580:. It is important to note that the distinction between the global and local versions is that the former relates morphisms between complexes of sheaves in the derived categories, whereas the latter relates internal Hom-complexes and so can be evaluated locally. Taking global sections of both sides in the local statement gives the global Verdier duality.
2247:
571:
2015:
2445:
1095:
is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasi-isomorphic to a sheaf concentrated in a single degree. From this perspective the derived category is necessary in the study of singular spaces.
426:
90:
from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying
2679:
1339:
2000:
1886:
1599:
440:
2906:
2784:
1396:
2242:{\displaystyle \mathrm {Hom} ^{\bullet }(\Gamma _{c}(X;I_{X}^{\bullet }),k)=\cdots \to \Gamma _{c}(X;I_{X}^{2})^{\vee }\to \Gamma _{c}(X;I_{X}^{1})^{\vee }\to \Gamma _{c}(X;I_{X}^{0})^{\vee }\to 0}
692:
2262:
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315:
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154:
2514:
1153:
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184:
1660:
1453:
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971:
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where the last non-zero term is in degree 0 and the ones to the left are in negative degree. Morphisms in the derived category are obtained from the
1798:
1520:
566:{\displaystyle R\,{\mathcal {H}}om(Rf_{!}{\mathcal {F}},{\mathcal {G}})\cong Rf_{\ast }R\,{\mathcal {H}}om({\mathcal {F}},f^{!}{\mathcal {G}})}
64:
1792:
To understand how
Poincaré duality is obtained from this statement, it is perhaps easiest to understand both sides piece by piece. Let
2806:
2690:
1616:
can be derived as a special case of
Verdier duality. Here one explicitly calculates cohomology of a space using the machinery of
1351:
3090:
3098:
3038:
3007:
2982:
2960:
2253:
872:
The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring
2440:{\displaystyle \cong H^{0}(\mathrm {Hom} ^{\bullet }(\Gamma _{c}(X;I_{X}^{\bullet }),k))=H_{c}^{0}(X;k_{X})^{\vee }.}
108:
3063:
3022:
2999:
641:
1192:
3071:
156:
of locally compact
Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports
3129:
999:
2450:
For the other side of the
Verdier duality statement above, we have to take for granted the fact that when
2996:
Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et
Fonctions L - (SGA 5)
2009:. Since morphisms between complexes of sheaves (or vector spaces) themselves form a complex we find that
86:
Verdier duality generalises the classical
Poincaré duality of manifolds in two directions: it applies to
421:{\displaystyle RHom(Rf_{!}{\mathcal {F}},{\mathcal {G}})\cong RHom({\mathcal {F}},f^{!}{\mathcal {G}}).}
613:
111:
3139:
697:
17:
1891:
be an injective resolution of the constant sheaf. Then by standard facts on right derived functors
915:
268:
224:
127:
2464:
1075:
to a point. Part of what makes
Verdier duality interesting in the singular setting is that when
107:
Verdier duality states that (subject to suitable finiteness conditions discussed below) certain
2991:
60:
1122:
3134:
1702:
1184:
56:
2519:
which is the dualizing complex for a manifold. Now we can re-express the right hand side as
775:
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949:
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8:
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92:
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40:
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36:
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2674:{\displaystyle ]\cong H^{n}(\mathrm {Hom} ^{\bullet }(k_{X},k_{X}))=H^{n}(X;k_{X}).}
3055:
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2932:
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1617:
1156:
214:
115:
96:
76:
1334:{\displaystyle D({\mathcal {F}})=R\,{\mathcal {H}}om({\mathcal {F}},\omega _{X}).}
3104:
3044:
1995:{\displaystyle Rp_{!}k_{X}=p_{!}I_{X}^{\bullet }=\Gamma _{c}(X;I_{X}^{\bullet })}
75:. It is thus (together with the said Ă©tale theory and for example Grothendieck's
2922:
87:
80:
3030:
3118:
1881:{\displaystyle k_{X}\to (I_{X}^{\bullet }=I_{X}^{0}\to I_{X}^{1}\to \cdots )}
610:
having finite cohomological dimension. This is the case if there is a bound
2948:
866:
583:
These results hold subject to the compactly supported direct image functor
3013:, Exposés I and II contain the corresponding theory in the étale situation
1594:{\displaystyle D(Rf_{!}({\mathcal {F}}))\cong Rf_{\ast }D({\mathcal {F}})}
24:
3083:(1965), "Dualité dans la cohomologie des espaces localement compacts",
2005:
is a complex whose cohomology is the compactly supported cohomology of
32:
1696:
be the constant map to a point. Global
Verdier duality then states
44:
2998:, Lecture notes in mathematics, vol. 589, Berlin, New York:
2901:{\displaystyle H_{c}^{i}(X;k_{X})^{\vee }\cong H^{n-i}(X;k_{X}).}
221:, in other words, for (complexes of) sheaves (of abelian groups)
2256:
of sheaves by taking the zeroth cohomology of the complex, i.e.
