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| In [[mathematics]], the '''derived category''' ''D''(''A'') of an [[abelian category]] ''A'' is a construction of [[homological algebra]] introduced to refine and in a certain sense to simplify the theory of [[derived functor]]s defined on ''A''. The construction proceeds on the basis that the [[Object (category theory)|objects]] of ''D''(''A'') should be [[chain complex]]es in ''A'', with two such chain complexes considered [[isomorphism|isomorphic]] when there is a [[Chain complex#Chain maps|chain map]] that induces an isomorphism on the level of [[homology (mathematics)|homology]] of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of [[hyperhomology|hypercohomology]]. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated [[spectral sequence]]s.
| | If you compare registry products there are a amount of elements to look out for. Because of the sheer amount of for registry products available found on the Internet at the moment it can be quite easy to be scammed. Something frequently overlooked is the fact that a few of these products usually the reality is end up damaging the PC. And the registry they state they have cleaned can just lead to more problems with a computer than the ones you started with.<br><br>Many of the reliable companies can provide a full funds back guarantee. This means which we have the chance to get a money back if you find the registry cleaning has not delivered what we expected.<br><br>Although this issue affects millions of computer users throughout the world, there is an simple method to fix it. You see, there's 1 reason for a slow loading computer, plus that's because a PC cannot read the files it must run. In a nutshell, this merely signifies that whenever you do anything on Windows, it requirements to read up on how to do it. It's traditionally a rather 'dumb' system, which has to have files to tell it to do everything.<br><br>First, constantly clean the PC plus keep it free of dust and dirt. Dirt clogs up all fans and can cause the PC to overheat. You additionally have to clean up disk room inside order to make the computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin plus remove programs we are not using.<br><br>The final step is to make certain which you clean the registry of the computer. The "registry" is a large database that shops important files, settings & choices, and info. Windows reads the files it needs inside purchase for it to run programs from this database. If the registry gets damaged, infected, or clogged up, then Windows can not be able to correctly access the files it needs for it to load up programs. As this occurs, difficulties plus mistakes like the d3d9.dll error happen. To fix this plus prevent future setbacks, you must download plus run a registry cleaning tool. The highly suggested software is the "Frontline [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]".<br><br>S/w connected error handling - If the blue screen physical memory dump occurs following the installation of s/w application or perhaps a driver it may be which there is program incompatibility. By booting into safe mode plus removing the software you are able to promptly fix this error. We may also try out a "program restore" to revert to an earlier state.<br><br>Most probably when you are experiencing a slow computer it will be a couple years older. You furthermore may not have been told that whilst you utilize the computer everyday; there are certain elements which it requires to continue running in its best performance. We also could not even own any diagnostic tools that usually receive your PC running like new again. So never allow which stop we from getting the program cleaned. With access to the internet you will find the tools which will help you get your program running like new again.<br><br>A registry cleaner is a program that cleans the registry. The Windows registry constantly gets flooded with junk information, info that has not been removed from uninstalled programs, erroneous file association and other computer-misplaced entries. These neat small system software tools are very usual nowadays plus you are able to find quite a few advantageous ones on the Internet. The advantageous ones provide you choice to keep, clean, update, backup, and scan the System Registry. When it finds supposedly unwelcome elements in it, the registry cleaner lists them plus recommends the consumer to delete or repair these orphaned entries and corrupt keys. |
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| The development of the derived category, by [[Alexander Grothendieck]] and his student [[Jean-Louis Verdier]] 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 [[#refVerdierAsterisque|Astérisque]] (a summary much earlier appeared in [[Grothendieck's Séminaire de géométrie algébrique|SGA 4½]]). The axiomatics required an innovation, the concept of [[triangulated category]], and the construction is based on [[localization of a category]], a generalization of [[localization of a ring]]. The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's [[coherent duality]] theory. Derived categories have since become indispensable also outside of [[algebraic geometry]], for example in the formulation of the theory of [[D-module]]s and [[microlocal analysis]].
