Erdős–Rényi model

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Template:Harvs for algebras over a field, and extended to algebras over more general rings by Template:Harvtxt.

Definition of Hochschild homology of algebras

Let k be a ring, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product A^e=A⊗A^o of A with its opposite algebra. Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as A^e-modules. Template:Harvtxt defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by

$HH_{n}(A,M)={\text{Tor}}_{n}^{A^{e}}(A,M)$

$HH^{n}(A,M)={\text{Ext}}_{A^{e}}^{n}(A,M)$

Hochschild complex

Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write A^⊗n for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by

$C_{n}(A,M):=M\otimes A^{\otimes n}$

with boundary operator d_i defined by

$d_{0}(m\otimes a_{1}\otimes \cdots \otimes a_{n})=ma_{1}\otimes a_{2}\cdots \otimes a_{n}$

$d_{i}(m\otimes a_{1}\otimes \cdots \otimes a_{n})=m\otimes a_{1}\otimes \cdots \otimes a_{i}a_{i+1}\otimes \cdots \otimes a_{n}$

$d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})=a_{n}m\otimes a_{1}\otimes \cdots \otimes a_{n-1}$

Here a_i is in A for all 1 ≤ i ≤ n and m ∈ M. If we let

$b=\sum _{i=0}^{n}(-1)^{i}d_{i},$

then b ° b = 0, so (C_n(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.

Remark

The maps d_i are face maps making the family of modules C_n(A,M) a simplicial object in the category of k-modules, i.e. a functor Δ^o → k-mod, where Δ is the simplicial category and k-mod is the category of k-modules. Here Δ^o is the opposite category of Δ. The degeneracy maps are defined by s_i(a₀ ⊗ ··· ⊗ a_n) = a₀ ⊗ ··· a_i ⊗ 1 ⊗ a_i+1 ⊗ ··· ⊗ a_n. Hochschild homology is the homology of this simplicial module.

Hochschild homology of functors

The simplicial circle S¹ is a simplicial object in the category Fin_* of finite pointed sets, i.e. a functor Δ^o → Fin_*. Thus, if F is a functor F: Fin → k-mod, we get a simplicial module by composing F with S¹

$\Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {F}{\longrightarrow }}k{\text{-}}\operatorname {mod} .$

The homology of this simplicial module is the Hochschild homology of the functor F. The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor.

Loday functor

A skeleton for the category of finite pointed sets is given by the objects

$n_{+}=\{0,1,\dots ,n\},\,$

where 0 is the basepoint, and the morphisms are the basepoint preserving set maps. Let A be a commutative k-algebra and M be a symmetric A-bimoduleTemplate:Elucidate. The Loday functor L(A,M) is given on objects in Fin_* by

$n_{+}\mapsto M\otimes A^{\otimes n}.\,$

A morphism

$f:m_{+}\rightarrow n_{+}$

is sent to the morphism f_* given by

$f_{*}(a_{0}\otimes \cdots \otimes a_{n})=(b_{0}\otimes \cdots \otimes b_{m})$

where

$b_{j}=\prod _{f(i)=j}a_{i},\,\,j=0,\dots ,n,$

and b_j = 1 if f ⁻¹(j) = ∅.

Another description of Hochschild homology of algebras

The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition

$\Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {{\mathcal {L}}(A,M)}{\longrightarrow }}k{\text{-}}\operatorname {mod} ,$

and this definition agrees with the one above.

See also

• Cyclic homology

References

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  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
  
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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
• Template:Eom
• 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
• Jean-Louis Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
• Richard S. Pierce, Associative Algebras, Graduate Texts in Mathematics (88), Springer, 1982.
• Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology

External links

• Dylan G.L. Allegretti, Differential Forms on Noncommutative Spaces. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms).
• Template:Nlab