Polynomial rings

Notice that we did not use the domain of the polynomial functions in our arguments in the previous paragraphs. A mathematician s natural reaction to such an observation is to think about a way to define the object in question without mentioning the unused information. These vector spaces (along with the natural multiplication of polynomials by polynomials) are studied in abstract algebra under the name of polynomial rings. Interested readers might consult Artin s book [Ar, Chapters 10-11] for more details and related ideas. 

Power series rings over k are examples of formally smooth k-algebras polynomial rings in a finite number of variables are smooth k-algebras. 

Suppose we have some family of polynomial equations over k. We can then form a most general possible solution of the equations as follows. Take a polynomial ring over k, with one indeterminate for each variable in the equations. Divide by the ideal generated by the relations which the equations express. Call the quotient algebra A. From the equation for SL2, for instance, we get A — X12, X2u X22 I[X X22 12 21 ) The... 

Every fc-algebra A arises in this way from some family of equations. To see this, take any set of generators x for A, and map the polynomial ring k[ X ] onto A by sending Xa to xa. Choose polynomials /, generating the kernel. (If we have finitely many generators and k noetherian, only finitely many/, are needed (A.5).) Clearly then x is the most general possible solution of the equations f = 0. In summary ... 

If the base ring k is not plain from context, we write explicitly 2 /t. Clearly we can also construct 2 for 4 not finitely generated just by extending the preliminary computation to polynomial rings in infinitely many variables. When S is a closed set in k", the elements of 2 s] are the (algebraic) differentials defined on S—combinations of the dxt multiplied by functions. In general therefore we call 2 the module of differentials of 4. 

Let G be an algebraic affine group scheme. By Noether normalization (A.7) we can write k[G] as a finite module over a polynomial ring k[Xu..., X ], The n occurring here is obviously unchanged by base extension. It is uniquely determined, since it is the transcendence degree of the fraction field of k[G°]/nilpotents. Intuitively it represents the number of independent parameters involved in expressing elements of G, and we call it the dimension of G. 

Theorem. Let R be noetherian. Then the polynomial ring R[Af] is also noetherian. 

Theorem. Let kbea field, R a finitely generated k-algebra. There is a subring S of R such that S is a polynomial ring and R is a finitely generated S-module. 

Proof. Write A as a finitely generated module over a polynomial ring k[x,... 

Hence the image is the locus of points satisfying Rij = RioRoj for all i,j > 1. This is certainly closed. Its affine coordinate ring is k[Rij / R — R Roj), which is clearly isomorphic to the polynomial ring k [Rio, Roj]- Under /, this is mapped isomorphically to k [Si,Tj] which is the affine coordinate ring of (P )x0 x (Pm)y0- Hence we do have an isomorphism. 

In this case. For could be translated to poljmomials in the polynomial ring Z2[X] (i.e. poljmomials with coefficients in the field Z2 and variables in the set X = xi, X2,... ). The translation function For —> Z2IX] is defined... 

Generalizing Boolean Circuits Via Polynomial Ring Calculus... 

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