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| ==Motivations==
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| In [[coherent sheaf]] theory, pushing to the limit of what could be done with [[Serre duality]] without the assumption of a [[non-singular]] [[scheme (mathematics)|scheme]], the need to take a whole complex of sheaves in place of a single ''dualizing sheaf'' became apparent. In fact the [[Cohen-Macaulay ring]] 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' [[tensor product]] and ''Hom'' functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.
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| Despite the level of abstraction, derived categories became accepted over the following decades, especially as a convenient setting for [[sheaf cohomology]]. Perhaps the biggest advance was the formulation of the [[Riemann-Hilbert correspondence]] in dimensions greater than 1 in derived terms, around 1980. The [[Mikio Sato|Sato]] school adopted the language of derived categories, and the subsequent history of [[D-module]]s was of a theory expressed in those terms.
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| A parallel development was the category of [[spectrum (homotopy theory)|spectra]] in [[homotopy theory]]. The homotopy category of spectra and the derived category of a ring are both examples of [[triangulated category|triangulated categories]].
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| == Definition ==
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| Let ''A'' be an [[abelian category]]. (Some basic examples are the category of [[module (mathematics)|modules]] over a [[ring (mathematics)|ring]], or the category of [[sheaf (mathematics)|sheaves]] of abelian groups on a topological space.) We obtain the derived category ''D''(''A'') in several steps:
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| * The basic object is the category Kom(''A'') of [[chain complexes]]
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| ::<math>
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| \cdots \to
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| X^{-1} \xrightarrow{d^{-1}}
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| X^0 \xrightarrow{d^0}
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| X^1 \xrightarrow{d^1}
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| X^2 \to \cdots</math>
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| in ''A''. Its objects will be the objects of the derived category but its morphisms will be altered.
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| * Pass to the [[homotopy category of chain complexes]] ''K''(''A'') by identifying morphisms which are [[chain homotopy|chain homotopic]].
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| * Pass to the derived category ''D''(''A'') by [[localization of a category|localizing]] at the set of [[quasi-isomorphism]]s. Morphisms in the derived category may be explicitly described as ''roofs'' ''X'' ← ''X' ''→ ''Y'', where ''X' ''→ ''X'' is a quasi-isomorphism and ''X' ''→ ''Y'' is any morphism of chain complexes.
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| The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism. But then the simple ''roof'' definition of morphisms must be replaced by a more complicated one using finite strings of morphisms (technically, it is no longer a ''calculus of fractions''). So the one-step construction is more efficient in a way, but more complicated.
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| From the point of view of [[Model category|model categories]], the derived category ''D''(''A'') is the true 'homotopy category' of the category of complexes, whereas ''K''(''A'') might be called the 'naive homotopy category'.
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| == Remarks ==
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| For certain purposes (see below) one uses ''bounded-below'' (''X<sup>n</sup>'' = 0 for ''n << 0''), ''bounded-above'' (''X<sup>n</sup>'' = 0 for ''n >> 0'') or ''bounded'' (''X<sup>n</sup>'' = 0 for ''|n| >> 0'') complexes instead of unbounded ones. The corresponding derived categories are usually denoted ''D<sup>+</sup>(A)'', ''D<sup>-</sup>(A)'' and ''D<sup>b</sup>(A)'', respectively.
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| If one adopts the classical point of view on categories, that there is a [[Set (mathematics)|set]] of morphisms from one object to another (not just a [[Class (set theory)|class]]), then one has to give an additional argument to prove this. If, for example, the abelian category ''A'' is small, i.e. has only a set of objects, then this issue will be no problem. Also, if ''A'' is a [[Grothendieck category|Grothendieck abelian category]], then the derived category ''D''(''A'') is equivalent to a full subcategory of the homotopy category ''K''(''A''), and hence has only a set of morphisms from one object to another.<ref>M. Kashiwara and P. Schapira. ''Categories and Sheaves.'' Springer-Verlag (2006). Theorem 14.3.1.</ref> 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.
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| 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.
