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Template:Distinguish In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality.

They are named for Template:Harvs, who proved the unmixedness theorem for polynomial rings, and for Template:Harvs, who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property.

Formal definition

A local Cohen–Macaulay ring is defined as a commutative noetherian local ring with Krull dimension equal to its depth. The depth is always bounded above by the Krull dimension; equality provides some interesting regularity conditions on the ring, enabling some powerful theorems to be proven in this rather general setting.

A non-local ring is called Cohen–Macaulay if all of its localizations at prime ideals are Cohen–Macaulay.

Examples

The following rings are Cohen–Macaulay

• regular local ring (e.g., field, K[[x]])
• Artinian ring
• 1-dimensional Noetherian reduced ring
• 2-dimensional normal ring
• Gorenstein ring. In particular, complete intersection rings.
• The ring of invariants $R^{G}$ when $R$ is a Cohen–Macaulay algebra over a field of characteristic zero and G is a finite group (more generally reductive algebraic group). This is the Hochster–Roberts theorem.

1. The ring K[x]/(x²) is a local Artinian ring and so is Cohen–Macaulay, but it is not regular.
2. K[[t², t³]], where t is an indeterminate, is an example of a 1-dimensional local ring which is not regular but is Gorenstein, so is Cohen–Macaulay.
3. K[[t³, t⁴, t⁵]], where t is an indeterminate, is an example of a 1-dimensional local ring which is not Gorenstein but is Cohen–Macaulay

Rational singularities are Cohen–Macaulay but not necessarily Gorenstein.

Properties

1. A local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay.
2. A ring R is Cohen–Macaulay if and only if the polynomial ring R[x] is Cohen–Macaulay.

A quotient of a Cohen–Macaulay ring is universally catenary.

Counterexamples

1. If K is a field, then the formal power series ring $K[[x,y]]/(x^{2},xy)$ (the completion of the local ring at the double point of a line with an embedded double point) is not Cohen–Macaulay, because it has depth zero but dimension 1.
2. If K is a field, then the ring $K[[x,y,z]]/(xy,xz)$ (the completion of the local ring at the intersection of a plane and a line) is not Cohen–Macaulay (it is not even equidimensional); quotienting by $x-z$ gives the previous example.
3. If K is a field, then the ring $K[[w,x,y,z]]/(wy,wz,xy,xz)$ (the completion of the local ring at the intersection of two planes meeting in a point) is not Cohen–Macaulay; quotienting by $w-x$ gives the previous example.

Consequences of the condition

One meaning of the Cohen–Macaulay condition is seen in coherent duality theory. Here the condition corresponds to case when the dualizing object, which a priori lies in a derived category, is represented by a single module (coherent sheaf). The finer Gorenstein condition is then expressed by this module being projective (an invertible sheaf). Non-singularity (regularity) is still stronger— it corresponds to the notion of smoothness of a geometric object at a particular point. Thus, in a geometric sense, the notions of Gorenstein and Cohen–Macaulay capture increasingly larger sets of points than the smooth ones, points which are not necessarily smooth but behave in many ways like smooth points.

The unmixedness theorem<unmixed>...</unmixed>

An ideal I of a Noetherian ring A is called unmixed if ht(I)= ht(P) for any associated prime P of A/I. The unmixedness theorem is said to hold for the ring A if every ideal I generated by ht(I) elements is unmixed. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it.

References

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  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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• David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (Springer), ISBN 0-387-94268-8 (hardcover), ISBN 0-387-94269-6 (soft cover)
• 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
• H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. (1986)