§ Nullstellensatz for schemes

§ System of equations

Consider a set of polynomials {F1,F2,Fm}\{F_1, F_2, \dots F_m\} subset K[T1,,Tn]K[T_1, \dots, T_n]. A system of equations XX for unknowns TT is the tuple (K[T1,,Tn],{F1,F2,Fm}K[T1,,Tn])(K[T_1, \dots, T_n], \{ F_1, F_2, \dots F_m \} \subset K[T_1, \dots, T_n]). We abbreviate this to (K[T],F)(K[\mathbf T], \mathbf F) where the bolded version implies that these are vectors of values.

§ Solutions to system of equations

Note that we often define equations, for example x2+1=0x^2 + 1 = 0 over a ring such as Z\mathbb Z. But its solutions live elsewhere: In this case, the solutions live in C\mathbb C, as well as in Z/2Z\mathbb Z/2Z. Hence, we should not restrict our solution space to be the ring where we defined our coefficients from! Rather, as long as we are able to interpret the polynomial fK[T]f \in K[\mathbf T] in some other ring AA, we can look for solutions in the ring AA. Some thought will tell us that all we need is a ring homomorphism ϕ:KL\phi: K \rightarrow L. Alternatively/equivalently, we need LL to be a KK-algebra . Let us consider the single-variable case with K[T]K[T]. This naturally extends to the multivariate case. Using ϕ\phi, we can map fK[T]f \in K[T] to ϕ(f)L[T]\phi(f) \in L[T] by taking f=ikiTif = \sum_i k_i T^i to ϕ(f)=iϕ(ki)Ti\phi(f) = \sum_i \phi(k_i) T^i. This clearly extends to the multivariate case. Thus, we can interpret solutions lLl \in L to an equation fK[T]f \in K[T] as f(l)=iϕ(ki)lif(l) = \sum_i \phi(k_i) l^i. Formally, the solution to a system X(K[T],F)X \equiv (K[\mathbf T], \mathbf F) in ring LL, written as Sol(X,L)Sol(X, L) is a set of elements lsLls \subseteq L such that Fi(l)=0F_i(l) = 0 for all ll in lsls and for all FiF_i in F\mathbf F.

§ Equivalent systems of equations

Two systems of equations X,YX, Y over the same ring KK are said to be equivalent over KK iff for all KK-algebras LL, we have Sol(X,L)=Sol(Y,L)Sol(X, L) = Sol(Y, L).

§ Biggest system of equations

For a given system of equations X(K[T],F)X \equiv (K[\mathbf T], \mathbf F) over the ring KK, we can generate the largest system of equations that still has the same solution: generate the ideal F=(F)\mathbf F' = (\mathbf F), and consider the system of equations X(K[T],F=(F))X' \equiv (K[\mathbf T], \mathbf F' = (\mathbf F) ).

§ Varieties and coordinate rings

Let gK[T1,Tn]g \in K[T_1, \dots T_n]. The polynomial is also a function which maps xKn\mathbf x \in K^n to KK through evaluation g(x)g(\mathbf x). Let us have a variety VknV \subseteq k^n defined by some set of polynomials FK[T1,,Tn]\mathbf F \in K[T_1, \dots, T_n]. So the variety is the vanishing set of F\mathbf F, and F\mathbf F is the largest such set of polynomials. Now, two functions g,hK[T1,,Tn]g, h \in K[T_1, \dots, T_n] are equal on the variety VV iff they differ by a function zz whose value is zero on the variety VV. Said differently, we have that gV=hVg|V = h|V iff hg=zh - g = z where zz vanishes on VV. We know that the polynomials in F\mathbf F vanish on VV, and is the largest set to do so. Hence we have that zFz \in \mathbf F. To wrap up, we have that two functions g,hg, h are equal on VV, that is, gV=hVg|V = h|V iff ghFg - h \in \mathbf F. So we can choose to build a ring where g,hg, h are "the same function". We do this by considering the ring K[V]K[T1,,Tn]/FK[V] \equiv K[T_1, \dots, T_n] / \mathbf F. This ring K[V]K[V] is called as the coordinate ring of the variety VV.

§ An aside: why is it called the "coordinate ring"?