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| Since ''K(A)'' is a [[triangulated category]], its localization ''D(A)'' is also triangulated. For an integer ''n'' and a complex ''X'', define<ref>S. Gelfand and Y. Manin. ''Methods of Homological Algebra.'' Springer-Verlag (2003). III.3.2.</ref> the complex ''X''[''n''] to be ''X'' shifted down by ''n'', so that
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| :<math>X[n]^{i} = X^{n+i},</math>
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| with differential
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| :<math>d_{X[n]} = (-1)^n d_X.</math>
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| By definition, a distinguished triangle in ''D(A)'' is a triangle that is isomorphic in ''D(A)'' to the triangle ''X'' → ''Y'' → Cone(''f'') → ''X''[1] for some map of complexes ''f'': ''X'' → ''Y''. Here Cone(''f'') denotes the [[mapping cone (homological algebra)|mapping cone]] of ''f''. In particular, for a short exact sequence
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| :<math>0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0</math>
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| in ''A'', the triangle ''X'' → ''Y'' → ''Z'' → ''X''[1] is distinguished in ''D(A)''. Verdier explained that the definition of the shift ''X''[1] is forced by requiring ''X''[1] to be the cone of the morphism ''X'' → 0.<ref>J.-L. Verdier. ''Astérisque'' 239. Soc. Math. de France (1996). Appendice to Ch. 1.</ref>
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| By viewing an object of ''A'' as a complex concentrated in degree zero, the derived category ''D(A)'' contains ''A'' as a [[subcategory|full subcategory]]. More interestingly, morphisms in the derived category include information about all [[Ext functor|Ext groups]]: for any objects ''X'' and ''Y'' in ''A'' and any integer ''j'',
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| :<math>\text{Hom}_{D(\mathcal{A})}(X,Y[j]) = \text{Ext}^j_{\mathcal{A}}(X,Y).</math>
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| == Projective and injective resolutions ==
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| One can easily show that a [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalence]] is a [[quasi-isomorphism]], 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
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| :<math>K(\mathcal A) \rightarrow D(\mathcal A).</math>
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| 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 [[injective resolution]]s. In both cases, the restriction of the above canonical functor to an appropriate subcategory will be an [[equivalence of categories]].
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| 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 [[derived functors]], which in turn have important applications in [[cohomology]] of [[sheaf (mathematics)|sheaves]] on [[topological space]]s or more advanced cohomology theories like [[étale cohomology]] or [[group cohomology]].
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| In order to apply this technique, one has to assume that the abelian category in question has ''enough injectives'', which means that every object ''X'' of the category admits a [[monomorphism]] to an [[injective object]] ''I''. (Neither the map nor the injective object has to be uniquely specified.) For example, every [[Grothendieck category|Grothendieck abelian category]] has enough injectives. Embedding ''X'' into some injective object ''I''<sup>0</sup>, the [[cokernel]] of this map into some injective ''I''<sup>1</sup> etc., one constructs an ''injective resolution'' of ''X'', i.e. an [[exact sequence|exact]] (in general infinite) sequence
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| :<math>0 \rightarrow X \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots, \, </math>
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| where the ''I''* are injective objects. This idea generalizes to give resolutions of bounded-below complexes ''X'', i.e. ''X<sup>n</sup> = 0'' for sufficiently small ''n''. 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.
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| This is the point where the homotopy category comes into play again: mapping an object ''X'' of ''A'' to (any) injective resolution ''I''* of ''A'' extends to a [[functor]]
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| :<math>D^+(\mathcal A) \rightarrow K^+(\mathrm{Inj}(\mathcal A))</math>
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| from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in ''A''.
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| 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(''X'',''Y'') in the derived category may be computed by resolving both ''X'' and ''Y'' and computing the morphisms in the homotopy category, which is at least theoretically easier. In fact, it is enough to resolve ''Y'': for any complex ''X'' and any bounded below complex ''Y'' of injectives,
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| :<math>\mathrm{Hom}_{D(A)}(X, Y) = \mathrm{Hom}_{K(A)}(X, Y).</math>
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| Dually, assuming that ''A'' has ''enough [[projective object|projectives]]'', i.e. for every object ''X'' there is an [[epimorphism]] from a projective object ''P'' to ''X'', one can use projective resolutions instead of injective ones.
