4129:
3474:. A theorem of Serpé, generalizing work of Grothendieck and of Spaltenstein, asserts that in a Grothendieck abelian category, every complex is quasi-isomorphic to a K-injective complex with injective terms, and moreover, this is functorial. In particular, we may define morphisms in the derived category by passing to K-injective resolutions and computing morphisms in the homotopy category. The functoriality of Serpé's construction ensures that composition of morphisms is well-defined. Like the construction using roofs, this construction also ensures suitable set theoretic properties for the derived category, this time because these properties are already satisfied by the homotopy category.
3807:
4124:{\displaystyle {\begin{matrix}0&\to &{\mathcal {E}}_{n}&\to &0&\to &\cdots &\to &0&\to &0\\\uparrow &&\uparrow &&\uparrow &&\cdots &&\uparrow &&\uparrow \\0&\to &{\mathcal {E}}_{n}&\to &{\mathcal {E}}_{n-1}&\to &\cdots &\to &{\mathcal {E}}_{1}&\to &0\\\downarrow &&\downarrow &&\downarrow &&\cdots &&\downarrow &&\downarrow \\0&\to &0&\to &0&\to &\cdots &\to &{\mathcal {E}}_{0}&\to &0\end{matrix}}}
3718:
3144:
is a
Grothendieck abelian category (meaning that it satisfies AB5 and has a set of generators), with the essential point being that only objects of bounded cardinality are relevant. In these cases, the limit may be calculated over a small subcategory, and this ensures that the result is a set. Then
2413:
is small, however, the construction by generators and relations generally results in a category whose structure is opaque, where morphisms are arbitrarily long paths subject to a mysterious equivalence relation. For this reason, it is conventional to construct the derived category more concretely
2385:
has a proper class of objects, all of which are isomorphic, then there is a proper class of paths between any two of these objects. The generators and relations construction therefore only guarantees that the morphisms between two objects form a proper class. However, the morphisms between two
3201:
There is a different approach based on replacing morphisms in the derived category by morphisms in the homotopy category. A morphism in the derived category with codomain being a bounded below complex of injective objects is the same as a morphism to this complex in the homotopy category; this
5327:. As remarked above, injective resolutions are not uniquely defined, but it is a fact that any two resolutions are homotopy equivalent to each other, i.e. isomorphic in the homotopy category. Moreover, morphisms of complexes extend uniquely to a morphism of two given injective resolutions.
183:
condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by
Grothendieck, this signified a need to reformulate. With it came the idea that the 'real'
5153:
In concrete situations, it is very difficult or impossible to handle morphisms in the derived category directly. Therefore, one looks for a more manageable category which is equivalent to the derived category. Classically, there are two (dual) approaches to this: projective and
3556:
545:
3077:
104:
shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides. The basic theory of
Verdier was written down in his dissertation, published finally in 1996 in
5948:
918:
4614:), and hence has only a set of morphisms from one object to another. Grothendieck abelian categories include the category of modules over a ring, the category of sheaves of abelian groups on a topological space, and many other examples.
5074:
4617:
Composition of morphisms, i.e. roofs, in the derived category is accomplished by finding a third roof on top of the two roofs to be composed. It may be checked that this is possible and gives a well-defined, associative composition.
715:
5620:) which immediately proved useful in the study of singular spaces; see, for example, the book by Kashiwara and Schapira (Categories and Sheaves) on various applications of unbounded derived category. Spaltenstein used so-called
5433:
5306:
3339:
2333:
is a small category, then there is a direct construction of the derived category by formally adjoining inverses of quasi-isomorphisms. This is an instance of the general construction of a category by generators and relations.
1154:
1313:
4351:
1389:
2454:. This is a collection of conditions that allow complicated paths to be rewritten as simpler ones. The Gabriel–Zisman theorem implies that localization at a multiplicative system has a simple description in terms of
2223:
1888:
5581:
4887:
4250:
3713:{\displaystyle 0\to {\mathcal {E}}_{n}{\overset {\phi _{n,n-1}}{\rightarrow }}{\mathcal {E}}_{n-1}{\overset {\phi _{n-1,n-2}}{\rightarrow }}\cdots {\overset {\phi _{1,0}}{\rightarrow }}{\mathcal {E}}_{0}\to 0}
5148:
3799:
1053:
2680:
2634:
2168:
1988:
1938:
1676:
1452:
413:
349:
421:
6119:
3517:
817:
766:
5879:
3406:
2962:
2496:
1774:
1640:
971:
5445:
It is not difficult to see that this functor is actually inverse to the restriction of the canonical localization functor mentioned in the beginning. In other words, morphisms Hom(
4789:
3196:
4163:
622:
3472:
3439:
3202:
follows from termwise injectivity. By replacing termwise injectivity by a stronger condition, one gets a similar property that applies even to unbounded complexes. A complex
3176:
2900:
2813:
2529:
2448:
2276:
2132:
1252:
313:
3118:
2780:
1488:
5634:
and May (2006) describe the derived category of modules over DG-algebras. Keller also gives applications to Koszul duality, Lie algebra cohomology, and
Hochschild homology.
4712:
3142:
2411:
2383:
2359:
2331:
1715:
1512:
1337:
1209:
576:
377:
264:
110:
2716:
3548:
3366:
3261:
3227:
2954:
2927:
2867:
2840:
2746:
2588:
2094:
2067:
1416:
1181:
822:
2225:
which is the identity on objects and which sends each morphism to its chain homotopy equivalence class. Since every chain homotopy equivalence is a quasi-isomorphism,
4561:
2040:
2014:
4525:
4462:
4399:
4488:
4425:
5092:, so the second step in the above construction may be omitted. The definition is usually given in this way because it reveals the existence of a canonical functor
2721:
Replacing chains of morphisms with roofs also enables the resolution of the set-theoretic issues involved in derived categories of large categories. Fix a complex
2561:
4973:
4183:
2243:
1600:
1580:
1552:
1532:
631:
5355:
5249:
3266:
1058:
2361:
is a large category, this construction does not work for set theoretic reasons. This construction builds morphisms as equivalence classes of paths. If
1257:
4258:
1342:
5165:
In the following we will describe the role of injective resolutions in the context of the derived category, which is the basis for defining right
2170:
but whose morphisms are equivalence classes of morphisms of complexes with respect to the relation of chain homotopy. There is a natural functor
2173:
1779:
5479:
4848:
4188:
203:
Despite the level of abstraction, derived categories became accepted over the following decades, especially as a convenient setting for
6232:
5098:
3726:
6268:
89:. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated
5838:), where the first equivalence of categories is described above. The classical derived functors are related to the total one via
5438:
from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in
976:
6755:
5461:
and computing the morphisms in the homotopy category, which is at least theoretically easier. In fact, it is enough to resolve
6586:
125:. The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's
6693:
6664:
6631:
6293:
1894:. It is straightforward to show that two homotopic morphisms induce identical morphisms on cohomology groups. We say that
540:{\displaystyle \cdots \to X^{-1}\xrightarrow {d^{-1}} X^{0}\xrightarrow {d^{0}} X^{1}\xrightarrow {d^{1}} X^{2}\to \cdots ,}
6227:
5085:
2098:
2639:
2593:
2137:
1947:
1897:
1645:
1421:
382:
318:
3801:
by truncating the complex above, shifting it, and using the obvious morphisms above. In particular, we have the picture
3120:
is potentially a large category, in some cases it is controlled by a small category. This is the case, for example, if
6731:
5943:{\displaystyle {\mathcal {A}}{\stackrel {F}{\rightarrow }}{\mathcal {B}}{\stackrel {G}{\rightarrow }}{\mathcal {C}},\,}
4835:
208:
6078:
6470:
6147:
with two vertices. They are very different abelian categories, but their (bounded) derived categories are equivalent.
3072:{\displaystyle \varinjlim _{I_{X^{\bullet }}}\operatorname {Hom} _{K({\mathcal {A}})}((X')^{\bullet },Y^{\bullet }),}
2869:
whose structure maps are quasi-isomorphisms. Then the multiplicative system condition implies that the morphisms in
2386:
objects in a category are usually required to be sets, and so this construction fails to produce an actual category.
771:
720:
3371:
2461:
1720:
1605:
3485:
146:
6412:
923:
5870:
6444:
6750:
4723:
3181:
85:
of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of
6439:
6210:
5760:
5158:. In both cases, the restriction of the above canonical functor to an appropriate subcategory will be an
5637:
More generally, carefully adapting the definitions, it is possible to define the derived category of an
4185:, the only non-trivial upward arrow is the equality morphism, and the only-nontrivial downward arrow is
4137:
581:
6247:
3550:
is a quasi-isomorphism. To get a better picture of what elements look like, consider an exact sequence
3444:
3411:
3148:
2872:
2785:
2501:
2420:
2248:
2104:
1224:
1215:
285:
223:
118:
6434:
4356:
in the derived category. One application of this observation is the construction of the Atiyah-class.
3089:
2751:
6237:
5159:
2417:
These other constructions go through the homotopy category. The collection of quasi-isomorphisms in
1457:
913:{\displaystyle f^{\bullet }\colon (X^{\bullet },d_{X}^{\bullet })\to (Y^{\bullet },d_{Y}^{\bullet })}
4659:
4586:), then one has to give an additional argument to prove this. If, for example, the abelian category
3123:
2392:
2364:
2340:
2312:
1688:
1493:
1318:
1190:
557:
358:
245:
2685:
54:
6330:
Markarian, Nikita (2009). "The Atiyah class, Hochschild cohomology and the
Riemann-Roch theorem".
5676:
left derived functors come from right exact functors and are calculated via projective resolutions
3344:
3239:
3205:
2932:
2905:
2845:
2842:
and whose morphisms are commutative diagrams. Equivalently, this is the category of objects over
2818:
2724:
2566:
2072:
2045:
1394:
1159:
5721:
5673:
right derived functors come from left exact functors and are calculated via injective resolutions
180:
6462:
6456:
6282:
Gabriel, Peter; Zisman, M. (6 December 2012). "1.2 The
Calculus of Fractions: Proposition 2.4".
5869:
Derived categories are, in a sense, the "right" place to study these functors. For example, the
4563:) complexes instead of unbounded ones. The corresponding derived categories are usually denoted
169:
122:
97:
82:
5315:* are injective objects. This idea generalizes to give resolutions of bounded-below complexes
5211:. (Neither the map nor the injective object has to be uniquely specified.) For example, every
5192:
In order to apply this technique, one has to assume that the abelian category in question has
4530:
3482:
As noted before, in the derived category the hom sets are expressed through roofs, or valleys
2019:
1993:
6068:
5212:
4626:
4595:
4497:
4434:
4371:
3522:
231:
141:. Recently derived categories have also become important in areas nearer to physics, such as
114:
6283:
4467:
4404:
6703:
6674:
6641:
6615:
6602:
6567:
6518:
6480:
6198:
6155:
5729:
5155:
5069:{\displaystyle {\text{Hom}}_{D({\mathcal {A}})}(X,Y)={\text{Ext}}_{\mathcal {A}}^{j}(X,Y).}
2534:
271:
230:. The homotopy category of spectra and the derived category of a ring are both examples of
172:
8:
6529:
6130:
5725:
5174:
279:
138:
42:
6726:. Cambridge Studies in Advanced Mathematics. Vol. 183. Cambridge University Press.