We can consider the ith coordinate function as one that takes K[T1,,Tn]K[T_1, \dots, T_n] to TiT_i So we have ϕiTiK[T1,,Tn]\phi_i \equiv T_i \in K[T_1, \dots, T_n] which defines a function ϕi:KnK\phi_i: K^n \rightarrow K which extracts the iith coordinate. Now the quotienting from the variety to build K[V]K[V], the coordinate ring of the variety VV will make sure to "modulo out" the coordinates that "do not matter" on the variety.

§ Notation for coordinate ring of solutions: Coord(X)Coord(X)

For a system X(K[T],F)X \equiv (K[\mathbf T], \mathbf F), we are interested in the solutions to F\mathbf F, which forms a variety V(F)V(\mathbf F). Furthermore, we are interested in the algebra of this variety, so we wish to talk about the coordinate ring k[V(F)]=K[T]/(F)k[V(\mathbf F)] = K[\mathbf T] / (\mathbf F). We will denote the ring k[T]/(F)k[\mathbf T] / (\mathbf F) as Coord(X)Coord(X).

§ Solutions for XX in LL: KK-algebra morphisms Coord(X)LCoord(X) \rightarrow L

Let's simplify to the single variable case. Multivariate follows similarly by recursing on the single variable case. X(K[T],FK[T])X \equiv (K[T], \mathbf F \subseteq K[T]). There is a one-to-one coorespondence between solutions to XX in LL and elements in HomK(Coord(X),L)Hom_K(Coord(X), L) where HomKHom_K is the set of KK-algebra morphisms. Expanding definitions, we need to establish a correspondence between
  • Points lLl \in L such that evall(f)=0eval_l(f) = 0 for all fFf \in F.
  • Morphisms K[T]/(F)LK[T] / (\mathbf F) \rightarrow L.

§ Forward: Solution to morphism

A solution for XX in LL is a point lL\mathbf l \in L such that FF vanishes on ll. Thus, the evaluation map eval[l]:K[T]Leval[l]: K[T] \rightarrow L has kernel (F)(\mathbf F). Hence, eval[l]eval[l] forms an honest to god morphism between K[T]/(F)K[T] / (\mathbf F) and LL.

§ Backward: morphism to solution

Assume we are given a morphism ϕ:Coord(X)L\phi: Coord(X) \rightarrow L. Expanding definitions, this means that ϕ:K[T]/(F)L\phi: K[T]/ (\mathbf F) \rightarrow L. We need to build a solution. We build the solution l=ϕ(T)l\star = \phi(T). Intuitively, we are thinking of ϕ\phi as eval[l]eval[l\star]. If we had an eval[l]eval[l\star], then we would learn the point lLl\star \in L by looking at eval[l](T)eval[l\star](T), since eval[l](T)=leval[l\star](T) = l\star. We can show that this point exists in the solution as follows:
eval[l](f)=iai(l)i=iaiϕ(T)iSince ϕ is ring homomorphism:=iaiϕ(Ti)Since ϕ is k-algebra homomorphism:=ϕ(iaiTi)=ϕ(f)Since fker(ϕ):=0 \begin{aligned} &eval[l\star](f) = \sum_i a_i (l\star)^i \\ &= \sum_i a_i \phi(T)^i \\ &\text{Since $\phi$ is ring homomorphism:} \\ &= \sum_i a_i \phi(T^i) \\ &\text{Since $\phi$ is $k$-algebra homomorphism:} \\ &= \phi(\sum_i a_i T^i) \\ &= \phi(f) \\ \text{Since $f \in ker(\phi)$:} \\ &= 0 \end{aligned}

§ Consistent and inconsistent system XX over ring LL

Fix a KK-algebra LL. The system XX is consistent over LL iff Sol(X,L)Sol(X, L) \neq \emptyset. the system XX over LL is inconsistent iff If Sol(X,L)=Sol(X, L) = \emptyset.

§ Geometric Language: Points

Let KK be the main ring, X(K[T1,Tn],F)X \equiv (K[T_1, \dots T_n], \mathbf F) a system of equations in nn unknowns T1,,TnT_1, \dots, T_n. For any KK-algebra LL, we consider the set Sol(X,L)Sol(X, L) as a collection of points in LnL^n. These points are solutions to the system XX.

§ The points of KK-algebra

§ References