2779:{\displaystyle H_{c}^{0}(X;k_{X})^{\vee }\cong H^{n}(X;k_{X}).}
2955:, Progress in Mathematics, Basel, Boston, Berlin: Birkhäuser,
1608:
1391:{\displaystyle D^{2}({\mathcal {F}})\cong {\mathcal {F}}}
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2693:
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1119:is a finite-dimensional locally compact space, and
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845:-dimensional manifolds or more generally at most
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2796:replaced with the same sheaf placed in degree
687:{\displaystyle H_{c}^{r}(X_{y},\mathbf {Z} )}
638:such that the compactly supported cohomology
2990:
2968:
2684:We finally have obtained the statement that
1246:{\displaystyle D\colon D^{b}(X)\to D^{b}(X)}
47:. Verdier duality was introduced in 1965 by
2789:By repeating this argument with the sheaf
1398:for sheaves with constructible cohomology.
1288:
912:-modules; the case above corresponds to
516:
447:
3079:
3016:
892:and (derived categories of) sheaves of
52:
3117:
2800:we get the classical Poincaré duality
1609:Relation to classical Poincaré duality
576:in the derived category of sheaves on
2947:
1041:{\displaystyle \omega _{X}=p^{!}(k),}
2254:homotopy category of chain complexes
65:Poincaré duality in étale cohomology
16:For duality over number fields, see
13:
3093:, pp. Exp. No. 300, 337–349,
3021:, Universitext, Berlin, New York:
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1583:
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1159:of sheaves of abelian groups over
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102:
57:locally compact topological spaces
14:
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1344:It has the following properties:
631:{\displaystyle d\in \mathbf {N} }
124:states that for a continuous map
79:) one instance of Grothendieck's
926:
677:
624:
1515:, then there is an isomorphism
798:. This holds if all the fibres
739:{\displaystyle X_{y}=f^{-1}(y)}
3091:Société Mathématique de France
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933:{\displaystyle A=\mathbf {Z} }
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282:{\displaystyle {\mathcal {G}}}
238:{\displaystyle {\mathcal {F}}}
149:{\displaystyle f\colon X\to Y}
140:
1:
2941:
2509:{\displaystyle p^{!}k=k_{X},}
7:
3058:; Schapira, Pierre (2002),
2911:
10:
3156:
1401:(Intertwining of functors
118:. There are two versions.
