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In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category.

Formal definition

Algebraic objects

We start with the definition of an inverse (or projective) system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (A_i)_i∈I be a family of groups and suppose we have a family of homomorphisms f_ij: A_j → A_i for all i ≤ j (note the order) with the following properties:

f_ii is the identity on A_i,
f_ik = f_ij o f_jk for all i ≤ j ≤ k.

Then the pair ((A_i)_i∈I, (f_ij)_{i≤ j∈I}) is called an inverse system of groups and morphisms over I, and the morphisms f_ij are called the transition morphisms of the system.

We define the inverse limit of the inverse system ((A_i)_i∈I, (f_ij)_{i≤ j∈I}) as a particular subgroup of the direct product of the A_i's:

\varprojlim _{i\in I}A_{i}={\Big \{}{\vec {a}}\in \prod _{i\in I}A_{i}\;{\Big |}\;a_{i}=f_{ij}(a_{j}){\mbox{ for all }}i\leq j{\mbox{ in }}I{\Big \}}.

The inverse limit, A, comes equipped with natural projections π_i: A → A_i which pick out the ith component of the direct product for each i in I. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the A_i's are sets,^[1] semigroups,^[1] topological spaces,^[1] rings, modules (over a fixed ring), algebras (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding category. The inverse limit will also belong to that category.

General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (X_i, f_ij) be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_i: X → X_i (called projections) satisfying π_i = f_ij o π_j for all i ≤ j. The pair (X, π_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u: Y → X making all the "obvious" identities true; i.e., the diagram

must commute for all i ≤ j. The inverse limit is often denoted

X=\varprojlim X_{i}

with the inverse system (X_i, f_ij) being understood.

The inverse limit might not exist in a category. If it does, however, it is unique in a strong sense: given any other inverse limit X′ there exists a unique isomorphism X′ → X commuting with the projection maps.

We note that an inverse system in a category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j. An inverse system is then just a contravariant functor I → C. And the inverse limit functor $\varprojlim :C^{I^{op}}\rightarrow C$ is a covariant functor.

Examples

The ring of p-adic integers is the inverse limit of the rings Z/pⁿZ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here.
The ring $\textstyle R[[t]]$ of formal power series over a commutative ring R can be thought of as the inverse limit of the rings $\textstyle R[t]/t^{n}R[t]$ , indexed by the natural numbers as usually ordered, with the morphisms from $\textstyle R[t]/t^{n+j}R[t]$ to $\textstyle R[t]/t^{n}R[t]$ given by the natural projection.
Pro-finite groups are defined as inverse limits of (discrete) finite groups.
Let the index set I of an inverse system (X_i, f_ij) have a greatest element m. Then the natural projection π_m: X → X_m is an isomorphism.
Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology.
- The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete, the limit space is totally disconnected. This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings).
Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product.
Let I consist of three elements i, j, and k with i ≤ j and i ≤ k (not directed). The inverse limit of any corresponding inverse system is the pullback.

Derived functors of the inverse limit

For an abelian category C, the inverse limit functor

\varprojlim :C^{I}\rightarrow C

is left exact. If I is ordered (not simply partially ordered) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms f_ij that ensures the exactness of $\varprojlim$ . Specifically, Eilenberg constructed a functor

\varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab}

(pronounced "lim one") such that if (A_i, f_ij), (B_i, g_ij), and (C_i, h_ij) are three projective systems of abelian groups, and

0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0

is a short exact sequence of inverse systems, then

0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{i}

is an exact sequence in Ab.

Mittag-Leffler condition

If the ranges of the morphisms of the inverse system of abelian groups (A_i, f_ij) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j : $f_{kj}(A_{j})=f_{ki}(A_{i})$ one says that the system satisfies the Mittag-Leffler condition. This condition implies that $\varprojlim {}^{1}A_{i}=0.$

For a discussion of the name "Mittag-Leffler" in its relation with the Mittag-Leffler theorem, see this thread on MathOverflow.

The following situations are examples where the Mittag-Leffler condition is satisfied:

a system in which the morphisms f_ij are surjective
a system of finite-dimensional vector spaces.

An example where this is non-zero is obtained by taking I to be the non-negative integers, letting A_i = pⁱZ, B_i = Z, and C_i = B_i / A_i = Z/pⁱZ. Then

\varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z}

where Z_p denotes the p-adic integers.

Further results

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C^I, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted

R^{n}\varprojlim :C^{I}\rightarrow C.

In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim¹ on Ab^I to series of functors limⁿ such that

\varprojlim {}^{n}\cong R^{n}\varprojlim .

It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim¹ A_i = 0 for (A_i, f_ij) an inverse system with surjective transition morphisms and I the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences"). However, in 2002, Amnon Neeman and Pierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim¹ A_i ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).

Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if I has cardinality $\aleph _{d}$ (the dth infinite cardinal), then Rⁿlim is zero for all n ≥ d + 2. This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which lim^n, on diagrams indexed by a countable set, is nonzero for n>1).

Related concepts and generalizations

The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.

Notes

↑ ^1.0 ^1.1 ^1.2 John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.

References

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
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.

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Section 3.5 of Template:Weibel IHA

de:Limes (Kategorientheorie)

[same-construction-1] 1.0 ^1.1 ^1.2 John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.

[1]

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+In [[mathematics]], the '''inverse limit''' (also called the '''projective limit''') is a construction which allows one to "glue together" several related [[mathematical object|objects]], the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any [[category (mathematics)|category]].
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+We start with the definition of an '''inverse''' (or '''projective''') '''system''' of [[group (mathematics)|groups]] and [[group homomorphism|homomorphisms]]. Let (''I'', ≤) be a [[directed set|directed]] [[poset]] (not all authors require ''I'' to be directed). Let (''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub> be a [[indexed family|family]] of groups and suppose we have a family of homomorphisms ''f''<sub>''ij''</sub>: ''A''<sub>''j''</sub> → ''A''<sub>''i''</sub> for all ''i'' ≤ ''j'' (note the order) with the following properties:
+# ''f''<sub>''ii''</sub> is the identity on ''A''<sub>''i''</sub>,
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+# ''f''<sub>''ik''</sub> = ''f''<sub>''ij''</sub> <small>o</small> ''f''<sub>''jk''</sub> for all ''i'' ≤ ''j'' ≤ ''k''.
+Then the pair ((''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub>, (''f''<sub>''ij''</sub>)<sub>''i''≤ ''j''∈''I''</sub>) is called an inverse system of groups and morphisms over ''I'', and the morphisms ''f''<sub>''ij''</sub> are called the transition morphisms of the system.
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+We define the '''inverse limit''' of the inverse system ((''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub>, (''f''<sub>''ij''</sub>)<sub>''i''≤ ''j''∈''I''</sub>) as a particular [[subgroup]] of the [[direct product]] of the ''A''<sub>''i''</sub>'s:
+:<math>\varprojlim_{i\in I} A_i = \Big\{\vec a \in \prod_{i\in I}A_i \;\Big|\; a_i = f_{ij}(a_j) \mbox{ for all } i \leq j \mbox{ in } I\Big\}.</math>
+The inverse limit, ''A'', comes equipped with ''natural projections'' π<sub>''i''</sub>: ''A'' → ''A''<sub>''i''</sub> which pick out the ''i''th component of the direct product for each ''i'' in ''I''. The inverse limit and the natural projections satisfy a [[universal property]] described in the next section.
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+This same construction may be carried out if the ''A''<sub>''i''</sub>'s are [[Set (mathematics)|sets]],<ref name="same-construction">John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.</ref> semigroups,<ref name="same-construction"/> topological spaces,<ref name="same-construction"/> [[ring (mathematics)|rings]], [[module (mathematics)|modules]] (over a fixed ring), [[algebra over a field|algebras]] (over a fixed field), etc., and the [[homomorphism]]s are homomorphisms in the corresponding [[category theory|category]]. The inverse limit will also belong to that category.
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+The inverse limit can be defined abstractly in an arbitrary [[category (mathematics)|category]] by means of a [[universal property]]. Let (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) be an inverse system of objects and [[morphism]]s  in a category ''C'' (same definition as above). The '''inverse limit''' of this system is an object ''X'' in ''C'' together with morphisms π<sub>''i''</sub>: ''X'' → ''X''<sub>''i''</sub> (called ''projections'') satisfying π<sub>''i''</sub> = ''f''<sub>''ij''</sub> <small>o</small> π<sub>''j''</sub> for all ''i'' ≤ ''j''. The pair (''X'', π<sub>''i''</sub>)  must be universal in the sense that for any other such pair (''Y'', ψ<sub>''i''</sub>) there exists a unique morphism ''u'': ''Y'' → ''X'' making all the "obvious" identities true; i.e., the diagram
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+<div style="text-align: center;">[[Image:InverseLimit-01.png]]</div>
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+must [[commutative diagram|commute]] for all ''i'' ≤ ''j''. The inverse limit is often denoted
+:<math>X = \varprojlim X_i</math>