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| In addition to these resolution techniques there are similar ones which apply to special cases, and which elegantly avoid the problem with bounded-above or -below restrictions: {{harvtxt|Spaltenstein|1988}} uses so-called ''K-injective'' and ''K-projective'' resolutions, {{harvtxt|May|2006}} and (in a slightly different language) {{harvtxt|Keller|1994}} introduced so called ''cell-modules'' and ''semi-free'' modules, respectively.
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| More generally, carefully adapting the definitions, it is possible to define the derived category of an [[exact category]] {{Harv|Keller|1996}}.
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| == The relation to derived functors ==
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| The derived category is a natural framework to define and study [[derived functors]]. In the following, let ''F'': ''A'' → ''B'' be a functor of abelian categories. There are two dual concepts:
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| * right derived functors come from left exact functors and are calculated via injective resolutions
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| * left derived functors come from right exact functors and are calculated via projective resolutions
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| In the following we will describe right derived functors. So, assume that ''F'' is left exact. Typical examples are ''F'': ''A'' → Ab given by ''X'' ↦ Hom(''X'', ''A'') or ''X'' ↦ Hom(''A'', ''X'') for some fixed object ''A'', or the [[global sections functor]] on [[sheaf (mathematics)|sheaves]] or the [[direct image functor]]. Their right derived functors are [[Ext functors|Ext<sup>''n''</sup>(–,''A'')]], Ext<sup>''n''</sup>(''A'',–), [[sheaf cohomology|''H''<sup>''n''</sup>(''X'', ''F'')]] or [[higher direct image functor|''R''<sup>''n''</sup>''f''<sub>∗</sub> (''F'')]], respectively.
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| The derived category allows us to encapsulate all derived functors ''R<sup>n</sup>F'' in one functor, namely the so-called ''total derived functor'' ''RF'': ''D''<sup>+</sup>(''A'') → ''D''<sup>+</sup>(''B''). It is the following composition: ''D''<sup>+</sup>(''A'') ≅ ''K''<sup>+</sup>(Inj(''A'')) → ''K''<sup>+</sup>(''B'') → ''D''<sup>+</sup>(''B''), where the first equivalence of categories is described above. The classical derived functors are related to the total one via ''R<sup>n</sup>F''(''X'') = ''H<sup>n</sup>''(''RF''(''X'')). One might say that the ''R<sup>n</sup>F'' forget the chain complex and keep only the cohomologies, whereas ''RF'' does keep track of the complexes.
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| Derived categories are, in a sense, the "right" place to study these functors. For example, the [[Grothendieck spectral sequence]] of a composition of two functors
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| :<math>\mathcal A \stackrel{F}{\rightarrow} \mathcal B \stackrel{G}{\rightarrow} \mathcal C, \,</math>
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| such that ''F'' maps [[injective object]]s in ''A'' to ''G''-acyclics (i.e. ''R''<sup>''i''</sup>''G''(''F''(''I'')) = 0 for all ''i'' > 0 and injective ''I''), is an expression of the following identity of total derived functors
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| :''R''(''G''∘''F'') ≅ ''RG''∘''RF''.
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| J.-L. Verdier showed how derived functors associated with an abelian category ''A'' can be viewed as [[Kan extension]]s along embeddings of ''A'' into suitable derived categories [Mac Lane].
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{springer|id=D/d031280|title=Derived category|last=Doorn|first=M.G.M. van }}
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| *{{Citation
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| | last = Keller
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| | first = Bernhard
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| | editor = Hazewinkel, M.
