Superdense coding

In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. A simple version states that for any field k, and any finitely generated commutative k-algebra A, there exists a nonnegative integer d and algebraically independent elements y₁, y₂, ..., y_d in A such that A is a finitely generated module over the polynomial ring S:=k[y₁, y₂, ..., y_d].

The integer d is the Krull dimension of A (since A and S have the same dimension.) When A is an integral domain, d is the transcendence degree of the field of fractions of A over k.

The theorem has geometric interpretation. Suppose A is integral. Let S be the coordinate ring of d-dimensional affine space ${\mathbb{𝔸}}_k^d$, and A as the coordinate ring of some other d-dimensional affine variety X. Then the inclusion map S → A induces a surjective finite morphism of affine varieties $X\to {\mathbb{𝔸}}_k^d$. The conclusion is that any affine variety is a branched covering of affine space. When k is infinite, such a branched covering map can be constructed by taking a general projection from an affine space containing X to a d-dimensional subspace.

More generally, in the language of schemes, the theorem can equivalently be stated as follows: every affine k-scheme (of finite type) X is finite over an affine n-dimensional space. The theorem can be refined to include a chain of prime ideals of R (equivalently, irreducible subsets of X) that are finite over the affine coordinate subspaces of the appropriate dimensions.

The form of the Noether normalization lemma stated above can be used as an important step in proving Hilbert's Nullstellensatz. This gives it further geometric importance, at least formally, as the Nullstellensatz underlies the development of much of classical algebraic geometry. The theorem is also an important tool in establishing the notions of Krull dimension for k-algebras.

Proof

The following proof is due to Nagata and is taken from Mumford's red book. A proof in the geometric flavor is also given in the page 127 of the red book and this mathoverflow thread.

The ring A in the lemma is generated as k-algebra by elements, say, $y_1,...,y_m$ such that $y_1,...,y_d$ (some d) are algebraically independent over k and the rest are algebraic over $k[y_1,...,y_d]$. We shall induct on m. If $m=d$, then the assertion is trivial. Assume now $m>d$. It is enough to show that there is a subring S of A that is generated by $m-1$ elements and is such that A is finite over S, for, by inductive hypothesis, we can find algebraically independent elements $x_1,...,x_d$ of S such that S is finite over $k[x_1,...,x_d]$. Since $m>d$, there is a nonzero polynomial f in m variables over k such that

$f(y_1,...,y_m)=0$.

Given an integer r which is determined later, set

$z_i=y_i-y_1^{r^{i-1}},2\le i\le m.$

Then the preceding reads:

$f(y_1,z_2+y_1^r,z_3+y_1^{r^2},...,z_m+y_1^{r^{m-1}})=0$.

Now, the highest term in $y_1$ of $a{y}_1^{{\alpha }_1}\prod _2^m(z_i+y_1^{r^{i-1}}{)}^{{\alpha }_i},a\in k,$ looks

$a{y}_1^{{\alpha }_1+r{\alpha }_2+...+{\alpha }_m{r}^{m-1}}.$

Thus, if r is larger than any exponent ${\alpha }_i$ appearing in f, then the highest term of $f(y_1,z_2+y_1^r,z_3+y_1^{r^2},...,z_m+y_1^{r^{m-1}})$ in $y_1$ also has the form as above. In other words, $y_1$ is integral over $S=k[z_2,...,z_m]$. Since $y_i=z_i+y_1^{r^{i-1}}$ are also integral over that ring, A is integral over S. It follows A is finite over S.

If A is an integral domain, then d is the transcendence degree of its field of fractions. Indeed, A and $S=k[y_1,...,y_d]$ have the same transcendence degree (i.e., the degree of the field of fractions) since the field of fractions of A is algebraic over that of S (as A is integral over S) and S obviously has transcendence degree d. Thus, it remains to show the Krull dimension of the polynomial ring S is d. (this is also a consequence of dimension theory.) We induct on d, $d=0$ being trivial. Since $0⊊(y_1)⊊(y_1,y_2)⊊\cdots ⊊(y_1,\dots ,y_d)$ is a chain of prime ideals, the dimension is at least d. To get the reverse estimate, let $0⊊{\mathfrak{p}}_1⊊\cdots ⊊{\mathfrak{p}}_m$ be a chain of prime ideals. Let $0\ne u\in {\mathfrak{p}}_1$. We apply the noether normalization and get $T=k[u,z_2,\dots ,z_d]$ (in the normalization process, we're free to choose the first variable) such that S is integral over T. By inductive hypothesis, $T/(u)$ has dimension d - 1. By incomparability, ${\mathfrak{p}}_i\cap T$ is a chain of length $m$ and then, in $T/({\mathfrak{p}}_1\cap T)$, it becomes a chain of length $m-1$. Since $\dim T/({\mathfrak{p}}_1\cap T)\le \dim T/(u)$, we have $m-1\le d-1$. Hence, $\dim S\le d$.

Notes

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References

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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.
  
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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