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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 functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.

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 Astérisque (a summary much earlier appeared in 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-modules and microlocal analysis.

Motivations

In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular 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.

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 Sato school adopted the language of derived categories, and the subsequent history of D-modules was of a theory expressed in those terms.

A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both examples of triangulated categories.

Definition

Let A be an abelian category. (Some basic examples are the category of modules over a ring, or the category of sheaves of abelian groups on a topological space.) We obtain the derived category D(A) in several steps:

The basic object is the category Kom(A) of chain complexes

\dots \to X^{- 1} \overset{d^{- 1}}{\to} X^{0} \overset{d^{0}}{\to} X^{1} \overset{d^{1}}{\to} X^{2} \to \dots

in A. Its objects will be the objects of the derived category but its morphisms will be altered.

Pass to the homotopy category of chain complexes K(A) by identifying morphisms which are chain homotopic.
Pass to the derived category D(A) by localizing at the set of quasi-isomorphisms. 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.

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.

From the point of view of 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'.

Remarks

For certain purposes (see below) one uses bounded-below (Xⁿ = 0 for n << 0), bounded-above (Xⁿ = 0 for n >> 0) or bounded (Xⁿ = 0 for |n| >> 0) complexes instead of unbounded ones. The corresponding derived categories are usually denoted D⁺(A), D^-(A) and D^b(A), respectively.

If one adopts the classical point of view on categories, that there is a set of morphisms from one object to another (not just a 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 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.^[1] 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.

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.

Since K(A) is a triangulated category, its localization D(A) is also triangulated. For an integer n and a complex X, define^[2] the complex X[n] to be X shifted down by n, so that

X [n]^{i} = X^{n + i},

with differential

d_{X [n]} = (- 1)^{n} d_{X} .

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 of f. In particular, for a short exact sequence

0 \to X \to Y \to Z \to 0

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.^[3]

By viewing an object of A as a complex concentrated in degree zero, the derived category D(A) contains A as a full subcategory. More interestingly, morphisms in the derived category include information about all Ext groups: for any objects X and Y in A and any integer j,

{Hom}_{D (𝒜)} (X, Y [j]) = {Ext}_{𝒜}^{j} (X, Y) .

Projective and injective resolutions

One can easily show that a 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

K (𝒜) \to D (𝒜) .

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 resolutions. In both cases, the restriction of the above canonical functor to an appropriate subcategory will be an equivalence of categories.

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 sheaves on topological spaces or more advanced cohomology theories like étale cohomology or group cohomology.

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 abelian category has enough injectives. Embedding X into some injective object I⁰, the cokernel of this map into some injective I¹ etc., one constructs an injective resolution of X, i.e. an exact (in general infinite) sequence

0 \to X \to I^{0} \to I^{1} \to \dots,

where the I* are injective objects. This idea generalizes to give resolutions of bounded-below complexes X, i.e. Xⁿ = 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.

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

D^{+} (𝒜) \to K^{+} (I n j (𝒜))

from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in A.

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,

{H o m}_{D (A)} (X, Y) = {H o m}_{K (A)} (X, Y) .

Dually, assuming that A has enough 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.

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: Template:Harvtxt uses so-called K-injective and K-projective resolutions, Template:Harvtxt and (in a slightly different language) Template:Harvtxt introduced so called cell-modules and semi-free modules, respectively.

More generally, carefully adapting the definitions, it is possible to define the derived category of an exact category Template:Harv.

The relation to derived functors

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:

right derived functors come from left exact functors and are calculated via injective resolutions
left derived functors come from right exact functors and are calculated via projective resolutions

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 sheaves or the direct image functor. Their right derived functors are Extⁿ(–,A), Extⁿ(A,–), Hⁿ(X, F) or Rⁿf_∗ (F), respectively.

The derived category allows us to encapsulate all derived functors RⁿF in one functor, namely the so-called total derived functor RF: D⁺(A) → D⁺(B). It is the following composition: D⁺(A) ≅ K⁺(Inj(A)) → K⁺(B) → D⁺(B), where the first equivalence of categories is described above. The classical derived functors are related to the total one via RⁿF(X) = Hⁿ(RF(X)). One might say that the RⁿF forget the chain complex and keep only the cohomologies, whereas RF does keep track of the complexes.

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

𝒜 \overset{F}{\to} ℬ \overset{G}{\to} 𝒞,

such that F maps injective objects in A to G-acyclics (i.e. RⁱG(F(I)) = 0 for all i > 0 and injective I), is an expression of the following identity of total derived functors

R(G∘F) ≅ RG∘RF.

J.-L. Verdier showed how derived functors associated with an abelian category A can be viewed as Kan extensions along embeddings of A into suitable derived categories [Mac Lane].