6213:. Varieties with equivalent derived categories of coherent sheaves are sometimes called
5182:
710:{\displaystyle H^{i}(X^{\bullet })=\operatorname {ker} d^{i}/\operatorname {im} d^{i-1}}
78:
6688:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
6574:
6357:
6339:
5330:
This is the point where the homotopy category comes into play again: mapping an object
4590:
is small, i.e. has only a set of objects, then this issue will be no problem. Also, if
4583:
4168:
2228:
1585:
1565:
1537:
1517:
275:
130:
101:
5428:{\displaystyle D^{+}({\mathcal {A}})\rightarrow K^{+}(\mathrm {Inj} ({\mathcal {A}}))}
5301:{\displaystyle 0\rightarrow X\rightarrow I^{0}\rightarrow I^{1}\rightarrow \cdots ,\,}
6727:
6707:
6689:
6660:
6627:
6590:
6555:
6506:
6466:
6289:
5593:
5178:
5089:
4579:
3334:{\displaystyle \operatorname {Hom} _{K({\mathcal {A}})}(X^{\bullet },I^{\bullet })=0}
90:
6361:
1218:
of the category of complexes with respect to quasi-isomorphisms. Specifically, the
1149:{\displaystyle H^{i}(f^{\bullet })\colon H^{i}(X^{\bullet })\to H^{i}(Y^{\bullet })}
6648:
6496:
6349:
6242:
6144:
5958:
5745:
5205:
5186:
2682:
is a chain homotopy equivalence class of morphisms. Conceptually, this represents
267:
204:
126:
35:
6719:
6699:
6670:
6656:
6637:
6623:
6598:
6563:
6543:
6514:
6476:
6452:
6190:
6126:
5654:
5166:
1308:{\displaystyle Q\colon \operatorname {Kom} ({\mathcal {A}})\to D({\mathcal {A}})}
227:
215:
school adopted the language of derived categories, and the subsequent history of
46:
6681:
6122:
5638:
5240:
4346:{\displaystyle \phi \in \mathbf {RHom} ({\mathcal {E}}_{0},{\mathcal {E}}_{n})}
2282:
185:
161:
6019:
J.-L. Verdier showed how derived functors associated with an abelian category
1384:{\displaystyle F\colon \operatorname {Kom} ({\mathcal {A}})\to {\mathcal {C}}}
192:
functors would be those existing on the derived level; with respect to those,
6744:
6594:
6559:
6510:
6024:
352:
165:
86:
66:
6711:
6538:
gives an interpretation of the derived category of modules over DG-algebras.
6525:
6166:
by reversing some arrows. In general, the categories of representations of
5733:
5201:
129:
theory. Derived categories have since become indispensable also outside of
6353:
5680:
In the following we will describe right derived functors. So, assume that
6064:
5602:
4948:
4944:
4578:
If one adopts the classical point of view on categories, that there is a
2218:{\displaystyle \operatorname {Kom} ({\mathcal {A}})\to K({\mathcal {A}})}
1883:{\displaystyle f^{i}-g^{i}=d_{Y}^{i-1}\circ h^{i}+h^{i+1}\circ d_{X}^{i}}
197:
193:
74:
20:
6501:
5992:), is an expression of the following identity of total derived functors
2309:
There are several possible constructions of the derived category. When
2293:) is the true 'homotopy category' of the category of complexes, whereas
5170:
2278:
can be equally well viewed as a localization of the homotopy category.
212:
5576:{\displaystyle \mathrm {Hom} _{D(A)}(X,Y)=\mathrm {Hom} _{K(A)}(X,Y).}
152:
Unbounded derived categories were introduced by
Spaltenstein in 1988.
45:
introduced to refine and in a certain sense to simplify the theory of
6344:
4882:{\displaystyle 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0}
1554:. Any two categories having this universal property are equivalent.
4245:{\displaystyle \phi _{1,0}:{\mathcal {E}}_{1}\to {\mathcal {E}}_{0}}
3341:. A straightforward consequence of this is that, for every complex
505:
478:
448:
6655:, Grundlehren der mathematischen Wissenschaften, Berlin, New York:
5779:
The derived category allows us to encapsulate all derived functors
5224:
216:
175:, the need to take a whole complex of sheaves in place of a single
134:
4947:. Morphisms in the derived category include information about all
5862:
forget the chain complex and keep only the cohomologies, whereas
5669:
be a functor of abelian categories. There are two dual concepts:
5347:
3083:
282:
of abelian groups on a topological space.) The derived category
142:
5653:
The derived category is a natural framework to define and study
5613:, one can use projective resolutions instead of injective ones.
315:
is defined by a universal property with respect to the category
211:
in dimensions greater than 1 in derived terms, around 1980. The
5143:{\displaystyle K({\mathcal {A}})\rightarrow D({\mathcal {A}}).}
4935:
as a complex concentrated in degree zero, the derived category
4606:) is equivalent to a full subcategory of the homotopy category
1214:
The universal property of the derived category is that it is a
3794:{\displaystyle \phi :{\mathcal {E}}_{0}\to {\mathcal {E}}_{n}}
6577:(1996), "Des Catégories Dérivées des Catégories Abéliennes",
5616:
In 1988 Spaltenstein defined an unbounded derived category (
5453:) in the derived category may be computed by resolving both
2718:. Two roofs are equivalent if they have a common overroof.
1048:{\displaystyle f_{i+1}\circ d_{X}^{i}=d_{Y}^{i}\circ f_{i}}
1055:. Such a morphism induces morphisms on cohomology groups
207:. Perhaps the biggest advance was the formulation of the
113:). The axiomatics required an innovation, the concept of
164:
theory, pushing to the limit of what could be done with
5079:
6143:-Rep be an abelian category of representations of the
3812:
6081:
6055:) are. Often this is an interesting relation between
5882:
5482:
5358:
5252:
5101:
4976:
4916:. Verdier explained that the definition of the shift
4851:
4726:
4662:
4533:
4500:
4470:
4437:
4407:
4374:
4261:
4191:
4171:
4140:
3810:
3729:
3559:
3525:
3488:
3447:
3414:
3374:
3347:
3269:
3242:
3208:
3184:
3151:
3126:
3092:
2965:
2935:
2908:
2875:
2848:
2821:
2788:
2754:
2727:
2688:
2642:
2596:
2569:
2537:
2504:
2464:
2423:
2395:
2367:
2343:
2315:
2251:
2231:
2176:
2140:
2107:
2075:
2048:
2022:
1996:
1950:
1900:
1782:
1723:
1691:
1648:
1608:
1588:
1568:
1540:
1520:
1496:
1460:
1424:
1397:
1345:
1321:
1260:
1227:
1193:
1162:
1061:
979:
926:
825:
774:
723:
634:
584:
560:
424:
385:
361:
321:
288:
248:
6609:
Four textbooks that discuss derived categories are:
4582:
of morphisms from one object to another (not just a
2675:{\displaystyle f\colon Z^{\bullet }\to Y^{\bullet }}
2629:{\displaystyle s\colon Z^{\bullet }\to X^{\bullet }}
2304:
2163:{\displaystyle \operatorname {Kom} ({\mathcal {A}})}
1983:{\displaystyle g\colon Y^{\bullet }\to X^{\bullet }}
1933:{\displaystyle f\colon X^{\bullet }\to Y^{\bullet }}
1671:{\displaystyle \operatorname {Kom} ({\mathcal {A}})}
1557:
1447:{\displaystyle \operatorname {Kom} ({\mathcal {A}})}
408:{\displaystyle \operatorname {Kom} ({\mathcal {A}})}
344:{\displaystyle \operatorname {Kom} ({\mathcal {A}})}
6531:
Derived categories from a topological point of view
6489:
6487:Keller, Bernhard (1994), "Deriving DG categories",
6047:
are not equivalent, but their derived categories D(
5648:
1315:, having the following universal property: Suppose
6113:
5942:
5575:
5427:
5300:
5142:
5068:
4881:
4783:
4706:
4555:
4519:
4482:
4456:
4419:
4393:
4345:
4244:
4177:
4157:
4123:
3793:
3712:
3542:
3511:
3466:
3433:
3400:
3360:
3333:
3255:
3221:
3190:
3170:
3136:
3112:
3071:
2948:
2921:
2894:
2861:
2834:
2807:
2774:
2740:
2710:
2674:
2628:
2582:
2555:
2523:
2490:
2442:
2405:
2377:
2353:
2325:
2270:
2237:
2217:
2162:
2126:
2088:
2061:
2034:
2008:
1982:
1932:
1882:
1768:
1709:
1670:
1634:
1594:
1574:
1546:
1526:
1506:
1482:
1446:
1410:
1383:
1339:is another category (not necessarily abelian) and
1331:
1307:
1246:
1203:
1175:
1148:
1047:
965:
912:
819:are two objects in this category, then a morphism
811:
760:
709:
616:
570:
539:
407:
371:
343:
307:
258:
133:, for example in the formulation of the theory of
53:. The construction proceeds on the basis that the
6647:
6374:
6063:. Such equivalences are related to the theory of
2301:) might be called the 'naive homotopy category'.
2042:are chain homotopic to the identity morphisms on
6742:
6114:{\displaystyle \mathrm {Coh} (\mathbb {P} ^{1})}
1187:if each of these morphisms is an isomorphism in
5169:, which in turn have important applications in
4252:. This diagram of complexes defines a morphism
812:{\displaystyle (Y^{\bullet },d_{Y}^{\bullet })}
761:{\displaystyle (X^{\bullet },d_{X}^{\bullet })}
4842:. In particular, for a short exact sequence
6281:
6613:
6541:
6386:
5617:
3401:{\displaystyle X^{\bullet }\to I^{\bullet }}
2491:{\displaystyle X^{\bullet }\to Y^{\bullet }}
2245:factors through this functor. Consequently
1769:{\displaystyle h^{i}\colon X^{i}\to Y^{i-1}}
1635:{\displaystyle X^{\bullet }\to Y^{\bullet }}
96:The development of the derived category, by
81:that induces an isomorphism on the level of
4794:By definition, a distinguished triangle in
3512:{\displaystyle X\rightarrow Y'\leftarrow Y}
222:A parallel development was the category of
73:, with two such chain complexes considered
6332:Journal of the London Mathematical Society
6039:It may happen that two abelian categories
5181:or more advanced cohomology theories like
4364:For certain purposes (see below) one uses
966:{\displaystyle f_{i}\colon X^{i}\to Y^{i}}
219:was of a theory expressed in those terms.