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| | title = Handbook of algebra
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| | year = 1996
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| | publisher = North Holland
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| | location = Amsterdam
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| | isbn = 0-444-82212-7
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| | oclc =
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| | doi =
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| | mr=1421815
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| | pages = 671–701
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| | chapter = Derived categories and their uses
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| | chapterurl=http://www.math.jussieu.fr/~keller/publ/dcu.ps
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| }}
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| * {{Citation | last1=Keller | first1=Bernhard | title=Deriving DG categories | url=http://www.numdam.org/numdam-bin/fitem?id=ASENS_1994_4_27_1_63_0 | mr=1258406 | year=1994 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=27 | issue=1 | pages=63–102}}
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| *{{Citation | last=May |first=J. P. |authorlink=J. Peter May |url=http://www.math.uchicago.edu/~may/MISC/DerivedCats.pdf |title=Derived categories from a topological point of view |year=2006}}
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| * {{Citation | last1=Spaltenstein | first1=N. | title=Resolutions of unbounded complexes | url=http://www.numdam.org/numdam-bin/fitem?id=CM_1988__65_2_121_0 | mr=932640 | year=1988 | journal=Compositio Mathematica | issn=0010-437X | volume=65 | issue=2 | pages=121–154}}
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| * {{Citation | ref=refVerdierAsterisque | last1=Verdier | first1=Jean-Louis | author1-link=Jean-Louis Verdier | title=Des Catégories Dérivées des Catégories Abéliennes | publisher=[[Société Mathématique de France]] | language = French | location=Paris | journal=Astérisque | issn=0303-1179 | volume=239 | mr=1453167 | year=1996}}
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| Three textbooks that discuss derived categories are:
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| * {{Citation | last1=Gelfand | first1=Sergei I. | last2=Manin | first2=Yuri Ivanovich | author2-link= Yuri Ivanovich Manin | title=Methods of Homological Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-43583-9 | mr=1950475 | year=2003}}
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| * {{Citation | last1=Kashiwara | first1=Masaki | author1-link= Masaki Kashiwara | last2=Schapira | first2=Pierre | title=Categories and Sheaves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der mathematischen Wissenschaften | isbn=978-3-540-27949-5 | mr=2182076 | year=2006}}
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| * {{Weibel IHA}}
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| [[Category:Homological algebra]]
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| [[Category:Category-theoretic categories]]
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If you compare registry products there are a amount of elements to look out for. Because of the sheer amount of for registry products available found on the Internet at the moment it can be quite easy to be scammed. Something frequently overlooked is the fact that a few of these products usually the reality is end up damaging the PC. And the registry they state they have cleaned can just lead to more problems with a computer than the ones you started with.
Many of the reliable companies can provide a full funds back guarantee. This means which we have the chance to get a money back if you find the registry cleaning has not delivered what we expected.
Although this issue affects millions of computer users throughout the world, there is an simple method to fix it. You see, there's 1 reason for a slow loading computer, plus that's because a PC cannot read the files it must run. In a nutshell, this merely signifies that whenever you do anything on Windows, it requirements to read up on how to do it. It's traditionally a rather 'dumb' system, which has to have files to tell it to do everything.
First, constantly clean the PC plus keep it free of dust and dirt. Dirt clogs up all fans and can cause the PC to overheat. You additionally have to clean up disk room inside order to make the computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin plus remove programs we are not using.
The final step is to make certain which you clean the registry of the computer. The "registry" is a large database that shops important files, settings & choices, and info. Windows reads the files it needs inside purchase for it to run programs from this database. If the registry gets damaged, infected, or clogged up, then Windows can not be able to correctly access the files it needs for it to load up programs. As this occurs, difficulties plus mistakes like the d3d9.dll error happen. To fix this plus prevent future setbacks, you must download plus run a registry cleaning tool. The highly suggested software is the "Frontline tuneup utilities".
S/w connected error handling - If the blue screen physical memory dump occurs following the installation of s/w application or perhaps a driver it may be which there is program incompatibility. By booting into safe mode plus removing the software you are able to promptly fix this error. We may also try out a "program restore" to revert to an earlier state.
Most probably when you are experiencing a slow computer it will be a couple years older. You furthermore may not have been told that whilst you utilize the computer everyday; there are certain elements which it requires to continue running in its best performance. We also could not even own any diagnostic tools that usually receive your PC running like new again. So never allow which stop we from getting the program cleaned. With access to the internet you will find the tools which will help you get your program running like new again.
A registry cleaner is a program that cleans the registry. The Windows registry constantly gets flooded with junk information, info that has not been removed from uninstalled programs, erroneous file association and other computer-misplaced entries. These neat small system software tools are very usual nowadays plus you are able to find quite a few advantageous ones on the Internet. The advantageous ones provide you choice to keep, clean, update, backup, and scan the System Registry. When it finds supposedly unwelcome elements in it, the registry cleaner lists them plus recommends the consumer to delete or repair these orphaned entries and corrupt keys.