Notes

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Three textbooks that discuss derived categories are:

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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Template:Weibel IHA

↑ M. Kashiwara and P. Schapira. Categories and Sheaves. Springer-Verlag (2006). Theorem 14.3.1.
↑ S. Gelfand and Y. Manin. Methods of Homological Algebra. Springer-Verlag (2003). III.3.2.
↑ J.-L. Verdier. Astérisque 239. Soc. Math. de France (1996). Appendice to Ch. 1.

[1] M. Kashiwara and P. Schapira. Categories and Sheaves. Springer-Verlag (2006). Theorem 14.3.1.

[2] S. Gelfand and Y. Manin. Methods of Homological Algebra. Springer-Verlag (2003). III.3.2.

[3] J.-L. Verdier. Astérisque 239. Soc. Math. de France (1996). Appendice to Ch. 1.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-== Air Max København Alligevel nedskæringerne gik ==
+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.
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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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+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 ==
+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>
+\cdots \to
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+X^{-1} \xrightarrow{d^{-1}}
+X^0 \xrightarrow{d^0}
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+X^1 \xrightarrow{d^1}
+X^2 \to \cdots</math>
- </ul>
+in ''A''.  Its objects will be the objects of the derived category but its morphisms will be altered.
+* Pass to the [[homotopy category of chain complexes]] ''K''(''A'') by identifying morphisms which are [[chain homotopy|chain homotopic]].
+* 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.
+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.
+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'.
+== Remarks ==
+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.
+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.
+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.
+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
+:<math>X[n]^{i} = X^{n+i},</math>
+with differential
+:<math>d_{X[n]} = (-1)^n d_X.</math>
+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
+:<math>0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0</math>
+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>
+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'',
+:<math>\text{Hom}_{D(\mathcal{A})}(X,Y[j]) = \text{Ext}^j_{\mathcal{A}}(X,Y).</math>
+== Projective and injective resolutions ==
+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
+:<math>K(\mathcal A) \rightarrow D(\mathcal A).</math>
+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]].
+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]].
+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
+:<math>0 \rightarrow X \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots, \, </math>
+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.
+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]]
+:<math>D^+(\mathcal A) \rightarrow K^+(\mathrm{Inj}(\mathcal A))</math>
+from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in ''A''.
+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,
+:<math>\mathrm{Hom}_{D(A)}(X, Y) = \mathrm{Hom}_{K(A)}(X, Y).</math>
+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.
+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.
+More generally, carefully adapting the definitions, it is possible to define the derived category of an [[exact category]] {{Harv|Keller|1996}}.
+== The relation to derived functors ==
+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:
+* right derived functors come from left exact functors and are calculated via injective resolutions
+* left derived functors come from right exact functors and are calculated via projective resolutions
+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>(&ndash;,''A'')]], Ext<sup>''n''</sup>(''A'',&ndash;), [[sheaf cohomology|''H''<sup>''n''</sup>(''X'', ''F'')]] or [[higher direct image functor|''R''<sup>''n''</sup>''f''<sub>&lowast;</sub> (''F'')]], respectively.
+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.
+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
+:<math>\mathcal A \stackrel{F}{\rightarrow} \mathcal B \stackrel{G}{\rightarrow} \mathcal C, \,</math>
+such that ''F'' maps [[injective object]]s in ''A'' to ''G''-acyclics (i.e. ''R''<sup>''i''</sup>''G''(''F''(''I''))&nbsp;=&nbsp;0 for all ''i''&nbsp;>&nbsp;0 and injective ''I''), is an expression of the following identity of total derived functors
+:''R''(''G''∘''F'') ≅ ''RG''∘''RF''.
+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].
+==Notes==
+{{reflist}}
+== References ==
+*{{springer|id=D/d031280|title=Derived category|last=Doorn|first=M.G.M. van }}
+*{{Citation
+| last = Keller
+| first = Bernhard
+| editor = Hazewinkel, M.
+| title = Handbook of algebra
+| year = 1996
+| publisher = North Holland
+| location = Amsterdam
+| isbn = 0-444-82212-7
+| oclc =
+| doi =
+| mr=1421815
+| pages = 671–701
+| chapter = Derived categories and their uses
+| chapterurl=http://www.math.jussieu.fr/~keller/publ/dcu.ps
+}}
+* {{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}}
+*{{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}}
+* {{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}}
+* {{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}}
+Three textbooks that discuss derived categories are:
+* {{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}}
+* {{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}}
+* {{Weibel IHA}}
+[[Category:Homological algebra]]
+[[Category:Category-theoretic categories]]

Luyten 726-8: Difference between revisions

Revision as of 15:23, 25 January 2014

Contents

Motivations

Definition

Remarks

Projective and injective resolutions

The relation to derived functors

Notes

References

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Luyten 726-8: Difference between revisions

Revision as of 15:23, 25 January 2014

Motivations

Definition

Remarks

Projective and injective resolutions

The relation to derived functors

Notes

References

Navigation menu

Search