1763:
1757:
1754:
1751:
1747:
1743:
1740:
1737:
1734:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1688:
1684:
1680:
1677:
1657:
1654:
1651:
1648:
1645:
1642:
1637:
1633:
1629:
1626:
1606:
1603:
1600:
1597:
1594:
1570:
1567:
1564:
1559:
1555:
1543:
1542:
1531:
1528:
1525:
1522:
1517:
1513:
1509:
1504:
1501:
1496:
1492:
1487:
1481:
1477:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1441:
1437:
1400:
1396:
1393:
1382:
1381:
1370:
1365:
1361:
1357:
1354:
1351:
1346:
1342:
1337:
1332:
1328:
1322:
1318:
1314:
1311:
1276:
1273:
1270:
1267:
1264:
1241:
1238:
1235:
1211:
1208:
1205:
1185:
1182:
1179:
1151:
1148:
1145:
1142:
1139:
1127:
1120:
1118:is any field.
1103:
1100:
1097:
1094:
1091:
1075:
1072:
1056:
1055:
1052:
1025:
1022:
1008:
985:
982:
979:
975:
952:
948:
924:
917:
912:
906:
885:
865:
861:
857:
852:
848:
842:
839:
836:
816:
813:
810:
799:
798:
784:
781:
777:
774:
771:
768:
764:
760:
755:
752:
747:
742:
736:
732:
727:
724:
720:
717:
712:
697:
696:
685:
680:
675:
670:
665:
660:
655:
651:
647:
643:
625:
624:
611:
608:
605:
601:
597:
590:
585:
579:
575:
569:
566:
561:
557:
552:
548:
532:consisting of
530:
529:
517:
511:
507:
503:
500:
497:
494:
491:
486:
482:
478:
475:
472:
466:
463:
459:
456:
453:
450:
446:
442:
437:
434:
429:
424:
418:
414:
409:
406:
402:
399:
394:
390:
385:
382:
379:
375:
371:
364:
359:
353:
349:
343:
340:
335:
331:
326:
322:
317:
298:Pierre Deligne
285:
282:
279:
259:
256:
253:
250:
247:
244:
241:
221:
218:
215:
212:
209:
206:
203:
176:correspondence
163:
160:
157:
154:
151:
131:
111:
108:
105:
102:
99:
96:
93:
45:(or sometimes
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8338:
8327:
8324:
8322:
8319:
8317:
8314:
8313:
8311:
8301:
8296:
8292:
8291:
8287:
8283:
8280:
8278:
8261:
8257:
8253:
8236:
8235:
8231:
8222:
8218:
8214:
8210:
8206:
8199:
8194:
8190:
8184:
8180:
8179:
8174:
8170:
8166:
8164:
8160:
8155:
8150:
8147:
8143:
8139:
8135:
8131:
8127:
8123:
8119:
8112:
8107:
8106:
8102:
8096:
8090:
8086:
8085:
8079:
8075:
8069:
8065:
8064:Mixed Motives
8060:
8057:
8053:
8049:
8043:
8039:
8035:
8034:
8029:
8024:
8021:
8015:
8011:
8006:
8001:
7997:
7994:
7990:
7987:
7983:
7982:
7980:
7976:
7972:
7966:
7962:
7958:
7953:
7950:
7946:
7942:
7936:
7932:
7927:
7926:
7922:
7916:
7911:
7907:
7903:
7898:
7894:on 2022-01-10
7890:
7886:
7882:
7878:
7874:
7870:
7863:
7858:
7854:
7850:
7846:
7842:
7838:
7834:
7827:
7822:
7819:
7816:
7812:
7808:
7803:
7798:
7794:
7793:
7788:
7784:
7783:
7779:
7774:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7746:
7742:
7740:
7738:
7734:
7730:
7726:
7722:
7718:
7714:
7710:
7705:
7703:
7700:
7696:
7692:
7685:
7681:
7655:
7648:
7644:
7640:
7636:
7632:
7628:
7624:
7619:
7573:
7572:Artin motives
7569:
7568:Galois theory
7565:
7561:
7557:
7550:
7546:
7542:
7538:
7537:
7536:
7534:
7526:
7524:
7522:
7518:
7514:
7492:
7486:
7483:
7477:
7472:
7468:
7457:
7442:
7436:
7433:
7430:
7422:
7417:
7415:
7391:
7387:
7369:
7363:
7360:
7357:
7350:
7312:
7304:
7300:
7296:
7291:
7289:
7285:
7281:
7279:
7275:
7271:
7263:
7259:
7255:
7251:
7247:
7243:
7239:
7235:
7231:
7227:
7223:
7219:
7215:
7211:
7206:
7204:
7200:
7194:
7192:
7184:
7182:
7180:
7176:
7161:
7153:
7126:
7120:
7104:
7085:
7069:
7065:
7046:
7037:
7028:
7017:
7014:
7008:
7003:
7000:
6996:
6992:
6983:
6972:
6969:
6961:
6957:
6953:
6947:
6944:
6941:
6933:
6929:
6921:
6920:
6919:
6917:
6913:
6912:
6904:
6902:
6899:
6897:
6890:
6886:
6879:
6875:
6867:
6866:
6865:
6863:
6858:
6856:
6830:
6811:
6807:
6798:
6795:
6787:
6778:
6769:
6765:
6749:
6741:
6731:
6728:
6720:
6716:
6712:
6706:
6698:
6694:
6685:
6678:
6675:
6672:
6668:
6664:
6662:
6658:
6655:
6629:
6626:
6622:
6618:
6617:
6616:
6613:
6609:
6604:
6602:
6598:
6594:
6590:
6588:
6583:
6579:
6571:
6568:
6566:
6565:
6564:
6562:
6554:
6552:
6550:
6546:
6542:
6538:
6535:
6531:
6527:
6522:
6520:
6516:
6512:
6508:
6504:
6500:
6495:
6491:
6483:
6481:
6467:
6444:
6433:
6416:
6414:
6397:
6391:
6368:
6337:
6317:
6312:
6290:
6287:
6284:
6278:
6275:
6269:
6266:
6263:
6257:
6251:
6244:
6239:
6236:
6220:
6214:
6211:
6208:
6205:
6202:
6199:
6196:
6188:
6179:
6176:
6173:
6167:
6161:
6158:
6155:
6146:
6132:
6124:
6123:
6122:
6097:
6094:
6091:
6066:
6063:
6031:
6025:
6019:
6016:
6013:
5990:
5984:
5957:
5951:
5943:
5924:
5921:
5910:
5904:
5898:
5875:
5869:
5866:
5858:
5822:
5819:
5812:
5801:
5795:
5792:
5778:
5753:
5752:
5751:
5745:
5743:
5726:
5723:
5720:
5714:
5708:
5702:
5696:
5684:
5682:
5664:
5637:
5631:
5628:
5625:
5592:
5570:
5559:
5555:
5551:
5546:
5542:
5537:
5529:
5523:
5517:
5507:
5503:
5499:
5495:
5491:
5487:
5483:
5478:
5470:
5467:
5464:
5454:
5453:
5452:
5432:
5418:
5408:
5405:
5395:
5394:
5393:
5391:
5372:
5322:
5318:
5310:
5302:
5300:
5286:
5280:
5277:
5274:
5254:
5251:
5248:
5245:
5240:
5223:
5221:
5182:
5163:
5160:
5157:
5149:
5145:
5116:
5092:
5089:
5086:
5083:
5080:
5072:
5065:
5058:
5050:
5048:
5034:
5030:
4995:
4991:
4952:
4944:
4941:
4921:
4916:
4914:
4910:
4908:
4904:
4900:
4896:
4891:
4889:
4888:
4882:
4878:
4852:
4846:
4843:
4835:
4831:
4823:
4822:
4821:
4820:
4816:
4812:
4807:
4805:
4783:
4780:
4777:
4771:
4766:
4761:
4758:
4754:
4746:
4745:
4744:
4724:
4718:
4715:
4706:
4700:
4697:
4690:
4689:
4688:
4671:
4665:
4662:
4655:
4651:
4650:mixed motives
4647:
4640:Mixed motives
4639:
4637:
4635:
4627:
4624:
4621:
4618:
4615:
4614:
4613:
4610:
4602:
4600:
4598:
4579:
4573:
4570:
4563:As intended,
4561:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4504:
4501:
4498:
4492:
4487:
4484:
4481:
4477:
4473:
4470:
4466:
4460:
4457:
4454:
4451:
4448:
4436:
4433:
4430:
4427:
4424:
4418:
4415:
4408:
4407:
4406:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4351:
4345:
4342:
4339:
4336:
4326:
4325:
4324:
4303:
4294:
4288:
4283:
4279:
4275:
4269:
4263:
4260:
4253:
4252:
4251:
4249:
4245:
4241:
4240:
4220:
4216:
4212:
4209:
4204:
4194:
4189:
4178:
4174:
4162:
4161:
4160:
4159:holds, since
4143:
4140:
4132:
4124:
4107:
4106:
4105:
4103:
4099:
4095:
4092:, called the
4091:
4066:
4056:
4051:
4036:
4032:
4028:
4023:
4013:
4010:
4007:
4004:
4001:
3997:
3991:
3988:
3983:
3970:
3967:
3960:
3959:
3958:
3956:
3952:
3948:
3944:
3936:
3932:
3930:
3928:
3921:
3911:
3904:
3897:
3890:
3886:
3861:
3857:
3851:
3846:
3842:
3838:
3833:
3829:
3823:
3818:
3814:
3810:
3805:
3801:
3797:
3792:
3788:
3783:
3775:
3771:
3767:
3762:
3758:
3751:
3743:
3739:
3735:
3730:
3726:
3719:
3711:
3707:
3703:
3698:
3694:
3687:
3679:
3675:
3671:
3666:
3662:
3655:
3650:
3646:
3642:
3637:
3633:
3625:
3624:
3623:
3621:
3614:
3604:
3597:
3593:
3586:
3579:
3572:
3568:
3561:
3554:
3547:
3540:
3533:
3526:
3519:
3512:
3505:
3498:
3491:
3472:
3469:
3466:
3463:
3454:
3451:
3448:
3442:
3436:
3433:
3430:
3424:
3419:
3415:
3404:
3401:
3398:
3395:
3386:
3383:
3380:
3374:
3368:
3365:
3362:
3356:
3351:
3347:
3339:
3338:
3337:
3320:
3314:
3309:
3304:
3300:
3296:
3293:
3288:
3283:
3279:
3275:
3272:
3269:
3266:
3260:
3254:
3251:
3245:
3236:
3230:
3227:
3221:
3208:
3207:
3206:
3204:
3185:
3181:
3177:
3174:
3171:
3168:
3164:
3160:
3157:
3154:
3150:
3145:
3141:
3135:
3132:
3126:
3117:
3111:
3108:
3102:
3089:
3088:
3087:
3085:
3081:
3076:
3074:
3070:
3066:
3061:
3033:
3030:
3027:
3024:
3019:
3011:
3005:
2994:
2982:
2974:
2971:
2965:
2959:
2948:
2938:
2932:
2894:
2888:
2885:
2879:
2874:
2871:
2864:
2863:
2862:
2859:
2857:
2853:
2849:
2845:
2841:
2822:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2784:
2781:
2778:
2775:
2772:
2766:
2757:
2754:
2751:
2745:
2739:
2736:
2733:
2724:
2721:
2714:
2700:
2694:
2691:
2688:
2685:
2682:
2669:
2663:
2660:
2657:
2651:
2648:
2645:
2642:
2639:
2633:
2627:
2624:
2621:
2612:
2608:
2601:
2595:
2590:
2586:
2581:
2577:
2574:
2567:
2566:
2565:
2563:
2559:
2555:
2551:
2525:
2519:
2516:
2510:
2507:
2504:
2501:
2498:
2495:
2489:
2483:
2478:
2474:
2466:
2465:
2464:
2447:
2441:
2438:
2430:
2422:
2418:
2402:
2398:
2393:
2389:
2386:
2383:
2379:
2375:
2371:
2367:
2364:
2361:
2357:
2352:
2346:
2342:
2332:
2329:
2326:
2318:
2314:
2310:
2304:
2301:
2298:
2290:
2286:
2282:
2276:
2273:
2270:
2264:
2261:
2258:
2255:
2248:
2247:
2246:
2244:
2239:
2235:
2231:
2227:
2222:
2217:
2213:
2209:
2205:
2185:
2179:
2176:
2169:the category
2156:
2150:
2144:
2141:
2109:
2098:
2091:
2085:
2075:
2069:
2066:
2057:
2051:
2048:
2042:
2037:
2034:
2027:
2026:
2025:
2011:
2005:
2002:
1999:
1979:
1976:
1973:
1970:
1965:
1951:
1934:
1928:
1925:
1905:
1899:
1893:
1890:
1881:
1864:
1858:
1855:
1852:
1846:
1841:
1838:
1835:
1831:
1827:
1823:
1816:
1808:
1803:
1800:
1796:
1792:
1786:
1778:
1773:
1770:
1766:
1761:
1755:
1752:
1749:
1745:
1741:
1738:
1735:
1732:
1725:
1724:
1723:
1709:
1703:
1700:
1697:
1691:
1686:
1682:
1678:
1675:
1652:
1649:
1646:
1640:
1635:
1631:
1627:
1624:
1604:
1601:
1598:
1595:
1592:
1584:
1565:
1557:
1553:
1529:
1523:
1520:
1515:
1511:
1502:
1499:
1494:
1490:
1485:
1479:
1475:
1471:
1465:
1462:
1459:
1450:
1444:
1439:
1435:
1427:
1426:
1425:
1423:
1419:
1415:
1394:
1391:
1368:
1363:
1359:
1355:
1352:
1349:
1344:
1340:
1335:
1330:
1326:
1320:
1316:
1312:
1309:
1302:
1301:
1300:
1298:
1294:
1290:
1271:
1265:
1262:
1253:
1239:
1236:
1233:
1225:
1209:
1206:
1203:
1183:
1177:
1169:
1165:
1146:
1140:
1137:
1125:
1121:
1119:
1117:
1098:
1092:
1089:
1081:
1073:
1071:
1069:
1063:
1061:
1053:
1050:
1049:
1048:
1046:
1042:
1040:
1035:
1031:
1023:
1021:
983:
980:
977:
973:
950:
946:
938:
922:
915:
904:
883:
855:
837:
834:
814:
811:
808:
775:
772:
769:
766:
762:
758:
753:
750:
745:
734:
730:
725:
722:
710:
702:
701:
700:
683:
678:
668:
663:
653:
645:
633:
632:
631:
630:
609:
606:
603:
599:
595:
588:
577:
573:
559:
555:
550:
546:
538:
537:
536:
535:
515:
509:
505:
501:
498:
495:
492:
489:
480:
476:
473:
470:
457:
454:
451:
448:
444:
440:
435:
432:
427:
416:
412:
407:
404:
392:
388:
383:
380:
377:
373:
369:
362:
351:
347:
333:
329:
324:
320:
315:
307:
306:
305:
303:
299:
283:
280:
277:
254:
251:
248:
245:
242:
216:
213:
210:
207:
204:
193:
189:
185:
181:
177:
161:
158:
155:
152:
149:
129:
106:
103:
100:
97:
94:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
8302:at Wikiquote
8221:the original
8208:
8204:
8177:
8153:
8121:
8117:
8087:. Springer.