112:image functors for sheaves
15:
3031:10.1007/978-3-642-82783-9
1662:is the constant sheaf on
1475:is a continuous map from
3017:Iversen, Birger (1986),
2454:is a compact orientable
1627:is a compact orientable
1148:{\displaystyle D^{b}(X)}
694:vanishes for all fibres
2992:Grothendieck, Alexandre
2953:Intersection cohomology
2458:-dimensional manifold
1782:{\displaystyle \cong .}
1631:-dimensional manifold,
3089:, vol. 9, Paris:
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1996:
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987:
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906:
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859:
839:
819:
792:
791:{\displaystyle r>d}
766:
765:{\displaystyle y\in Y}
740:
688:
632:
604:
567:
422:
303:
283:
259:
239:
207:
180:
179:{\displaystyle Rf_{!}}
150:
122:Global Verdier duality
61:Alexander Grothendieck
49:Jean-Louis Verdier
3019:Cohomology of sheaves
2971:Manin, Yuri Ivanovich
2903:
2781:
2676:
2511:
2442:
2244:
1997:
1883:
1784:
1691:
1666:with coefficients in
1657:
1655:{\displaystyle k_{X}}
1596:
1510:
1490:
1470:
1450:
1448:{\displaystyle f_{!}}
1423:
1421:{\displaystyle f_{*}}
1393:
1336:
1248:
1185:contravariant functor
1174:
1150:
1114:
1090:
1070:
1043:
988:
968:
966:{\displaystyle D_{X}}
935:
907:
887:
860:
840:
820:
818:{\displaystyle X_{y}}
793:
767:
741:
689:
633:
605:
603:{\displaystyle f_{!}}
568:
432:Local Verdier duality
423:
304:
284:
260:
240:
208:
206:{\displaystyle f^{!}}
181:
151:
18:Artin–Verdier duality
3060:Sheaves on Manifolds
3002:, pp. xii+484,
2977:, Berlin: Springer,
2969:Gelfand, Sergei I.;
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2016:
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186:has a right adjoint
160:
128:
3130:Homological algebra
3081:Verdier, Jean-Louis
2975:Homological algebra
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2708:
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2371:
2222:
2175:
2128:
2073:
1988:
1951:
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1832:
1689:{\displaystyle f=p}
659:
55:) as an analog for
3086:SĂ©minaire Bourbaki
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73:algebraic geometry
37:algebraic topology
3100:978-2-85629-042-2
3056:Kashiwara, Masaki
3040:978-3-540-16389-3
3009:978-3-540-08248-4
2984:978-3-540-65378-3
2962:978-0-8176-3274-8
1508:{\displaystyle Y}
1488:{\displaystyle X}
1468:{\displaystyle f}
1172:{\displaystyle X}
1112:{\displaystyle X}
1088:{\displaystyle X}
1068:{\displaystyle X}
993:is defined to be
986:{\displaystyle X}
945:dualizing complex
905:{\displaystyle A}
885:{\displaystyle A}
858:{\displaystyle d}
838:{\displaystyle d}
302:{\displaystyle Y}
258:{\displaystyle X}
39:that generalizes
3147:
3140:Duality theories
3111:
3076:
3051:
3012:
2987:
2965:
2933:Derived category
2928:Coherent duality
2918:Poincaré duality
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1614:Poincaré duality
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116:adjoint functors
97:perverse sheaves
77:coherent duality
41:Poincaré duality
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3023:Springer-Verlag
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3000:Springer-Verlag
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1806:
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1797:
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1704:
1701:
1700:
1675:
1672:
1671:
1646:
1642:
1640:
1637:
1636:
1635:is a field and
1611:
1606:
1582:
1581:
1569:
1565:
1547:
1546:
1537:
1533:
1522:
1519:
1518:
1500:
1497:
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1480:
1477:
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1412:
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1382:
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1369:
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1355:
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1349:
1319:
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1290:
1289:
1273:
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1264:
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1228:
1224:
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1194:
1191:
1190:
1164:
1161:
1160:
1130:
1126:
1124:
1121:
1120:
1104:
1101:
1100:
1080:
1077:
1076:
1060:
1057:
1056:
1020:
1016:
1007:
1003:
1001:
998:
997:
978:
975:
974:
957:
953:
951:
948:
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917:
914:
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893:
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874:
873:
850:
847:
846:
830:
827:
826:
809:
805:
803:
800:
799:
777:
774:
773:
751:
748:
747:
718:
714:
705:
701:
699:
696:
695:
676:
667:
663:
654:
649:
643:
640:
639:
623:
615:
612:
611:
594:
590:
588:
585:
584:
554:
553:
547:
543:
534:
533:
518:
517:
507:
503:
488:
487:
478:
477:
471:
467:
449:
448:
442:
439:
438:
406:
405:
399:
395:
386:
385:
358:
357:
348:
347:
341:
337:
317:
314:
313:
294:
291:
290:
273:
272:
270:
267:
266:
250:
247:
246:
229:
228:
226:
223:
222:
197:
193:
191:
188:
187:
170:
166:
161:
158:
157:
129:
126:
125:
105:
103:Verdier duality
88:continuous maps
29:Verdier duality
21:
12:
11:
5:
3153:
3143:
3142:
3137:
3132:
3127:
3113:
3112:
3099:
3077:
3072:
3052:
3039:
3014:
3008:
2988:
2983:
2966:
2961:
2943:
2940:
2936:
2935:
2930:
2925:
2923:Six operations
2920:
2913:
2910:
2909:
2908:
2897:
2894:
2889:
2885:
2881:
2878:
2875:
2870:
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2275:
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2268:
2250:
2249:
2238:
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2215:
2211:
2207:
2204:
2201:
2196:
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2110:
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2102:
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2062:
2058:
2055:
2052:
2047:
2043:
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2034:
2029:
2026:
2023:
2003:
2002:
1991:
1986:
1981:
1977:
1973:
1970:
1967:
1962:
1958:
1954:
1949:
1944:
1940:
1934:
1930:
1926:
1921:
1917:
1911:
1907:
1903:
1889:
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956:
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921:
901:
881:
854:
834:
812:
808:
787:
784:
781:
761:
758:
755:
735:
732:
729:
724:
721:
717:
713:
708:
704:
683:
679:
675:
670:
666:
662:
657:
652:
648:
626:
622:
619:
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593:
574:
573:
562:
557:
550:
546:
542:
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529:
526:
521:
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510:
506:
502:
499:
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491:
486:
481:
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446:
429:
428:
417:
414:
409:
402:
398:
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389:
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366:
361:
356:
351:
344:
340:
336:
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330:
327:
324:
321:
298:
276:
254:
232:
200:
196:
173:
169:
165:
145:
142:
139:
136:
133:
104:
101:
81:six operations
9:
6:
4:
3:
2:
3152:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3122:
3120:
3110:
3106:
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3096:
3092:
3088:
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3082:
3078:
3075:
3069:
3065:
3061:
3057:
3053:
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3042:
3036:
3032:
3028:
3024:
3020:
3015:
3011:
3005:
3001:
2997:
2993:
2989:
2986:
2980:
2976:
2972:
2967:
2964:
2958:
2954:
2950:
2949:Borel, Armand
2946:
2945:
2939:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2915:
2895:
2887:
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2803:
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2704:
2699:
2695:
2687:
2686:
2685:
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2652:
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2641:
2637:
2633:
2622:
2618:
2614:
2609:
2605:
2596:
2575:
2571:
2567:
2558:
2550:
2546:
2542:
2537:
2533:
2522:
2521:
2520:
2503:
2497:
2489:
2485:
2481:
2478:
2473:
2469:
2461:
2460:
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2457:
2453:
2434:
2429:
2419:
2415:
2411:
2408:
2400:
2395:
2391:
2387:
2378:
2375:
2367:
2362:
2358:
2354:
2351:
2343:
2330:
2309:
2305:
2301:
2295:
2292:
2287:
2283:
2277:
2273:
2269:
2259:
2258:
2257:
2255:
2236:
2228:
2218:
2213:
2209:
2205:
2202:
2194:
2181:
2171:
2166:
2162:
2158:
2155:
2147:
2134:
2124:
2119:
2115:
2111:
2108:
2100:
2089:
2086:
2080:
2077:
2069:
2064:
2060:
2056:
2053:
2045:
2032:
2012:
2011:
2010:
2008:
1984:
1979:
1975:
1971:
1968:
1960:
1952:
1947:
1942:
1938:
1932:
1928:
1924:
1919:
1915:
1909:
1905:
1901:
1894:
1893:
1892:
1872:
1864:
1859:
1855:
1846:
1841:
1837:
1833:
1828:
1823:
1819:
1807:
1803:
1795:
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1793:
1776:
1770:
1765:
1761:
1757:
1752:
1748:
1741:
1735:
1732:
1727:
1723:
1717:
1713:
1709:
1699:
1698:
1697:
1683:
1680:
1677:
1669:
1665:
1647:
1643:
1634:
1630:
1626:
1621:
1619:
1615:
1575:
1570:
1566:
1562:
1559:
1538:
1534:
1530:
1524:
1517:
1516:
1502:
1482:
1462:
1440:
1436:
1413:
1409:
1400:
1378:
1360:
1356:
1348:
1347:
1345:
1328:
1320:
1316:
1312:
1299:
1296:
1285:
1282:
1266:
1259:
1258:
1257:
1237:
1229:
1225:
1215:
1207:
1203:
1199:
1196:
1189:
1188:
1187:
1186:
1182:
1166:
1158:
1139:
1131:
1127:
1106:
1097:
1082:
1062:
1054:
1035:
1029:
1021:
1017:
1013:
1008:
1004:
996:
995:
994:
980:
958:
954:
946:
941:
922:
919:
899:
879:
870:
868:
865:-dimensional
852:
832:
810:
806:
785:
782:
779:
759:
756:
753:
730:
722:
719:
715:
711:
706:
702:
673:
668:
664:
655:
650:
646:
620:
617:
595:
591:
581:
579:
548:
544:
540:
527:
524:
513:
508:
504:
500:
497:
484:
472:
468:
464:
458:
455:
444:
437:
436:
435:
433:
415:
400:
396:
392:
379:
376:
373:
370:
367:
354:
342:
338:
334:
328:
325:
322:
319:
312:
311:
310:
296:
252:
220:
216:
198:
194:
171:
167:
163:
143:
137:
134:
131:
123:
119:
117:
114:are actually
113:
110:
100:
98:
94:
93:constructible
89:
84:
82:
78:
74:
70:
66:
63:'s theory of
62:
58:
54:
50:
46:
42:
38:
34:
33:cohomological
30:
26:
19:
3135:Sheaf theory
3085:
3059:
3018:
2995:
2974:
2952:
2937:
2797:
2790:
2788:
2683:
2518:
2455:
2451:
2449:
2251:
2006:
2004:
1890:
1791:
1667:
1663:
1632:
1628:
1624:
1622:
1612:
1343:
1255:
1181:Verdier dual
1180:
1155:the bounded
1098:
1052:
1050:
944:
942:
871:
867:CW-complexes
825:are at most
582:
577:
575:
434:states that
431:
430:
121:
120:
106:
85:
28:
22:
1256:defined by
1179:, then the
83:formalism.
35:duality in
25:mathematics
3119:Categories
3073:3540518614
3062:, Berlin:
2942:References
2866:−
2855:≅
2850:∨
2739:≅
2734:∨
2597:∙
2568:≅
2430:∨
2368:∙
2340:Γ
2331:∙
2302:≅
2234:→
2229:∨
2191:Γ
2187:→
2182:∨
2144:Γ
2140:→
2135:∨
2097:Γ
2093:→
2090:⋯
2070:∙
2042:Γ
2033:∙
1985:∙
1957:Γ
1948:∙
1873:⋯
1870:→
1852:→
1829:∙
1813:→
1742:≅
1571:∗
1560:≅
1414:∗
1379:≅
1317:ω
1222:→
1200::
1005:ω
757:∈
720:−
621:∈
509:∗
498:≅
368:≅
309:we have
141:→
135::
45:manifolds
3125:Topology
3064:Springer
2994:(1977),
2973:(1999),
2951:(1984),
2912:See also
1623:Suppose
3109:1610971
3049:0842190
746:(where
219:sheaves
213:in the
109:derived
69:schemes
51: (
3107:
3097:
3070:
3047:
3037:
3006:
2981:
2959:
1670:. Let
1455:). If
1051:where
772:) and
1183:is a
31:is a
3095:ISBN
3068:ISBN
3035:ISBN
3004:ISBN
2979:ISBN
2957:ISBN
1428:and
943:The
783:>
265:and
217:of
67:for
59:of
53:1965
43:for
3027:doi
1495:to
1099:If
973:on
940:.
869:.
289:on
245:on
95:or
71:in
23:In
3121::
3105:MR
3103:,
3066:,
3045:MR
3043:,
3033:,
3025:,
1620:.
99:.
27:,
3029::
2896:.
2893:)
2888:X
2884:k
2880:;
2877:X
2874:(
2869:i
2863:n
2859:H
2846:)
2840:X
2836:k
2832:;
2829:X
2826:(
2821:i
2816:c
2812:H
2798:i
2794:X
2791:k
2774:.
2771:)
2766:X
2762:k
2758:;
2755:X
2752:(
2747:n
2743:H
2730:)
2724:X
2720:k
2716:;
2713:X
2710:(
2705:0
2700:c
2696:H
2669:.