6718:
6500:
6343:
6329:
6285:Calculus of Fractions and Homotopy Theory
6233:Derived noncommutative algebraic geometry
6098:
5939:
5297:
3178:may be defined to have these sets as its
2134:is the category with the same objects as
6269:Categories for the Working Mathematician
3723:We can use this to construct a morphism
2782:whose objects are quasi-isomorphisms in
200:become more like computational devices.
6573:
6398:
1254:is a category, together with a functor
920:is defined to be a family of morphisms
6743:
6686:An introduction to homological algebra
6680:
6486:
6450:
6410:
6308:
6034:
5642:
5631:
2414:even when set theory is not at issue.
2099:homotopy category of cochain complexes
628:th cohomology group of the complex is
6461:, Amsterdam: North Holland, pp.
6432:
5783:in one functor, namely the so-called
4633:is also triangulated. For an integer
6544:"Resolutions of unbounded complexes"
6228:Homotopy category of chain complexes
5806:). It is the following composition:
5684:is left exact. Typical examples are
5080:Projective and injective resolutions
4798:is a triangle that is isomorphic in
4784:{\displaystyle d_{X}=(-1)^{n}d_{X}.}
3191:{\displaystyle \operatorname {Hom} }
270:. (Examples include the category of
6524:
6453:"Derived categories and their uses"
6031:into suitable derived categories .
5732:. Their right derived functors are
3477:
117:, and the construction is based on
109:(a summary had earlier appeared in
106:
13:
6089:
6086:
6083:
5931:
5908:
5885:
5873:of a composition of two functors
5866:does keep track of the complexes.
5536:
5533:
5530:
5491:
5488:
5485:
5414:
5405:
5402:
5399:
5374:
5129:
5110:
5037:
4992:
4305:
4288:
4231:
4214:
4158:{\displaystyle {\mathcal {E}}_{0}}
4144:
4096:
3991:
3954:
3933:
3828:
3756:
3739:
3693:
3612:
3569:
3456:
3441:are the same as such morphisms in
3423:
3283:
3160:
3129:
3013:
2884:
2797:
2513:
2432:
2398:
2370:
2346:
2318:
2260:
2207:
2188:
2152:
2116:
1660:
1499:
1436:
1376:
1363:
1324:
1297:
1278:
1236:
1196:
617:{\displaystyle d^{i+1}\circ d^{i}}
563:
397:
364:
333:
297:
251:
14:
6767:
5215:has enough injectives. Embedding
3467:{\displaystyle D({\mathcal {A}})}
3434:{\displaystyle K({\mathcal {A}})}
3171:{\displaystyle D({\mathcal {A}})}
2895:{\displaystyle D({\mathcal {A}})}
2808:{\displaystyle K({\mathcal {A}})}
2524:{\displaystyle D({\mathcal {A}})}
2443:{\displaystyle K({\mathcal {A}})}
2305:Constructing the derived category
2271:{\displaystyle D({\mathcal {A}})}
2127:{\displaystyle K({\mathcal {A}})}
1558:Relation to the homotopy category
1391:is a functor such that, whenever
1247:{\displaystyle D({\mathcal {A}})}
308:{\displaystyle D({\mathcal {A}})}
6413:"Derived categories and tilting"
6178:-Rep) is always equivalent to D(
5988: > 0 and injective
5649:The relation to derived functors
5227:of this map into some injective
5196:, which means that every object
4278:
4275:
4272:
4269:
3113:{\displaystyle I_{X^{\bullet }}}
2775:{\displaystyle I_{X^{\bullet }}}
5243:(in general infinite) sequence
4924:to be the cone of the morphism
1483:{\displaystyle F(f^{\bullet })}
6620:Methods of Homological Algebra
6587:Société Mathématique de France
6404:
6392:
6380:
6368:
6323:
6314:
6302:
6275:
6260:
6108:
6093:
5917:
5894:
5871:Grothendieck spectral sequence
5567:
5555:
5550:
5544:
5522:
5510:
5505:
5499:
5469:and any bounded below complex
5422:
5419:
5409:
5395:
5382:
5379:
5369:
5338:to (any) injective resolution
5288:
5275:
5262:
5256:
5134:
5124:
5118:
5115:
5105:
5060:
5048:
5023:
5020:
5014:
5002:
4997:
4987:
4873:
4867:
4861:
4855:
4759:
4749:
4741:
4735:
4707:{\displaystyle X^{i}=X^{n+i},}
4673:
4666:
4543:
4535:
4340:
4337:
4334:
4322:
4316:
4282:
4225:
4109:
4088:
4078:
4068:
4058:
4046:
4040:
4028:
4022:
4016:
4004:
3983:
3973:
3946:
3925:
3913:
3907:
3895:
3889:
3883:
3871:
3861:
3851:
3841:
3820:
3788:
3785:
3773:
3767:
3750:
3704:
3669:
3631:
3582:
3563:
3529:
3503:
3492:
3461:
3451:
3428:
3418:
3385:
3322:
3296:
3288:
3278:
3236:if, for every acyclic complex
3165:
3155:
3137:{\displaystyle {\mathcal {A}}}
3063:
3041:
3029:
3026:
3018:
3008:
2889:
2879:
2802:
2792:
2659:
2613:
2550:
2538:
2518:
2508:
2475:
2437:
2427:
2406:{\displaystyle {\mathcal {A}}}
2378:{\displaystyle {\mathcal {A}}}
2354:{\displaystyle {\mathcal {A}}}
2326:{\displaystyle {\mathcal {A}}}
2265:
2255:
2212:
2202:
2196:
2193:
2183:
2157:
2147:
2121:
2111:
1967:
1917:
1747:
1710:{\displaystyle h\colon f\to g}
1701:
1665:
1655:
1619:
1507:{\displaystyle {\mathcal {C}}}
1477:
1464:
1441:
1431:
1371:
1368:
1358:
1332:{\displaystyle {\mathcal {C}}}
1302:
1292:
1286:
1283:
1273:
1241:
1231:
1204:{\displaystyle {\mathcal {A}}}
1143:
1130:
1117:
1114:
1101:
1085:
1072:
950:
907:
876:
873:
870:
839:
806:
775:
755:
724:
658:
645:
571:{\displaystyle {\mathcal {A}}}
528:
428:
402:
392:
372:{\displaystyle {\mathcal {A}}}
338:
328:
302:
292:
259:{\displaystyle {\mathcal {A}}}
209:Riemann–Hilbert correspondence
155:
1:
6756:Categories in category theory
6426:
6375:Kashiwara & Schapira 2006
5213:Grothendieck abelian category
4596:Grothendieck abelian category
4134:where the bottom complex has
2711:{\displaystyle f\circ s^{-1}}
1717:is a collection of morphisms
237:
179:became apparent. In fact the
4598:, then the derived category
3361:{\displaystyle X^{\bullet }}
3256:{\displaystyle X^{\bullet }}
3222:{\displaystyle I^{\bullet }}
2949:{\displaystyle Y^{\bullet }}
2922:{\displaystyle X^{\bullet }}
2862:{\displaystyle X^{\bullet }}
2835:{\displaystyle X^{\bullet }}
2741:{\displaystyle X^{\bullet }}
2583:{\displaystyle Z^{\bullet }}
2089:{\displaystyle Y^{\bullet }}
2062:{\displaystyle X^{\bullet }}
1411:{\displaystyle f^{\bullet }}
1176:{\displaystyle f^{\bullet }}
168:without the assumption of a
7:
6651:; Schapira, Pierre (2006),
6455:, in Hazewinkel, M. (ed.),
6440:Encyclopedia of Mathematics
6433:Doorn, M.G.M. van (2001) ,
6221:
5858:)). One might say that the
5219:into some injective object
5084:One can easily show that a
2636:is a quasi-isomorphism and
2531:may be described as a pair
578:and each of the composites
10:
6772:
6311:, remark 10.4.5 and errata
6248:Derived algebraic geometry
6205:)) is equivalent to D(Coh(
6162:be a quiver obtained from
6121:be an abelian category of
6071:. Here are some examples.
4818:for some map of complexes
4359:
2748:and consider the category
2281:From the point of view of
1942:chain homotopy equivalence
1418:is a quasi-isomorphism in
119:localization of a category
6542:Spaltenstein, N. (1988),
6451:Keller, Bernhard (1996),
6411:Keller, Bernhard (2003).
6320:Stacks Project, tag 079P.
6238:Coherent sheaf cohomology
5984:)) = 0 for all
5605:from a projective object
5200:of the category admits a
5160:equivalence of categories
3086:is in fact a set. While
2563:, where for some complex
6387:Gelfand & Manin 2003
6288:. Springer. p. 14.
