## § Nullstellensatz for schemes

#### § System of equations

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

#### § Solutions to system of equations

Note that we often define equations, for example $x^2 + 1 = 0$ over a ring such as $\mathbb Z$. But its solutions live elsewhere: In this case, the solutions live in $\mathbb C$, as well as in $\mathbb Z/2Z$. 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 $f \in K[\mathbf T]$ in some other ring $A$, we can look for solutions in the ring $A$. Some thought will tell us that all we need is a ring homomorphism $\phi: K \rightarrow L$. Alternatively/equivalently, we need $L$ to be a $K$-algebra . Let us consider the single-variable case with $K[T]$. This naturally extends to the multivariate case. Using $\phi$, we can map $f \in K[T]$ to $\phi(f) \in L[T]$ by taking $f = \sum_i k_i T^i$ to $\phi(f) = \sum_i \phi(k_i) T^i$. This clearly extends to the multivariate case. Thus, we can interpret solutions $l \in L$ to an equation $f \in K[T]$ as $f(l) = \sum_i \phi(k_i) l^i$. Formally, the solution to a system $X \equiv (K[\mathbf T], \mathbf F)$ in ring $L$, written as $Sol(X, L)$ is a set of elements $ls \subseteq L$ such that $F_i(l) = 0$ for all $l$ in $ls$ and for all $F_i$ in $\mathbf F$.

#### § Equivalent systems of equations

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

#### § Biggest system of equations

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

#### § Varieties and coordinate rings

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

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

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

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

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

#### § Solutions for $X$ in $L$: $K$-algebra morphisms $Coord(X) \rightarrow L$

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

#### § Forward: Solution to morphism

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

#### § Backward: morphism to solution

Assume we are given a morphism $\phi: Coord(X) \rightarrow L$. Expanding definitions, this means that $\phi: K[T]/ (\mathbf F) \rightarrow L$. We need to build a solution. We build the solution $l\star = \phi(T)$. Intuitively, we are thinking of $\phi$ as $eval[l\star]$. If we had an $eval[l\star]$, then we would learn the point $l\star \in L$ by looking at $eval[l\star](T)$, since $eval[l\star](T) = l\star$. We can show that this point exists in the solution as follows:
\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 $X$ over ring $L$

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

#### § Geometric Language: Points

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