8083:
8063:
8032:
8012:, Springer,
8009:
7999:
7992:
7991:S. Kleiman:
7985:
7960:
7930:
7905:
7889:the original
7868:
7836:
7832:
7820:- J.S. Milne
7802:math/0604004
7791:
7721:Galois group
7713:Hodge theory
7706:
7701:
7687:
7683:
7679:
7650:
7646:
7622:
7620:
7571:
7563:
7559:
7555:
7553:
7548:
7544:
7540:
7532:
7530:
7520:
7517:Galois group
7418:
7349:full functor
7302:
7292:
7284:Conjecture D
7283:
7282:
7278:Hodge theory
7273:
7265:
7261:
7257:
7253:
7249:
7245:
7241:
7237:
7233:
7229:
7225:
7221:
7217:
7213:
7209:
7207:
7195:
7188:
7175:
7102:
7067:
7063:
7061:
6909:
6908:
6900:
6895:
6888:
6884:
6877:
6873:
6871:
6861:
6859:
6854:
6828:
6763:
6681:
6667:Galois group
6660:
6611:
6607:
6605:
6593:finite field
6586:
6575:
6560:
6558:
6548:
6544:
6536:
6525:
6523:
6518:
6515:vector space
6506:
6487:
6420:
6321:
6318:Tate motives
5941:
5843:
5749:
5685:
5588:
5450:
5306:
5227:
5180:
5114:
5054:
4925:
4911:
4903:Fields Medal
4894:
4892:
4884:
4880:
4876:
4874:
4814:
4808:
4801:
4742:
4649:
4645:
4643:
4633:
4631:
4606:
4562:
4559:
4404:
4322:
4247:
4243:
4237:
4235:
4158:
4101:
4097:
4093:
4089:
4087:
3954:
3946:
3940:
3934:
3923:
3916:
3909:
3902:
3895:
3888:
3881:
3879:
3616:
3609:
3602:
3595:
3591:
3584:
3577:
3570:
3566:
3559:
3552:
3545:
3538:
3531:
3524:
3517:
3510:
3503:
3496:
3489:
3487:
3335:
3200:
3079:
3077:
3072:
3068:
3064:
3059:
3056:
2860:
2855:
2851:
2847:
2843:
2839:
2837:
2561:
2557:
2553:
2549:
2547:
2426:
2420:
2237:
2233:
2229:
2225:
2215:
2211:
2207:
2203:
2133:
1952:
1882:
1879:
1582:
1544:
1421:
1417:
1413:
1383:
1296:
1292:
1288:
1254:
1163:
1129:
1123:
1115:
1077:
1064:
1057:
1038:
1027:
1024:Introduction
800:
698:
626:
531:
301:
179:
83:
49:, following
46:
42:
36:
8238:Musings on
8211:: 755–799.