6253:
6211:Fourier–Mukai transforms
5716:) for some fixed object
5657:. In the following, let
5597:, i.e. for every object
5231:etc., one constructs an
4931:By viewing an object of
4556:{\displaystyle |n|\gg 0}
2035:{\displaystyle f\circ g}
2009:{\displaystyle g\circ f}
16:Homological construction
6069:triangulated categories
5722:global sections functor
5323:for sufficiently small
4920:is forced by requiring
4520:{\displaystyle X^{n}=0}
4457:{\displaystyle X^{n}=0}
4394:{\displaystyle X^{n}=0}
4165:concentrated in degree
3543:{\displaystyle Y\to Y'}
2285:, the derived category
232:triangulated categories
6653:Categories and Sheaves
6548:Compositio Mathematica
6215:Fourier–Mukai partners
6115:
5944:
5586:Dually, assuming that
5577:
5429:
5302:
5144:
5070:
4883:
4785:
4708:
4557:
4521:
4484:
4483:{\displaystyle n\gg 0}
4458:
4421:
4420:{\displaystyle n\ll 0}
4395:
4347:
4246:
4179:
4159:
4125:
3795:
3714:
3544:
3513:
3468:
3435:
3402:
3362:
3335:
3257:
3223:
3192:
3172:
3138:
3114:
3073:
2950:
2923:
2896:
2863:
2836:
2809:
2776:
2742:
2712:
2676:
2630:
2584:
2557:
2525:
2492:
2444:
2407:
2379:
2355:
2327:
2272:
2239:
2219:
2164:
2128:
2090:
2063:
2036:
2010:
1984:
1934:
1884:
1770:
1711:
1672:
1636:
1596:
1576:
1548:
1528:
1508:
1484:
1448:
1412:
1385:
1333:
1309:
1248:
1205:
1177:
1150:
1049:
967:
914:
813:
762:
711:
618:
572:
541:
409:
373:
345:
309:
260:
123:localization of a ring
121:, a generalization of
98:Alexander Grothendieck
6616:Manin, Yuri Ivanovich
6174:are different, but D(
6116:
5945:
5785:total derived functor
5578:
5430:
5303:
5156:injective resolutions
5145:
5071:
4884:
4786:
4709:
4641:, define the complex
4627:triangulated category
4558:
4522:
4485:
4459:
4422:
4396:
4348:
4247:
4180:
4160:
4126:
3796:
3715:
3545:
3514:
3469:
3436:
3403:
3363:
3336:
3258:
3224:
3193:
3173:
3139:
3115:
3074:
2951:
2924:
2897:
2864:
2837:
2810:
2777:
2743:
2713:
2677:
2631:
2585:
2558:
2556:{\displaystyle (s,f)}
2526:
2493:
2452:multiplicative system
2445:
2408:
2380:
2356:
2328:
2273:
2240:
2220:
2165:
2129:
2096:, respectively. The
2091:
2064:
2037:
2011:
1985:
1935:
1885:
1771:
1712:
1673:
1637:
1597:
1577:
1549:
1529:
1509:
1490:is an isomorphism in
1485:
1449:
1413:
1386:
1334:
1310:
1249:
1206:
1178:
1151:
1050:
968:
915:
814:
763:
712:
619:
573:
542:
410:
374:
346:
310:
261:
115:triangulated category
41:is a construction of
6622:, Berlin, New York:
6614:Gelfand, Sergei I.;
6401:, Appendice to Ch. 1
6209:)) by the theory of
6199:dual abelian variety
6079:
6027:along embeddings of
5880:
5730:direct image functor
5480:
5356:
5250:
5233:injective resolution
5099:
5086:homotopy equivalence
4974:
4912:is distinguished in
4849:
4724:
4660:
4531:
4498:
4468:
4435:
4405:
4372:
4259:
4189:
4169:
4138:
3808:
3727:
3557:
3523:
3486:
3445:
3412:
3372:
3345:
3267:
3240:
3206:
3182:
3149:
3124:
3090:
2963:
2933:
2906:
2873:
2846:
2819:
2786:
2752:
2725:
2686:
2640:
2594:
2567:
2535:
2502:
2462:
2421:
2393:
2365:
2341:
2313:
2249:
2229:
2174:
2138:
2105:
2073:
2046:
2020:
1994:
1948:
1898:
1780:
1721:
1689:
1646:
1606:
1586:
1566:
1538:
1518:
1494:
1458:
1422:
1395:
1343:
1319:
1258:
1225:
1191:
1160:
1059:
977:
924:
823:
772:
721:
632:
582:
558:
422:
383:
359:
319:
286:
278:and the category of
246:
6751:Homological algebra
6575:Verdier, Jean-Louis
6502:10.24033/asens.1689
6458:Handbook of algebra
6354:10.1112/jlms/jdn064
6035:Derived equivalence
5618:Spaltenstein (1988)
5047:
4629:, its localization
3082:assuming that this
1879:
1829:
1031:
1013:
906:
869:
805:
754:
516:
489:
462:
181:Cohen–Macaulay ring
139:microlocal analysis
43:homological algebra
6724:Derived Categories
6682:Weibel, Charles A.
6435:"Derived category"
6111:
5940:
5573:
5465:: for any complex
5425:
5298:
5179:topological spaces
5140:
5066:
5029:
4951:: for any objects
4879:
4781:
4717:with differential
4704:
4553:
4517:
4480:
4454:
4417:
4391:
4343:
4242:
4175:
4155:
4121:
4119:
3791:
3710:
3540:
3509:
3464:
3431:
3398:
3358:
3331:
3253:
3219:
3188:
3168:
3134:
3110:
3069:
2996:
2975:
2946:
2919:
2892:
2859:
2832:
2805:
2772:
2738:
2708:
2672:
2626:
2580:
2553:
2521:
2488:
2440:
2403:
2375:
2351:
2323:
2268:
2235:
2215:
2160:
2124:
2086:
2059:
2032:
2006:
1980:
1930:
1880:
1865:
1809:
1766:
1707:
1668:
1632:
1602:are two morphisms
1592:
1572:
1544:
1524:
1504:
1480:
1444:
1408:
1381:
1329:
1305:
1244:
1201:
1173:
1146:
1045:
1017:
999:
963:
910:
892:
855:
809:
791:
758:
740:
707:
614:
568:
537:
405:
379:. The objects of
369:
341:
305:
256:
131:algebraic geometry
102:Jean-Louis Verdier
91:spectral sequences
6695:978-0-521-55987-4
6666:978-3-540-27949-5
6649:Kashiwara, Masaki
6633:978-3-540-43583-9
6295:978-3-642-85844-4
6023:can be viewed as
5959:injective objects
5926:
5903:
5194:enough injectives
5090:quasi-isomorphism
5033:
4981:
4178:{\displaystyle 0}
3688:
3662:
3607:
2968:
2966:
2238:{\displaystyle Q}
1595:{\displaystyle g}
1575:{\displaystyle f}
1547:{\displaystyle Q}
1527:{\displaystyle F}
1185:quasi-isomorphism
517:
490:
463:
353:cochain complexes
6763:
6737:
6720:Yekutieli, Amnon
6715:
6677:
6644:
6605:
6570:
6537:
6536:
6521:
6504:
6483:
6447:
6420:
6419:
6417:
6408:
6402:
6396:
6390:
6384:
6378:
6377:, Theorem 14.3.1
6372:
6366:
6365:
6347:
6327:
6321:
6318:
6312:
6306:
6300:
6299:
6279:
6273:
6264:
6243:Coherent duality
6145:Kronecker quiver
6123:coherent sheaves
6120:
6118:
6117:
6112:
6107:
6106:
6101:
6092:
5969:-acyclics (i.e.
5949:
5947:
5946:
5941:
5935:
5934:
5928:
5927:
5925:
5920:
5915:
5912:
5911:
5905:
5904:
5902:
5897:
5892:
5889:
5888:
5776:, respectively.
5655:derived functors
5582:
5580:
5579:
5574:
5554:
5553:
5539:
5509:
5508:
5494:
5434:
5432:
5431:
5426:
5418:
5417:
5408:
5394:
5393:
5378:
5377:
5368:
5367:
5307:
5305:
5304:
5299:
5287:
5286:
5274:
5273:
5206:injective object
5187:group cohomology
5183:étale cohomology
5167:derived functors
5149:
5147:
5146:
5141:
5133:
5132:
5114:
5113:
5075:
5073:
5072:
5067:
5046:
5041:
5040:
5034:
5031:
5001:
5000:
4996:
4995:
4982:
4979:
4963:and any integer
4945:full subcategory
4888:
4886:
4885:
4880:
4802:to the triangle
4790:
4788:
4787:
4782:
4777:
4776:
4767:
4766:
4745:
4744:
4713:
4711:
4710:
4705:
4700:
4699:
4681:
4680:
4649:shifted down by
4575:, respectively.
4562:
4560:
4559:
4554:
4546:
4538:
4526:
4524:
4523:
4518:
4510:
4509:
4489:
4487:
4486:
4481:
4463:
4461:
4460:
4455:
4447:
4446:
4426:
4424:
4423:
4418:
4400:
4398:
4397:
4392:
4384:
4383:
4352:
4350:
4349:
4344:
4315:
4314:
4309:
4308:
4298:
4297:
4292:
4291:
4281:
4251:
4249:
4248:
4243:
4241:
4240:
4235:
4234:
4224:
4223:
4218:
4217:
4207:
4206:
4184:
4182:
4181:
4176:
4164:
4162:
4161:
4156:
4154:
4153:
4148:
4147:
4130:
4128:
4127:
4122:
4120:
4106:
4105:
4100:
4099:
4044:
4038:
4032:
4026:
4020:
4001:
4000:
3995:
3994:
3970:
3969:
3958:
3957:
3943:
3942:
3937:
3936:
3911:
3905:
3899:
3893:
3887:
3838:
3837:
3832:
3831:
3800:
3798:
3797:
3792:
3766:
3765:
3760:
3759:
3749:
3748:
3743:
3742:
3719:
3717:
3716:
3711:
3703:
3702:
3697:
3696:
3689:
3687:
3686:
3668:
3663:
3661:
3660:
3630:
3628:
3627:
3616:
3615:
3608:
3606:
3605:
3581:
3579:
3578:
3573:
3572:
3549:
3547:
3546:
3541:
3539:
3518:
3516:
3515:
3510:
3502:
3478:Derived Hom-sets
3473:
3471:
3470:
3465:
3460:
3459:
3440:
3438:
3437:
3432:
3427:
3426:
3407:
3405:
3404:
3399:
3397:
3396:
3384:
3383:
3367:
3365:
3364:
3359:
3357:
3356:
3340:
3338:
3337:
3332:
3321:
3320:
3308:
3307:
3292:
3291:
3287:
3286:
3262:
3260:
3259:
3254:
3252:
3251:
3228:
3226:
3225:
3220:
3218:
3217:
3197:
3195:
3194:
3189:
3177:
3175:
3174:
3169:
3164:
3163:
3143:
3141:
3140:
3135:
3133:
3132:
3119:
3117:
3116:
3111:
3109:
3108:
3107:
3106:
3078:
3076:
3075:
3070:
3062:
3061:
3049:
3048:
3039:
3022:
3021:
3017:
3016:
2995:
2994:
2993:
2992:
2991:
2976:
2955:
2953:
2952:
2947:
2945:
2944:
2928:
2926:
2925:
2920:
2918:
2917:
2901:
2899:
2898:
2893:
2888:
2887:
2868:
2866:
2865:
2860:
2858:
2857:
2841:
2839:
2838:
2833:
2831:
2830:
2814:
2812:
2811:
2806:
2801:
2800:
2781:
2779:
2778:
2773:
2771:
2770:
2769:
2768:
2747:
2745:
2744:
2739:
2737:
2736:
2717:
2715:
2714:
2709:
2707:
2706:
2681:
2679:
2678:
2673:
2671:
2670:
2658:
2657:
2635:
2633:
2632:
2627:
2625:
2624:
2612:
2611:
2589:
2587:
2586:
2581:
2579:
2578:
2562:
2560:
2559:
2554:
2530:
2528:
2527:
2522:
2517:
2516:
2497:
2495:
2494:
2489:
2487:
2486:
2474:
2473:
2449:
2447:
2446:
2441:
2436:
2435:
2412:
2410:
2409:
2404:
2402:
2401:
2384:
2382:
2381:
2376:
2374:
2373:
2360:
2358:
2357:
2352:
2350:
2349:
2332:
2330:
2329:
2324:
2322:
2321:
2283:model categories
2277:
2275:
2274:
2269:
2264:
2263:
2244:
2242:
2241:
2236:
2224:
2222:
2221:
2216:
2211:
2210:
2192:
2191:
2169:
2167:
2166:
2161:
2156:
2155:
2133:
2131:
2130:
2125:
2120:
2119:
2095:
2093:
2092:
2087:
2085:
2084:
2068:
2066:
2065:
2060:
2058:
2057:
2041:
2039:
2038:
2033:
2015:
2013:
2012:
2007:
1989:
1987:
1986:
1981:
1979:
1978:
1966:
1965:
1944:if there exists
1939:
1937:
1936:
1931:
1929:
1928:
1916:
1915:
1889:
1887:
1886:
1881:
1878:
1873:
1861:
1860:
1842:
1841:
1828:
1817:
1805:
1804:
1792:
1791:
1775:
1773:
1772:
1767:
1765:
1764:
1746:
1745:
1733:
1732:
1716:
1714:
1713:
1708:
1677:
1675:
1674:
1669:
1664:
1663:
1641:
1639:
1638:
1633:
1631:
1630:
1618:
1617:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1553:
1551:
1550:
1545:
1534:factors through
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1503:
1502:
1489:
1487:
1486:
1481:
1476:
1475:
1453:
1451:
1450:
1445:
1440:
1439:
1417:
1415:
1414:
1409:
1407:
1406:
1390:
1388:
1387:
1382:
1380:
1379:
1367:
1366:
1338:
1336:
1335:
1330:
1328:
1327:
1314:
1312:
1311:
1306:
1301:
1300:
1282:
1281:
1253:
1251:
1250:
1245:
1240:
1239:
1220:derived category
1210:
1208:
1207:
1202:
1200:
1199:
1182:
1180:
1179:
1174:
1172:
1171:
1155:
1153:
1152:
1147:
1142:
1141:
1129:
1128:
1113:
1112:
1100:
1099:
1084:
1083:
1071:
1070:
1054:
1052:
1051:
1046:
1044:
1043:
1030:
1025:
1012:
1007:
995:
994:
972:
970:
969:
964:
962:
961:
949:
948:
936:
935:
919:
917:
916:
911:
905:
900:
888:
887:
868:
863:
851:
850:
835:
834:
818:
816:
815:
810:
804:
799:
787:
786:
767:
765:
764:
759:
753:
748:
736:
735:
716:
714:
713:
708:
706:
705:
684:
679:
678:
657:
656:
644:
643:
623:
621:
620:
615:
613:
612:
600:
599:
577:
575:
574:
569:
567:
566:
554:is an object of
546:
544:
543:
538:
527:
526:
515:
514:
501:
500:
499:
488:
487:
474:
473:
472:
461:
460:
444:
443:
442:
415:are of the form
414:
412:
411:
406:
401:
400:
378:
376:
375:
370:
368:
367:
350:
348:
347:
342:
337:
336:
314:
312:
311:
306:
301:
300:
268:abelian category
265:
263:
262:
257:
255:
254:
205:sheaf cohomology
127:coherent duality
100:and his student
77:when there is a
47:derived functors
36:abelian category
25:derived category
6771:
6770:
6766:
6765:
6764:
6762:
6761:
6760:
6741:
6740:
6734:
6696:
6667:
6657:Springer-Verlag
6634:
6624:Springer-Verlag
6534:
6473:
6429:
6424:
6423:
6415:
6409:
6405:
6397:
6393:
6385:
6381:
6373:
6369:
6328:
6324:
6319:
6315:
6307:
6303:
6296:
6280:
6276:
6265:
6261:
6256:
6224:
6191:abelian variety
6142:
6127:projective line
6102:
6097:
6096:
6082:
6080:
6077:
6076:
6037:
5930:
5929:
5921:
5916:
5914:
5913:
5907:
5906:
5898:
5893:
5891:
5890:
5884:
5883:
5881:
5878:
5877:
5769:
5651:
5540:
5529:
5528:
5495:
5484:
5483:
5481:
5478:
5477:
5473:of injectives,
5413:
5412:
5398:
5389:
5385:
5373:
5372:
5363:
5359:
5357:
5354:
5353:
5282:
5278:
5269:
5265:
5251:
5248:
5247:
5128:
5127:
5109:
5108:
5100:
5097:
5096:
5082:
5042:
5036:
5035:
5030:
4991:
4990:
4983:
4978:
4977:
4975:
4972:
4971:
4896:, the triangle
4850:
4847:
4846:
4772:
4768:
4762:
4758:
4731:
4727:
4725:
4722:
4721:
4689:
4685:
4676:
4672:
4661:
4658:
4657:
4542:
4534:
4532:
4529:
4528:
4505:
4501:
4499:
4496:
4495:
4469:
4466:
4465:
4442:
4438:
4436:
4433:
4432:
4406:
4403:
4402:
4379:
4375:
4373:
4370:
4369:
4362:
4310:
4304:
4303:
4302:
4293:
4287:
4286:
4285:
4268:
4260:
4257:
4256:
4236:
4230:
4229:
4228:
4219:
4213:
4212:
4211:
4196:
4192:
4190:
4187:
4186:
4170:
4167:
4166:
4149:
4143:
4142:
4141:
4139:
4136:
4135:
4118:
4117:
4112:
4107:
4101:
4095:
4094:
4093:
4091:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4050:
4049:
4043:
4037:
4031:
4025:
4019:
4013:
4012:
4007:
4002:
3996:
3990:
3989:
3988:
3986:
3981:
3976:
3971:
3959:
3953:
3952:
3951:
3949:
3944:
3938:
3932:
3931:
3930:
3928:
3923:
3917:
3916:
3910:
3904:
3898:
3892:
3886:
3880:
3879:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3839:
3833:
3827:
3826:
3825:
3823:
3818:
3811:
3809:
3806:
3805:
3761:
3755:
3754:
3753:
3744:
3738:
3737:
3736:
3728:
3725:
3724:
3698:
3692:
3691:
3690:
3676:
3672:
3667:
3638:
3634:
3629:
3617:
3611:
3610:
3609:
3589:
3585:
3580:
3574:
3568:
3567:
3566:
3558:
3555:
3554:
3532:
3524:
3521:
3520:
3495:
3487:
3484:
3483:
3480:
3455:
3454:
3446:
3443:
3442:
3422:
3421:
3413:
3410:
3409:
3392:
3388:
3379:
3375:
3373:
3370:
3369:
3352:
3348:
3346:
3343:
3342:
3316:
3312:
3303:
3299:
3282:
3281:
3274:
3270:
3268:
3265:
3264:
3247:
3243:
3241:
3238:
3237:
3213:
3209:
3207:
3204:
3203:
3183:
3180:
3179:
3159:
3158:
3150:
3147:
3146:
3128:
3127:
3125:
3122:
3121:
3102:
3098:
3097:
3093:
3091:
3088:
3087:
3057:
3053:
3044:
3040:
3032:
3012:
3011:
3004:
3000:
2987:
2983:
2982:
2978:
2977:
2967:
2964:
2961:
2960:
2940:
2936:
2934:
2931:
2930:
2913:
2909:
2907:
2904:
2903:
2883:
2882:
2874:
2871:
2870:
2853:
2849:
2847:
2844:
2843:
2826:
2822:
2820:
2817:
2816:
2796:
2795:
2787:
2784:
2783:
2764:
2760:
2759:
2755:
2753:
2750:
2749:
2732:
2728:
2726:
2723:
2722:
2699:
2695:
2687:
2684:
2683:
2666:
2662:
2653:
2649:
2641:
2638:
2637:
2620:
2616:
2607:
2603:
2595:
2592:
2591:
2574:
2570:
2568:
2565:
2564:
2536:
2533:
2532:
2512:
2511:
2503:
2500:
2499:
2482:
2478:
2469:
2465:
2463:
2460:
2459:
2431:
2430:
2422:
2419:
2418:
2397:
2396:
2394:
2391:
2390:
2369:
2368:
2366:
2363:
2362:
2345:
2344:
2342:
2339:
2338:
2317:
2316:
2314:
2311:
2310:
2307:
2259:
2258:
2250:
2247:
2246:
2230:
2227:
2226:
2206:
2205:
2187:
2186:
2175:
2172:
2171:
2151:
2150:
2139:
2136:
2135:
2115:
2114:
2106:
2103:
2102:
2080:
2076:
2074:
2071:
2070:
2053:
2049:
2047:
2044:
2043:
2021:
2018:
2017:
1995:
1992:
1991:
1974:
1970:
1961:
1957:
1949:
1946:
1945:
1924:
1920:
1911:
1907:
1899:
1896:
1895:
1874:
1869:
1850:
1846:
1837:
1833:
1818:
1813:
1800:
1796:
1787:
1783:
1781:
1778:
1777:
1754:
1750:
1741:
1737:
1728:
1724:
1722:
1719:
1718:
1690:
1687:
1686:
1659:
1658:
1647:
1644:
1643:
1626:
1622:
1613:
1609:
1607:
1604:
1603:
1587:
1584:
1583:
1567:
1564:
1563:
1560:
1539:
1536:
1535:
1519:
1516:
1515:
1498:
1497:
1495:
1492:
1491:
1471:
1467:
1459:
1456:
1455:
1435:
1434:
1423:
1420:
1419:
1402:
1398:
1396:
1393:
1392:
1375:
1374:
1362:
1361:
1344:
1341:
1340:
1323:
1322:
1320:
1317:
1316:
1296:
1295:
1277:
1276:
1259:
1256:
1255:
1235:
1234:
1226:
1223:
1222:
1195:
1194:
1192:
1189:
1188:
1167:
1163:
1161:
1158:
1157:
1137:
1133:
1124:
1120:
1108:
1104:
1095:
1091:
1079:
1075:
1066:
1062:
1060:
1057:
1056:
1039:
1035:
1026:
1021:
1008:
1003:
984:
980:
978:
975:
974:
957:
953:
944:
940:
931:
927:
925:
922:
921:
901:
896:
883:
879:
864:
859:
846:
842:
830:
826:
824:
821:
820:
800:
795:
782:
778:
773:
770:
769:
749:
744:
731:
727:
722:
719:
718:
695:
691:
680:
674:
670:
652:
648:
639:
635:
633:
630:
629:
608:
604:
589:
585:
583:
580:
579:
562:
561:
559:
556:
555:
522:
518:
510:
506:
495:
491:
483:
479:
468:
464:
453:
449:
435:
431:
423:
420:
419:
396:
395:
384:
381:
380:
363:
362:
360:
357:
356:
332:
331:
320:
317:
316:
296:
295:
287:
284:
283:
250:
249:
247:
244:
243:
240:
228:homotopy theory
177:dualizing sheaf
158:
147:mirror symmetry
87:hypercohomology
67:chain complexes
17:
12:
11:
5:
6769:
6759:
6758:
6753:
6739:
6738:
6733:978-1108419338
6732:
6716:
6694:
6678:
6665:
6645:
6632:
6607:
6606:
6571:
6554:(2): 121–154,
6539:
6522:
6484:
6471:
6448:
6428:
6425:
6422:
6421:
6403:
6391:
6379:
6367:
6322:
6313:
6301:
6294:
6274:
6258:
6257:
6255:
6252:
6251:
6250:
6245:
6240:
6235:
6230:
6223:
6220:
6219:
6218:
6183:
6148:
6140:
6110:
6105:
6100:
6095:
6091:
6088:
6085:
6036:
6033:
6025:Kan extensions
6017:
6016:
5951:
5950:
5938:
5933:
5924:
5919:
5910:
5901:
5896:
5887:
5767:
5692:→ Ab given by
5678:
5677:
5674:
5650:
5647:
5639:exact category
5584:
5583:
5572:
5569:
5566:
5563:
5560:
5557:
5552:
5549:
5546:
5543:
5538:
5535:
5532:
5527:
5524:
5521:
5518:
5515:
5512:
5507:
5504:
5501:
5498:
5493:
5490:
5487:
5436:
5435:
5424:
5421:
5416:
5411:
5407:
5404:
5401:
5397:
5392:
5388:
5384:
5381:
5376:
5371:
5366:
5362:
5309:
5308:
5296:
5293:
5290:
5285:
5281:
5277:
5272:
5268:
5264:
5261:
5258:
5255:
5151:
5150:
5139:
5136:
5131:
5126:
5123:
5120:
5117:
5112:
5107:
5104:
5081:
5078:
5077:
5076:
5065:
5062:
5059:
5056:
5053:
5050:
5045:
5039:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
4999:
4994:
4989:
4986:
4890:
4889:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4834:) denotes the
4792:
4791:
4780:
4775:
4771:
4765:
4761:
4757:
4754:
4751:
4748:
4743:
4740:
4737:
4734:
4730:
4715:
4714:
4703:
4698:
4695:
4692:
4688:
4684:
4679:
4675:
4671:
4668:
4665:
4637:and a complex
4552:
4549:
4545:
4541:
4537:
4516:
4513:
4508:
4504:
4479:
4476:
4473:
4453:
4450:
4445:
4441:
4416:
4413:
4410:
4390:
4387:
4382:
4378:
4361:
4358:
4354:
4353:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4313:
4307:
4301:
4296:
4290:
4284:
4280:
4277:
4274:
4271:
4267:
4264:
4239:
4233:
4227:
4222:
4216:
4210:
4205:
4202:
4199:
4195:
4174:
4152:
4146:
4132:
4131:
4116:
4113:
4111:
4108:
4104:
4098:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4014:
4011:
4008:
4006:
4003:
3999:
3993:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3968:
3965:
3962:
3956:
3950:
3948:
3945:
3941:
3935:
3929:
3927:
3924:
3922:
3919:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3881:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3836:
3830:
3824:
3822:
3819:
3817:
3814:
3813:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3764:
3758:
3752:
3747:
3741:
3735:
3732:
3721:
3720:
3709:
3706:
3701:
3695:
3685:
3682:
3679:
3675:
3671:
3666:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3637:
3633:
3626:
3623:
3620:
3614:
3604:
3601:
3598:
3595:
3592:
3588:
3584:
3577:
3571:
3565:
3562:
3538:
3535:
3531:
3528:
3508:
3505:
3501:
3498:
3494:
3491:
3479:
3476:
3463:
3458:
3453:
3450:
3430:
3425:
3420:
3417:
3395:
3391:
3387:
3382:
3378:
3355:
3351:
3330:
3327:
3324:
3319:
3315:
3311:
3306:
3302:
3298:
3295:
3290:
3285:
3280:
3277:
3273:
3250:
3246:
3216:
3212:
3187:
3167:
3162:
3157:
3154:
3131:
3105:
3101:
3096:
3080:
3079:
3068:
3065:
3060:
3056:
3052:
3047:
3043:
3038:
3035:
3031:
3028:
3025:
3020:
3015:
3010:
3007:
3003:
2999:
2990:
2986:
2981:
2974:
2971:
2943:
2939:
2916:
2912:
2891:
2886:
2881:
2878:
2856:
2852:
2829:
2825:
2815:with codomain
2804:
2799:
2794:
2791:
2767:
2763:
2758:
2735:
2731:
2705:
2702:
2698:
2694:
2691:
2669:
2665:
2661:
2656:
2652:
2648:
2645:
2623:
2619:
2615:
2610:
2606:
2602:
2599:
2577:
2573:
2552:
2549:
2546:
2543:
2540:
2520:
2515:
2510:
2507:
2485:
2481:
2477:
2472:
2468:
2458:. A morphism
2439:
2434:
2429:
2426:
2400:
2372:
2348:
2320:
2306:
2303:
2267:
2262:
2257:
2254:
2234:
2214:
2209:
2204:
2201:
2198:
2195:
2190:
2185:
2182:
2179:
2159:
2154:
2149:
2146:
2143:
2123:
2118:
2113:
2110:
2083:
2079:
2056:
2052:
2031:
2028:
2025:
2005:
2002:
1999:
1977:
1973:
1969:
1964:
1960:
1956:
1953:
1927:
1923:
1919:
1914:
1910:
1906:
1903:
1877:
1872:
1868:
1864:
1859:
1856:
1853:
1849:
1845:
1840:
1836:
1832:
1827:
1824:
1821:
1816:
1812:
1808:
1803:
1799:
1795:
1790:
1786:
1763:
1760:
1757:
1753:
1749:
1744:
1740:
1736:
1731:
1727:
1706:
1703:
1700:
1697:
1694:
1680:chain homotopy
1667:
1662:
1657:
1654:
1651:
1629:
1625:
1621:
1616:
1612:
1591:
1571:
1559:
1556:
1543:
1523:
1501:
1479:
1474:
1470:
1466:
1463:
1443:
1438:
1433:
1430:
1427:
1405:
1401:
1378:
1373:
1370:
1365:
1360:
1357:
1354:
1351:
1348:
1326:
1304:
1299:
1294:
1291:
1288:
1285:
1280:
1275:
1272:
1269:
1266:
1263:
1243:
1238:
1233:
1230:
1198:
1170:
1166:
1145:
1140:
1136:
1132:
1127:
1123:
1119:
1116:
1111:
1107:
1103:
1098:
1094:
1090:
1087:
1082:
1078:
1074:
1069:
1065:
1042:
1038:
1034:
1029:
1024:
1020:
1016:
1011:
1006:
1002:
998:
993:
990:
987:
983:
960:
956:
952:
947:
943:
939:
934:
930:
909:
904:
899:
895:
891:
886:
882:
878:
875:
872:
867:
862:
858:
854:
849:
845:
841:
838:
833:
829:
808:
803:
798:
794:
790:
785:
781:
777:
757:
752:
747:
743:
739:
734:
730:
726:
704:
701:
698:
694:
690:
687:
683:
677:
673:
669:
666:
663:
660:
655:
651:
647:
642:
638:
624:is zero. The
611:
607:
603:
598:
595:
592:
588:
565:
548:
547:
536:
533:
530:
525:
521:
513:
509:
504:
498:
494:
486:
482:
477:
471:
467:
459:
456:
452:
447:
441:
438:
434:
430:
427:
404:
399:
394:
391:
388:
366:
355:with terms in
340:
335:
330:
327:
324:
304:
299:
294:
291:
253:
239:
236:
186:tensor product
162:coherent sheaf
157:
154:
15:
9:
6:
4:
3:
2:
6768:
6757:
6754:
6752:
6749:
6748:
6746:
6735:
6729:
6725:
6721:
6717:
6713:
6709:
6705:
6701:
6697:
6691:
6687:
6683:
6679:
6676:
6672:
6668:
6662:
6658:
6654:
6650:
6646:
6643:
6639:
6635:
6629:
6625:
6621:
6617:
6612:
6611:
6610:
6604:
6600:
6596:
6592:
6588:
6584:
6581:(in French),
6580:
6576:
6572:
6569:
6565:
6561:
6557:
6553:
6549:
6545:
6540:
6533:
6532:
6527:
6523:
6520:
6516:
6512:
6508:
6503:
6498:
6495:(1): 63–102,
6494:
6490:
6485:
6482:
6478:
6474:
6472:0-444-82212-7
6468:
6464:
6460:
6459:
6454:
6449:
6446:
6442:
6441:
6436:
6431:
6430:
6414:
6407:
6400:
6395:
6388:
6383:
6376:
6371:
6363:
6359:
6355:
6351:
6346:
6341:
6337:
6333:
6326:
6317:
6310:
6305:
6297:
6291:
6287:
6286:
6278:
6272:
6270:
6263:
6259:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6231:
6229:
6226:
6225:
6216:
6212:
6208:
6204:
6201:. Then D(Coh(
6200:
6196:
6192:
6188:
6184:
6181:
6177:
6173:
6169:
6165:
6161:
6157:
6153:
6149:
6146:
6139:
6135:
6132:
6128:
6124:
6103:
6074:
6073:
6072:
6070:
6066:
6062:
6058:
6054:
6050:
6046:
6042:
6032:
6030:
6026:
6022:
6014:
6010:
6006:
6002:
5998:
5995:
5994:
5993:
5991:
5987:
5983:
5979:
5975:
5972:
5968:
5964:
5960:
5956:
5936:
5922:
5899:
5876:
5875:
5874:
5872:
5867:
5865:
5861:
5857:
5853:
5849:
5845:
5841:
5837:
5833:
5829:
5825:
5821:
5817:
5813:
5809:
5805:
5801:
5797:
5793:
5789:
5786:
5782:
5777:
5775:
5773:
5766:
5763:
5758:
5756:
5752:
5748:
5743:
5739:
5737:
5731:
5727:
5723:
5719:
5715:
5711:
5707:
5703:
5699:
5695:
5691:
5687:
5683:
5675:
5672:
5671:
5670:
5668:
5664:
5660:
5656:
5646:
5644:
5640:
5635:
5633:
5632:Keller (1994)
5629:
5628:resolutions.