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+with the inverse system (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) being understood.
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+The inverse limit might not exist in a category. If it does, however, it is unique in a strong sense: given any other inverse limit ''X''&prime; there exists a ''unique'' [[isomorphism]] ''X''&prime; → ''X'' commuting with the projection maps.
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+We note that an inverse system in a category ''C'' admits an alternative description in terms of [[functor]]s. Any partially ordered set ''I'' can be considered as a [[small category]] where the morphisms consist of arrows ''i'' → ''j'' [[if and only if]] ''i'' ≤ ''j''. An inverse system is then just a [[contravariant functor]] ''I'' → ''C''. And the inverse limit functor
+<math>\varprojlim:C^{I^{op}}\rightarrow C</math> is a [[covariant functor]].
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+== Examples ==
- </ul>
+* The ring of [[p-adic number|''p''-adic integers]] is the inverse limit of the rings '''Z'''/''p''<sup>''n''</sup>'''Z''' (see [[modular arithmetic]]) with the index set being the [[natural number]]s with the usual order, and the morphisms being "take remainder". The natural topology on the ''p''-adic integers is the same as the one described here.
+* The ring <math>\textstyle R[[t]]</math> of [[formal power series]] over a commutative ring ''R'' can be thought of as the inverse limit of the rings <math>\textstyle R[t]/t^nR[t]</math>, indexed by the natural numbers as usually ordered, with the morphisms from <math>\textstyle R[t]/t^{n+j}R[t]</math> to <math>\textstyle R[t]/t^nR[t]</math> given by the natural projection.
+* [[Pro-finite group]]s are defined as inverse limits of (discrete) finite groups.
+* Let the index set ''I'' of an inverse system (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) have a [[greatest element]] ''m''. Then the natural projection π<sub>''m''</sub>: ''X'' → ''X''<sub>''m''</sub> is an isomorphism.
+* Inverse limits in the [[category of topological spaces]] are given by placing the [[initial topology]] on the underlying set-theoretic inverse limit.  This is known as the '''limit topology'''.
+** The set of infinite [[String (computer science)|strings]] is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are [[discrete topology|discrete]], the limit space is [[totally disconnected]]. This is one way of realizing the [[p-adic|''p''-adic numbers]] and the [[Cantor set]] (as infinite strings).
+* Let (''I'', =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the [[product (category theory)|product]].
+* Let ''I'' consist of three elements ''i'', ''j'', and ''k'' with ''i'' ≤ ''j'' and ''i'' ≤ ''k'' (not directed). The inverse limit of any corresponding inverse system is the [[pullback (category theory)|pullback]].
+==Derived functors of the inverse limit==
+For an [[abelian category]] ''C'', the inverse limit functor
+:<math>\varprojlim:C^I\rightarrow C</math>
+is [[Exact functor|left exact]]. If ''I'' is ordered (not simply partially ordered) and [[countable]], and ''C'' is the category '''Ab''' of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms ''f''<sub>''ij''</sub> that ensures the exactness of <math>\varprojlim</math>. Specifically, [[Samuel Eilenberg|Eilenberg]] constructed a functor
+:<math>\varprojlim{}^1:\operatorname{Ab}^I\rightarrow\operatorname{Ab}</math>
+(pronounced "lim one") such that if (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>), (''B''<sub>''i''</sub>, ''g''<sub>''ij''</sub>), and (''C''<sub>''i''</sub>, ''h''<sub>''ij''</sub>) are three projective systems of abelian groups, and
+:<math>0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0</math>
+is a [[short exact sequence]] of inverse systems, then
+:<math>0\rightarrow\varprojlim A_i\rightarrow\varprojlim B_i\rightarrow\varprojlim C_i\rightarrow\varprojlim{}^1A_i</math>
+is an exact sequence in '''Ab'''.
+===Mittag-Leffler condition===
+If the ranges of the morphisms of the inverse system of abelian groups (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) are ''stationary'', that is, for every ''k'' there exists ''j'' ≥ ''k'' such that for all ''i'' ≥ ''j'' :<math> f_{kj}(A_j)=f_{ki}(A_i)</math> one says that the system satisfies the '''Mittag-Leffler condition'''. This condition implies that <math>\varprojlim{}^1A_i=0.</math>
+For a discussion of the name "Mittag-Leffler" in its relation with the [[Mittag-Leffler theorem]], see this [http://mathoverflow.net/questions/14717/mittag-leffler-condition-whats-the-origin-of-its-name thread] on [[MathOverflow]].
+The following situations are examples where the Mittag-Leffler condition is satisfied:
+* a system in which the morphisms ''f''<sub>''ij''</sub> are surjective
+* a system of finite-dimensional vector spaces.