2666:)
2661:X
2657:k
2653:;
2650:X
2647:(
2642:n
2638:H
2634:=
2631:)
2628:)
2623:X
2619:k
2615:,
2610:X
2606:k
2602:(
2592:m
2589:o
2586:H
2581:(
2576:n
2572:H
2565:]
2562:]
2559:n
2556:[
2551:X
2547:k
2543:,
2538:X
2534:k
2530:[
2504:,
2501:]
2498:n
2495:[
2490:X
2486:k
2482:=
2479:k
2474:!
2470:p
2456:n
2452:X
2435:.
2426:)
2420:X
2416:k
2412:;
2409:X
2406:(
2401:0
2396:c
2392:H
2388:=
2385:)
2382:)
2379:k
2376:,
2373:)
2363:X
2359:I
2355:;
2352:X
2349:(
2344:c
2336:(
2326:m
2323:o
2320:H
2315:(
2310:0
2306:H
2299:]
2296:k
2293:,
2288:X
2284:k
2278:!
2274:p
2270:R
2267:[
2237:0
2225:)
2219:0
2214:X
2210:I
2206:;
2203:X
2200:(
2195:c
2178:)
2172:1
2167:X
2163:I
2159:;
2156:X
2153:(
2148:c
2131:)
2125:2
2120:X
2116:I
2112:;
2109:X
2106:(
2101:c
2087:=
2084:)
2081:k
2078:,
2075:)
2065:X
2061:I
2057:;
2054:X
2051:(
2046:c
2038:(
2028:m
2025:o
2022:H
2007:X
1990:)
1980:X
1976:I
1972:;
1969:X
1966:(
1961:c
1953:=
1943:X
1939:I
1933:!
1929:p
1925:=
1920:X
1916:k
1910:!
1906:p
1902:R
1876:)
1865:1
1860:X
1856:I
1847:0
1842:X
1838:I
1834:=
1824:X
1820:I
1816:(
1808:X
1804:k
1777:.
1774:]
1771:k
1766:!
1762:p
1758:,
1753:X
1749:k
1745:[
1739:]
1736:k
1733:,
1728:X
1724:k
1718:!
1714:p
1710:R
1707:[
1684:p
1681:=
1678:f
1668:k
1664:X
1648:X
1644:k
1633:k
1629:n
1625:X
1601:.
1589:)
1584:F
1579:(
1576:D
1567:f
1563:R
1557:)
1554:)
1549:F
1544:(
1539:!
1535:f
1531:R
1528:(
1525:D
1503:Y
1483:X
1463:f
1441:!
1437:f
1410:f
1384:F
1376:)
1371:F
1366:(
1361:2
1357:D
1329:.
1326:)
1321:X
1313:,
1308:F
1303:(
1300:m
1297:o
1292:H
1286:R
1283:=
1280:)
1275:F
1270:(
1267:D
1241:)
1238:X
1235:(
1230:b
1226:D
1219:)
1216:X
1213:(
1208:b
1204:D
1197:D
1167:X
1143:)
1140:X
1137:(
1132:b
1128:D
1107:X
1083:X
1063:X
1053:p
1036:,
1033:)
1030:k
1027:(
1022:!
1018:p
1014:=
1009:X
981:X
959:X
955:D
927:Z
923:=
920:A
900:A
880:A
853:d
833:d
811:y
807:X
786:d
780:r
760:Y
754:y
734:)
731:y
728:(
723:1
716:f
712:=
707:y
703:X
682:)
678:Z
674:,
669:y
665:X
661:(
656:r
651:c
647:H
625:N
618:d
596:!
592:f
578:Y
561:)
556:G
549:!
545:f
541:,
536:F
531:(
528:m
525:o
520:H
514:R
505:f
501:R
495:)
490:G
485:,
480:F
473:!
469:f
465:R
462:(
459:m
456:o
451:H
445:R
416:.
413:)
408:G
401:!
397:f
393:,
388:F
383:(
380:m
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374:H
371:R
365:)
360:G
355:,
350:F
343:!
339:f
335:R
332:(
329:m
326:o
323:H
320:R
297:Y
275:G
253:X
231:F
199:!
195:f
172:!
168:f
164:R
144:Y
138:X
132:f
20:.
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