5627:
5623:
5619:
5614:
5612:
5608:
5604:
5600:
5596:
5595:
5589:
5570:
5564:
5561:
5558:
5547:
5541:
5525:
5519:
5516:
5513:
5502:
5496:
5476:
5475:
5474:
5472:
5468:
5464:
5460:
5456:
5452:
5448:
5443:
5441:
5390:
5386:
5364:
5360:
5352:
5351:
5350:
5349:
5346:extends to a
5345:
5341:
5337:
5333:
5328:
5326:
5322:
5318:
5314:
5294:
5291:
5283:
5279:
5270:
5266:
5259:
5253:
5246:
5245:
5244:
5242:
5238:
5234:
5230:
5226:
5222:
5218:
5214:
5210:
5207:
5203:
5199:
5195:
5190:
5188:
5184:
5180:
5176:
5172:
5168:
5163:
5161:
5157:
5137:
5121:
5102:
5095:
5094:
5093:
5091:
5087:
5063:
5057:
5054:
5051:
5043:
5026:
5017:
5011:
5008:
5005:
4984:
4970:
4969:
4968:
4966:
4962:
4958:
4954:
4950:
4946:
4942:
4938:
4934:
4929:
4927:
4923:
4919:
4915:
4911:
4907:
4903:
4899:
4895:
4876:
4870:
4864:
4858:
4852:
4845:
4844:
4843:
4841:
4837:
4833:
4829:
4825:
4821:
4817:
4813:
4809:
4805:
4801:
4797:
4778:
4773:
4769:
4763:
4755:
4752:
4746:
4738:
4732:
4728:
4720:
4719:
4718:
4701:
4696:
4693:
4690:
4686:
4682:
4677:
4669:
4663:
4656:
4655:
4654:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4624:
4619:
4615:
4613:
4609:
4605:
4601:
4597:
4593:
4589:
4585:
4581:
4576:
4574:
4570:
4566:
4550:
4547:
4539:
4514:
4511:
4506:
4502:
4493:
4477:
4474:
4471:
4451:
4448:
4443:
4439:
4430:
4429:bounded-above
4414:
4411:
4408:
4388:
4385:
4380:
4376:
4367:
4366:bounded-below
4357:
4331:
4328:
4325:
4319:
4311:
4299:
4294:
4265:
4262:
4255:
4254:
4253:
4237:
4220:
4208:
4203:
4200:
4197:
4193:
4172:
4150:
4114:
4102:
4083:
4073:
4063:
4053:
4034:
4009:
3997:
3978:
3966:
3963:
3960:
3939:
3920:
3901:
3876:
3866:
3856:
3846:
3834:
3815:
3804:
3803:
3802:
3782:
3779:
3776:
3770:
3762:
3745:
3733:
3730:
3707:
3699:
3683:
3680:
3677:
3673:
3664:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3635:
3624:
3621:
3618:
3602:
3599:
3596:
3593:
3590:
3586:
3575:
3560:
3553:
3552:
3551:
3536:
3533:
3526:
3506:
3499:
3496:
3489:
3475:
3448:
3415:
3393:
3389:
3380:
3376:
3353:
3349:
3328:
3325:
3317:
3313:
3309:
3304:
3300:
3293:
3275:
3271:
3248:
3244:
3235:
3233:
3214:
3210:
3199:
3185:
3152:
3103:
3099:
3094:
3085:
3066:
3058:
3054:
3050:
3045:
3036:
3033:
3023:
3005:
3001:
2997:
2988:
2984:
2979:
2972:
2969:
2959:
2958:
2957:
2941:
2937:
2914:
2910:
2876:
2854:
2850:
2827:
2823:
2789:
2765:
2761:
2756:
2733:
2729:
2719:
2703:
2700:
2696:
2692:
2689:
2667:
2663:
2654:
2650:
2646:
2643:
2621:
2617:
2608:
2604:
2600:
2597:
2575:
2571:
2547:
2544:
2541:
2505:
2483:
2479:
2470:
2466:
2457:
2453:
2424:
2415:
2387:
2335:
2302:
2300:
2296:
2292:
2288:
2284:
2279:
2252:
2232:
2199:
2180:
2177:
2144:
2141:
2108:
2101:
2100:
2081:
2077:
2054:
2050:
2029:
2026:
2023:
2003:
2000:
1997:
1975:
1971:
1962:
1958:
1954:
1951:
1943:
1925:
1921:
1912:
1908:
1904:
1901:
1893:
1875:
1870:
1866:
1862:
1857:
1854:
1851:
1847:
1843:
1838:
1834:
1830:
1825:
1822:
1819:
1814:
1810:
1806:
1801:
1797:
1793:
1788:
1784:
1761:
1758:
1755:
1751:
1742:
1738:
1734:
1729:
1725:
1704:
1698:
1695:
1692:
1685:
1681:
1652:
1649:
1627:
1623:
1614:
1610:
1589:
1569:
1555:
1541:
1521:
1472:
1468:
1461:
1428:
1425:
1403:
1399:
1355:
1352:
1349:
1346:
1289:
1270:
1267:
1264:
1261:
1228:
1221:
1217:
1212:
1186:
1168:
1164:
1138:
1134:
1125:
1121:
1109:
1105:
1096:
1092:
1088:
1080:
1076:
1067:
1063:
1040:
1036:
1032:
1027:
1022:
1018:
1014:
1009:
1004:
1000:
996:
991:
988:
985:
981:
958:
954:
945:
941:
937:
932:
928:
902:
897:
893:
889:
884:
880:
865:
860:
856:
852:
847:
843:
836:
831:
827:
801:
796:
792:
788:
783:
779:
750:
745:
741:
737:
732:
728:
702:
699:
696:
692:
688:
685:
681:
675:
671:
667:
664:
661:
653:
649:
640:
636:
627:
609:
605:
601:
596:
593:
590:
586:
553:
534:
531:
523:
519:
511:
507:
502:
496:
492:
484:
480:
475:
469:
465:
457:
454:
450:
445:
439:
436:
432:
425:
418:
417:
416:
389:
386:
354:
325:
322:
289:
281:
277:
273:
269:
235:
233:
229:
225:
220:
218:
214:
210:
206:
201:
199:
195:
191:
187:
182:
178:
174:
171:
167:
166:Serre duality
163:
153:
150:
148:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
37:
33:
29:
26:
22:
6723:
6685:
6652:
6619:
6608:
6582:
6578:
6551:
6547:
6530:
6492:
6488:
6457:
6438:
6406:
6399:Verdier 1996
6394:
6382:
6370:
6345:math/0610553
6335:
6331:
6325:
6316:
6304:
6284:
6277:
6267:
6262:
6214:
6206:
6202:
6194:
6186:
6179:
6175:
6171:
6167:
6163:
6159:
6151:
6137:
6133:
6065:t-structures
6060:
6056:
6052:
6048:
6044:
6040:
6038:
6028:
6020:
6018:
6012:
6008:
6004:
6000:
5996:
5989:
5985:
5981:
5977:
5973:
5970:
5966:
5962:
5954:
5952:
5868:
5863:
5859:
5855:
5851:
5847:
5843:
5839:
5835:
5831:
5827:
5823:
5819:
5815:
5811:
5807:
5803:
5799:
5795:
5791:
5787:
5784:
5780:
5778:
5771:
5764:
5761:
5754:
5750:
5746:
5741:
5735:
5734:Ext(–,
5717:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5685:
5681:
5679:
5666:
5662:
5658:
5652:
5636:
5630:
5626:K-projective
5625:
5621:
5615:
5610:
5606:
5601:there is an
5598:
5591:
5587:
5585:
5470:
5466:
5462:
5458:
5454:
5450:
5446:
5444:
5439:
5437:
5343:
5339:
5335:
5331:
5329:
5324:
5320:
5316:
5312:
5310:
5236:
5232:
5228:
5220:
5216:
5208:
5202:monomorphism
5197:
5193:
5191:
5164:
5152:
5083:
4964:
4960:
4956:
4952:
4940:
4936:
4932:
4930:
4925:
4921:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4891:
4839:
4836:mapping cone
4831:
4830:. Here Cone(
4827:
4823:
4819:
4815:
4811:
4807:
4803:
4799:
4795:
4793:
4716:
4650:
4646:
4642:
4638:
4634:
4630:
4622:
4620:
4616:
4611:
4607:
4603:
4599:
4591:
4587:
4577:
4572:
4568:
4564:
4491:
4428:
4365:
4363:
4355:
4133:
3722:
3481:
3368:, morphisms
3231:
3230:
3200:
3081:
2720:
2455:
2451:
2416:
2388:
2336:
2308:
2298:
2294:
2290:
2286:
2280:
2097:
1941:
1891:
1683:
1679:
1561:
1454:, its image
1219:
1216:localization
1213:
1184:
1183:is called a
625:
551:
549:
241:
221:
202:
189:
176:
170:non-singular
159:
151:
95:
70:
65:) should be
62:
58:
50:
38:
31:
27:
24:
18:
6491:, Série 4,
6338:: 129–143.
6309:Weibel 1994
5744:,–),
5643:Keller 1996
5622:K-injective
5603:epimorphism
5594:projectives
4653:, so that
550:where each
156:Motivations
49:defined on
21:mathematics
6745:Categories
6579:Astérisque
6526:May, J. P.
6427:References
6266:Mac Lane,
5953:such that
5311:where the
5239:, i.e. an
5171:cohomology
4949:Ext groups
3263:, we have
3234:-injective
2389:Even when
1990:such that
1890:for every
1776:such that
1682:or simply
973:such that
238:Definition
107:Astérisque
75:isomorphic
6595:0303-1179
6585:, Paris:
6560:0010-437X
6511:0012-9593
6445:EMS Press
6389:, III.3.2
5918:→
5895:→
5720:, or the
5383:→
5292:⋯
5289:→
5276:→
5263:→
5257:→
5119:→
4939:contains
4874:→
4868:→
4862:→
4856:→
4753:−
4548:≫
4475:≫
4412:≪
4329:−
4266:∈
4263:ϕ
4226:→
4194:ϕ
4110:→
4089:→
4084:⋯
4079:→
4069:→
4059:→
4047:↓
4041:↓
4035:⋯
4029:↓
4023:↓
4017:↓
4005:→
3984:→
3979:⋯
3974:→
3964:−
3947:→
3926:→
3914:↑
3908:↑
3902:⋯
3896:↑
3890:↑
3884:↑
3872:→
3862:→
3857:⋯
3852:→
3842:→
3821:→
3780:−
3751:→
3731:ϕ
3705:→
3674:ϕ
3670:→
3665:⋯
3655:−
3643:−
3636:ϕ
3632:→
3622:−
3600:−
3587:ϕ
3583:→
3564:→
3530:→
3504:←
3493:→
3394:∙
3386:→
3381:∙
3354:∙
3318:∙
3305:∙
3294:
3249:∙
3215:∙
3104:∙
3059:∙
3046:∙
3024:
2998:
2989:∙
2973:→
2942:∙
2915:∙
2855:∙
2828:∙
2766:∙
2734:∙
2701:−
2693:∘
2668:∙
2660:→
2655:∙
2647::
2622:∙
2614:→
2609:∙
2601::
2576:∙
2484:∙
2476:→
2471:∙
2197:→
2181:
2145:
2082:∙
2055:∙
2027:∘
2001:∘
1976:∙
1968:→
1963:∙
1955::
1926:∙
1918:→
1913:∙
1905::
1863:∘
1831:∘
1823:−
1794:−
1759:−
1748:→
1735::
1702:→
1696::
1678:, then a
1653:
1628:∙
1620:→
1615:∙
1473:∙
1429:
1404:∙
1372:→
1356:
1350::
1287:→
1271:
1265::
1169:∙
1139:∙
1118:→
1110:∙
1089::
1081:∙
1033:∘
997:∘
951:→
938::
903:∙
885:∙
874:→
866:∙
848:∙
837::
832:∙
802:∙
784:∙
751:∙
733:∙
700:−
689:
668:
654:∙
602:∘
532:⋯
529:→
455:−
437:−
429:→
426:⋯
390:
326:
217:D-modules
135:D-modules
79:chain map
34:) of an
6722:(2019).