8124:: 447–452,
7998:A. Scholl:
7737:Lie algebra
7554:which maps
6768:affine line
5942:Tate motive
4887:t-structure
4239:Tate motive
2241:). It is a
1224:Chow cycles
8310:Categories
7984:L. Breen:
7869:Astérisque
7775:References
7412:(rational
6519:cohomology
4879:back from
2554:idempotent
2134:Just like
1617:. For any
1060:CW-complex
8146:120799359
7915:1108.3787
7877:0303-1179
7845:0002-9920
7487:
7478:
7473:ℓ
7465:→
7458:ℓ
7385:→
7159:→
7009:
6788:∗
6766:with the
6732:×
6721:∗
6713:≅
6699:∗
6595:, and in
6543:group of
6494:morphisms
6434:⊕
6252:
6212:−
6203:−
6200:≥
6147:
6017:⊗
5922:−
5870:
5864:→
5796:
5787:→
5766:→
5724:×
5703:⊗
5552:−
5524:⊕
5468:∩
5427:→
5409:×
5284:→
5252:×
5246:⊂
5237:Γ
5090:×
5084:⊂
4772:
4767:∗
4713:→
4701:
4574:
4539:∘
4521:∘
4493:
4485:−
4474:∈
4443:→
4382:∈
4370:⊢
4346:
4340:∈
4289:
4264:
4246: :=
4210:×
4175:≅
4141:⊕
4057:×
4029:∈
4014:×
4002:λ
3992:λ
3858:γ
3852:∗
3843:π
3839:⋅
3830:γ
3824:∗
3815:π
3798:⊗
3772:β
3752:⊗
3740:β
3720:⊢
3708:α
3688:⊗
3676:α
3643:⊗
3530: : (
3495: : (
3467:×
3461:→
3452:×
3443:×
3434:×
3416:π
3399:×
3393:→
3384:×
3375:×
3366:×
3348:π
3315:β
3310:∗
3301:π
3297:⋅
3294:α
3289:∗
3280:π
3270:×
3255:β
3237:⊗
3231:α
3178:β
3172:α
3158:∐
3136:β
3118:⊕
3112:α
3031:×
3025:⊂
3016:Γ
3000:⟼
2979:Δ
2954:⟼
2933:
2903:⟶
2889:
2814:∘
2811:β
2799:α
2796:∘
2782:⊢
2758:β
2740:α
2725:
2695:α
2689:α
2686:∘
2683:α
2664:
2658:∈
2649:⊢
2640:α
2634:∣
2628:α
2596:
2578:
2520:
2484:
2442:
2387:∐
2376:×
2365:∐
2347:∗
2339:↪
2330:×
2319:∗
2311:⊕
2302:×
2291:∗
2283:∈
2277:β
2271:α
2262:β
2256:α
2232: :=
2210: :=
2180:
2145:
2106:Γ
2102:⟼
2089:⟼
2070:
2064:⟶
2052:
2009:→
1977:×
1971:⊆
1962:Γ
1929:
1894:
1847:
1828:∈
1817:β
1809:∗
1797:π
1793:⋅
1787:α
1779:∗
1767:π
1756:∗
1746:π
1739:α
1736:∘
1733:β
1692:
1679:∈
1676:β
1641:
1628:∈
1625:α
1602:⊢
1593:α
1521:×
1476:⨁
1445:
1395:∈
1356:
1317:∐
1266:
1237:×
1207:×
1181:→
1141:
1093:
947:ϕ
884:ϕ
851:¯
838:
770:
627:over the
506:ϕ
499:ϕ
485:∞
452:
281:−
159:⊢
8217:17160833
7862:"Motifs"
7743:See also
7637:and the
6625:integers
6572:= + +
6490:category
5813:→
5538:→
5508:′
5492:′
5479:→
5390:localize
5388:. If we
5137:-module
5071:integral
5069:call an
5055:Given a
4934:and let
4922:Notation
4885:motivic
4875:Getting
3945:to Chow(
3887: :
2850: :
1114:, where
1080:category
1054:= + +
876:-action
192:morphism
188:category
122:, where
61:such as
8126:Bibcode
8056:2242284
7979:1265518
7961:Motives
7949:2115000
7885:1144336
7853:2104916
7807:Bibcode
7274:weights
7203:Deligne
7101:is the
6898:) etc.
6006:denote
5977:we let
5064:variety
4811:Deligne
3622:. Then
3082:) is a
3057:where Δ
534:modules
184:integer
79:variety
43:motives
8215:
8185:
8144:
8091:
8070:
8054:
8044:
8016:
7977:
7967:
7947:
7937:
7883:
7875:
7851:
7843:
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8224:(PDF)
8213:S2CID
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7923:Books
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7797:arXiv
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