+An example where this is non-zero is obtained by taking ''I'' to be the non-negative [[integer]]s, letting ''A''<sub>''i''</sub> = ''p''<sup>''i''</sup>'''Z''', ''B''<sub>''i''</sub> = '''Z''', and ''C''<sub>''i''</sub> = ''B''<sub>''i''</sub> / ''A''<sub>''i''</sub> = '''Z'''/''p''<sup>''i''</sup>'''Z'''. Then
+:<math>\varprojlim{}^1A_i=\mathbf{Z}_p/\mathbf{Z}</math>
+where '''Z'''<sub>''p''</sub> denotes the [[p-adic integers]].
+===Further results===
+More generally, if ''C'' is an arbitrary abelian category that has [[Injective object#Enough injectives|enough injectives]], then so does ''C''<sup>''I''</sup>, and the right [[derived functors]] of the inverse limit functor can thus be defined. The ''n''th right derived functor is denoted
+:<math>R^n\varprojlim:C^I\rightarrow C.</math>
+In the case where ''C'' satisfies [[Grothendieck]]'s axiom [[Abelian category#Grothendieck's axioms|(AB4*)]], [[Jan-Erik Roos]] generalized the functor lim<sup>1</sup> on '''Ab'''<sup>''I''</sup> to series of functors lim<sup>n</sup> such that
+:<math>\varprojlim{}^n\cong R^n\varprojlim.</math>
+It was thought for almost 40 years that Roos had proved (in ''Sur les foncteurs dérivés de lim. Applications. '') that lim<sup>1</sup> ''A''<sub>''i''</sub> = 0 for (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) an inverse system with surjective transition morphisms and ''I'' the set of non-negative integers (such inverse systems are often called "[[Mittag-Leffler]] sequences"). However, in 2002, [[Amnon Neeman]] and [[Pierre Deligne]] constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim<sup>1</sup> ''A''<sub>''i''</sub> ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if ''C'' has a set of generators (in addition to satisfying (AB3) and (AB4*)).
+[[Barry Mitchell (mathematician)|Barry Mitchell]] has shown (in "The cohomological dimension of a directed set") that if ''I'' has [[cardinality]] <math>\aleph_d</math> (the ''d''th [[Aleph number|infinite cardinal]]), then ''R''<sup>''n''</sup>lim is zero for all ''n'' ≥ ''d'' + 2. This applies to the ''I''-indexed diagrams in the category of ''R''-modules, with ''R'' a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which lim^n, on diagrams indexed by a countable set, is nonzero for n>1).
+== Related concepts and generalizations ==
+The [[dual (category theory)|categorical dual]] of an inverse limit is a [[direct limit]] (or inductive limit). More general concepts are the [[limit (category theory)|limits and colimits]] of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.
+==See also==
+*[[Direct limit|Direct, or inductive limit]]
+== Notes ==
+<references />
+==References==
+*{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Algebra I|publisher=Springer|year=1989|isbn=978-3-540-64243-5|oclc=40551484}}
+*{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=General topology: Chapters 1-4|publisher=Springer|year=1989|isbn=978-3-540-64241-1|oclc=40551485}}
+*{{citation|first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=[[Categories for the Working Mathematician]] | edition=2nd |date=September 1998 |publisher=Springer|isbn=0-387-98403-8}}
+*{{Citation | last=Mitchell | first=Barry | author-link=Barry Mitchell (mathematician) | title=Rings with several objects | journal=[[Advances in Mathematics]] | mr=0294454  | year=1972 | volume=8 | pages=1–161 | doi=10.1016/0001-8708(72)90002-3}}
+*{{Citation | last=Neeman | first=Amnon | author-link=Amnon Neeman | title=A counterexample to a 1962 "theorem" in homological algebra (with appendix by Pierre Deligne) | journal=[[Inventiones Mathematicae]] | mr=1906154  | year=2002 | volume=148 | issue=2 | pages=397–420 | doi=10.1007/s002220100197}}
+*{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Sur les foncteurs dérivés de lim. Applications | journal=C. R. Acad. Sci. Paris | mr=0132091  | year=1961 | volume=252 | pages=3702–3704}}
+*{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Derived functors of inverse limits revisited | journal=[[London Mathematical Society|J. London Math. Soc. (2)]] | mr=2197371  | year=2006 | volume=73 | issue=1 | pages=65–83 | doi=10.1112/S0024610705022416}}
+* Section 3.5 of {{Weibel IHA}}
+[[Category:Limits (category theory)]]
+[[Category:Abstract algebra]]
+[[de:Limes (Kategorientheorie)]]

Ionosphere: Difference between revisions

Revision as of 01:56, 2 February 2014

Contents

Formal definition

Algebraic objects

General definition

Examples

Derived functors of the inverse limit

Mittag-Leffler condition

Further results

Related concepts and generalizations

See also

Notes

References

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Ionosphere: Difference between revisions

Revision as of 01:56, 2 February 2014

Formal definition

Algebraic objects

General definition

Examples

Derived functors of the inverse limit

Mittag-Leffler condition

Further results

Related concepts and generalizations

See also

Notes

References

Navigation menu

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