6712:36131259
6684:(1994).
6618:(2003),
6528:(2006),
6362:16236000
6222:See also
6051:) and D(
5225:cokernel
3537:′
3519:, where
3500:′
3037:′
2450:forms a
1684:homotopy
503:→
476:→
446:→
143:D-branes
83:homology
6704:1269324
6675:2182076
6642:1950475
6603:1453167
6568:0932640
6519:1258406
6481:1421815
6463:671–701
6154:be any
6129:over a
6125:on the
5768:∗
5728:or the
5726:sheaves
5592:enough
5348:functor
5319:, i.e.
5175:sheaves
4810:→ Cone(
4492:bounded
4360:Remarks
3084:colimit
1514:; then
280:sheaves
274:over a
272:modules
224:spectra
55:objects
6730:
6710:
6702:
6692:
6673:
6663:
6640:
6630:
6601:
6593:
6566:
6558:
6517:
6509:
6479:
6469:
6360:
6292:
6189:be an
6182:-Rep).
6156:quiver
6136:. Let
5740:, Ext(
5708:↦ Hom(
5696:↦ Hom(
5223:, the
5204:to an
4645:to be
4621:Since
3198:sets.
1156:, and
717:. If
266:be an
173:scheme
111:SGA 4½
23:, the
6535:(PDF)
6416:(PDF)
6358:S2CID
6340:arXiv
6254:Notes
6131:field
5957:maps
5822:)) →
5818:(Inj(
5704:) or
5342:* of
5321:X = 0
5241:exact
5088:is a
4943:as a
4928:→ 0.
4625:is a
4594:is a
4584:class
4490:) or
2902:from
2456:roofs
2337:When
1940:is a
6728:ISBN
6708:OCLC
6690:ISBN
6661:ISBN
6628:ISBN
6591:ISSN
6556:ISSN
6507:ISSN
6467:ISBN
6290:ISBN
6197:its
6185:Let
6170:and
6158:and
6150:Let
6075:Let
6059:and
6043:and
6007:) ≅
5846:) =
5830:) →
5814:) ≅
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5624:and
5590:has
5457:and
4955:and
4937:D(A)
4914:D(A)
4814:) →
4800:D(A)
4796:D(A)
4631:D(A)
4623:K(A)
4573:D(A)
4571:and
4569:D(A)
4565:D(A)
4527:for
4464:for
4401:for
2956:are
2069:and
2016:and
1582:and
768:and
276:ring
242:Let
213:Sato
196:and
188:and
145:and
137:and
6583:239
6497:doi
6350:doi
6067:in
5965:to
5961:in
5759:or
5724:on
5645:).
5609:to
5334:of
5235:of
5185:or
5177:on
5173:of
5032:Ext
4980:Hom
4959:in
4892:in
4838:of
4580:set
4427:),
3408:in
3272:Hom
3229:is
3186:Hom
3002:Hom
2970:lim
2929:to
2498:in
2178:Kom
2142:Kom
1650:Kom
1642:in
1562:If
1426:Kom
1353:Kom
1268:Kom
665:ker
387:Kom
351:of
323:Kom
226:in
198:Ext
194:Tor
190:Hom
160:In
69:in
57:of
19:In
6747::
6706:.
6700:MR
6698:.
6671:MR
6669:,
6659:,
6638:MR
6636:,
6626:,
6599:MR
6597:,
6589:,
6564:MR
6562:,
6552:65
6550:,
6546:,
6515:MR
6513:,
6505:,
6493:27
6477:MR
6475:,
6465:,
6443:,
6437:,
6356:.
6348:.
6336:79
6334:.
6193:,
6013:RF
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5864:RF
5860:RF
5852:RF
5840:RF
5790::
5788:RF
5781:RF
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5712:,
5700:,
5688::
5665:→
5661::
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5162:.
4967:,
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2590:,
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686:im
234:.
149:.
93:.
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6714:.
6499::
6418:.
6364:.
6352::
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6217:.
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6087:o
6084:C
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6057:A
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6029:A
6021:A
6015:.
6011:∘
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6003:∘
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5997:R
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5986:i
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5980:(
5978:F
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5526:=
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5511:(
5506:)
5503:A
5500:(
5497:D
5492:m
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5440:A
5423:)
5420:)
5415:A
5410:(
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5271:0
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5125:(
5122:D
5116:)
5111:A
5106:(
5103:K
5064:.
5061:)
5058:Y
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5052:X
5049:(
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5038:A
5027:=
5024:)
5021:]
5018:j
5015:[
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5009:,
5006:X
5003:(
4998:)
4993:A
4988:(
4985:D
4965:j
4961:A
4957:Y
4953:X
4941:A
4933:A
4926:X
4922:X
4918:X
4910:X
4906:Z
4902:Y
4898:X
4894:A
4877:0
4871:Z
4865:Y
4859:X
4853:0
4840:f
4832:f
4828:Y
4824:X
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4808:Y
4804:X
4779:.
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4760:)
4756:1
4750:(
4747:=
4742:]
4739:n
4736:[
4733:X
4729:d
4702:,
4697:i
4694:+
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4683:=
4678:i
4674:]
4670:n
4667:[
4664:X
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4635:n
4612:A
4610:(
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4604:A
4602:(
4600:D
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4588:A
4551:0
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4540:n
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4515:0
4512:=
4507:n
4503:X
4494:(
4478:0
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4449:=
4444:n
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4431:(
4415:0
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4389:0
4386:=
4381:n
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4368:(
4341:)
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4335:)
4332:1
4326:n
4323:(
4320:+
4317:[
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4283:(
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4273:H
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4238:0
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4209::
4204:0
4201:,
4198:1
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4064:0
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3998:1
3992:E
3967:1
3961:n
3955:E
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3934:E
3921:0
3877:0
3867:0
3847:0
3835:n
3829:E
3816:0
3789:]
3786:)
3783:1
3777:n
3774:(
3771:+
3768:[
3763:n
3757:E
3746:0
3740:E
3734::
3708:0
3700:0
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3684:0
3681:,
3678:1
3658:2
3652:n
3649:,
3646:1
3640:n
3625:1
3619:n
3613:E
3603:1
3597:n
3594:,
3591:n
3576:n
3570:E
3561:0
3534:Y
3527:Y
3507:Y
3497:Y
3490:X
3462:)
3457:A
3452:(
3449:D
3429:)
3424:A
3419:(
3416:K
3390:I
3377:X
3350:X
3329:0
3326:=
3323:)
3314:I
3310:,
3301:X
3297:(
3289:)
3284:A
3279:(
3276:K
3245:X
3232:K
3211:I
3166:)
3161:A
3156:(
3153:D
3130:A
3100:X
3095:I
3067:,
3064:)
3055:Y
3051:,
3042:)
3034:X
3030:(
3027:(
3019:)
3014:A
3009:(
3006:K
2985:X
2980:I
2938:Y
2911:X
2890:)
2885:A
2880:(
2877:D
2851:X
2824:X
2803:)
2798:A
2793:(
2790:K
2762:X
2757:I
2730:X
2704:1
2697:s
2690:f
2664:Y
2651:Z
2644:f
2618:X
2605:Z
2598:s
2572:Z
2551:)
2548:f
2545:,
2542:s
2539:(
2519:)
2514:A
2509:(
2506:D
2480:Y
2467:X
2438:)
2433:A
2428:(
2425:K
2399:A
2371:A
2347:A
2319:A
2299:A
2297:(
2295:K
2291:A
2289:(
2287:D
2266:)
2261:A
2256:(
2253:D
2233:Q
2213:)
2208:A
2203:(
2200:K
2194:)
2189:A
2184:(
2158:)
2153:A
2148:(
2122:)
2117:A
2112:(
2109:K
2078:Y
2051:X
2030:g
2024:f
2004:f
1998:g
1972:X
1959:Y
1952:g
1922:Y
1909:X
1902:f
1892:i
1876:i
1871:X
1867:d
1858:1
1855:+
1852:i
1848:h
1844:+
1839:i
1835:h
1826:1
1820:i
1815:Y
1811:d
1807:=
1802:i
1798:g
1789:i
1785:f
1762:1
1756:i
1752:Y
1743:i
1739:X
1730:i
1726:h
1705:g
1699:f
1693:h
1666:)
1661:A
1656:(
1624:Y
1611:X
1590:g
1570:f
1542:Q
1522:F
1500:C
1478:)
1469:f
1465:(
1462:F
1442:)
1437:A
1432:(
1400:f
1377:C
1369:)
1364:A
1359:(
1347:F
1325:C
1303:)
1298:A
1293:(
1290:D
1284:)
1279:A
1274:(
1262:Q
1242:)
1237:A
1232:(
1229:D
1197:A
1165:f
1144:)
1135:Y
1131:(
1126:i
1122:H
1115:)
1106:X
1102:(
1097:i
1093:H
1086:)
1077:f
1073:(
1068:i
1064:H
1041:i
1037:f
1028:i
1023:Y
1019:d
1015:=
1010:i
1005:X
1001:d
992:1
989:+
986:i
982:f
959:i
955:Y
946:i
942:X
933:i
929:f
908:)
898:Y
894:d
890:,
881:Y
877:(
871:)
861:X
857:d
853:,
844:X
840:(
828:f
807:)
797:Y
793:d
789:,
780:Y
776:(
756:)
746:X
742:d
738:,
729:X
725:(
703:1
697:i
693:d
682:/
676:i
672:d
662:=
659:)
650:X
646:(
641:i
637:H
626:i
610:i
606:d
597:1
594:+
591:i
587:d
564:A
552:X
535:,
524:2
520:X
512:1
508:d
497:1
493:X
485:0
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470:0
466:X
458:1
451:d
440:1
433:X
403:)
398:A
393:(
365:A
339:)
334:A
329:(
303:)
298:A
293:(
290:D
252:A
71:A
63:A
61:(
59:D
51:A
39:A
32:A
30:(
28:D
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