Optical properties of carbon nanotubes

In mathematics, particularly in abstract algebra and homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects in an abelian category), which is used to describe the structure of a specific module or object of this category. In particular, projective and injective resolutions induce a quasi-isomorphism between the exact sequence and the module, which may be regarded as a weak equivalence, with the resolution having nicer properties as a space.

Generally, the objects in the sequence are restricted to have some property P (for example to be free). Thus one speaks of a P resolution: for example, a flat resolution, a free resolution, an injective resolution, a projective resolution. The sequence is supposed to be infinite to the left (to the right for a coresolution). However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero.

Resolutions of modules

Definitions

Given a module M over a ring R, a left resolution (or simply resolution) of M is an exact sequence (possibly infinite) of R-modules

$\cdots \stackrel{d_{n+1}}{⟶}{E}_n\stackrel{d_n}{⟶}\cdots \stackrel{d_3}{⟶}{E}_2\stackrel{d_2}{⟶}{E}_1\stackrel{d_1}{⟶}{E}_0\stackrel{ϵ}{⟶}M⟶0,$

with all the E_i modules over R. The homomorphisms d_i 's are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as

$E_{\bullet }\stackrel{ϵ}{⟶}M⟶0.$

The dual notion is that of a right resolution (or coresolution, or simply resolution). Specifically, given a module M over a ring R, a right resolution is a possibly infinite exact sequence of R-modules

$0⟶M\stackrel{ϵ}{⟶}{C}^0\stackrel{d^0}{⟶}{C}^1\stackrel{d^1}{⟶}{C}^2\stackrel{d^2}{⟶}\cdots \stackrel{d^{n-1}}{⟶}{C}^n\stackrel{d^n}{⟶}\cdots ,$

where each Cⁱ is an R-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as

$0⟶M\stackrel{ϵ}{⟶}{C}^{\bullet }.$

A (co)resolution is said to be finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index n labeling a nonzero module in the finite resolution.

Free, projective, injective, and flat resolutions

In many circumstances conditions are imposed on the modules E_i resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules E_i are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the E_i are projective and flat R-modules, respectively. Injective resolutions are right resolutions whose Cⁱ are all injective modules.

Every R-module possesses a free left resolution. A fortiori, every module also admits projective and flat resolutions. The proof idea is to define E₀ to be the free R-module generated by the elements of M, and then E₁ to be the free R-module generated by the elements of the kernel of the natural map E₀ → M etc. Dually, every R-module possesses an injective resolution. Flat resolutions can be used to compute Tor functors.

Projective resolution of a module M is unique up to a chain homotopy, i.e., given two projective resolution P₀ → M and P₁ → M of M there exists a chain homotopy between them.

Resolutions are used to define homological dimensions. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd(M). For example, a module has projective dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative local ring R, the projective dimension is finite if and only if R is regular and in this case it coincides with the Krull dimension of R. Analogously, the injective dimension id(M) and flat dimension fd(M) are defined for modules also.

The injective and projective dimensions are used on the category of right R modules to define a homological dimension for R called the right global dimension of R. Similarly, flat dimension is used to define weak global dimension. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a semisimple ring, and a ring has weak global dimension 0 if and only if it is a von Neumann regular ring.

Graded modules and algebras

Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules E_i may be graded in such a way that the d_i and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each E_i is minimal. The number of basis elements of each E_i and their degrees are the same for all the minimal free resolutions of a graded module.

If I is a homogeneous ideal in a polynomial ring over a field, the Castelnuovo-Mumford regularity of the projective algebraic set defined by I is the minimal integer r such that the degrees of the basis elements of the E_i in a minimal free resolution of I are all lower than r-i.

Examples

A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence in a graded algebra finitely generated over a field.

Let X be an aspherical space, i.e., its universal cover E is contractible. Then every singular (or simplicial) chain complex of E is a free resolution of the module Z not only over the ring Z but also over the group ring Z [π₁(X)].

Resolutions in abelian categories

The definition of resolutions of an object M in an abelian category A is the same as above, but the E_i and Cⁱ are objects in A, and all maps involved are morphisms in A.

The analogous notion of projective and injective modules are projective and injective objects, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category A. If every object of A has a projective (resp. injective) resolution, then A is said to have enough projectives (resp. enough injectives). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every R-module has an injective resolution, but this resolution is not functorial, i.e., given a homomorphism M → M' , together with injective resolutions

$0\to M\to I_{*},0\to M^{\prime }\to I{\text{'}}_{*},$

there is in general no functorial way of obtaining a map between $I_{*}$ and $I{\text{'}}_{*}$.

Acyclic resolution

In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given functor. Therefore, in many situations, the notion of acyclic resolutions is used: given a left exact functor F: A → B between two abelian categories, a resolution

$0\to M\to E_0\to E_1\to E_2\to \dots$

of an object M of A is called F-acyclic, if the derived functors R_iF(E_n) vanish for all i>0 and n≥0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution.

For example, given a R module M, the tensor product   ${\otimes }_{R}M$ is a right exact functor Mod(R) → Mod(R). Every flat resolution is acyclic with respect to this functor. A flat resolution is acyclic for the tensor product by every M. Similarly, resolutions that are acyclic for all the functors Hom( ⋅ , M) are the projective resolutions and those that are acyclic for the functors Hom(M,  ⋅ ) are the injective resolutions.

Any injective (projective) resolution is F-acyclic for any left exact (right exact, respectively) functor.

The importance of acyclic resolutions lies in the fact that the derived functors R_iF (of a left exact functor, and likewise L_iF of a right exact functor) can be obtained from as the homology of F-acyclic resolutions: given an acyclic resolution $E_{*}$ of an object M, we have

$R_{i}F(M)=H_{i}F(E_{*}),$

where right hand side is the i-th homology object of the complex $F(E_{*}).$

This situation applies in many situations. For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves ${\mathcal{𝒞}}^{*}(M)$ of smooth differential forms: $0\to R\subset {\mathcal{𝒞}}^0(M)\stackrel{d}{\to }{\mathcal{𝒞}}^1(M)\stackrel{d}{\to }\dots {\mathcal{𝒞}}^{dimM}(M)\to 0.$ The sheaves ${\mathcal{𝒞}}^{*}(M)$ are fine sheaves, which are known to be acyclic with respect to the global section functor $\mathrm{\Gamma }:\mathcal{ℱ}\mapsto \mathcal{ℱ}(M)$. Therefore, the sheaf cohomology, which is the derived functor of the global section functor Γ is computed as ${\mathrm{H}}^i(M,\mathbf{R})={\mathrm{H}}^i({\mathcal{𝒞}}^{*}(M)).$

Similarly Godement resolutions are acyclic with respect to the global sections functor.

See also

• Resolution (disambiguation)
• Hilbert–Burch theorem

Notes

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References

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  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
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