[ag/3] Schemes II

Paris, Chennai

The final ingredient in the definition of an affine scheme is the structure sheaf $\mathscr{O}_{\text{Spec } A}$, which we will think of as the sheaf of algebraic functions. For example, in $\mathbb{A}^2$, we expect that on the open set $D(xy)$, $(3x^4 + y + 7)/(x^3 y^8)$ is an algebraic function. The function will have values at points, but won't be determined by their values at points; instead, they will be determined by their germs. It suffices to describe the structure sheaf as the sheaf of rings on the base of distinguished open sets.

Define $\mathscr{O}_{\text{Spec } A}(D(f))$ to be the localization of ring $A$ at the multiplicative sets of all functions that do not vanish outside of $V(f)$; i.e. those $g$ for which $V(g) \subset V(f)$ or equivalently $I(f) \subset I(g)$. This depends only on $D(f)$ and not on $f$ itself.

If $D(f') \subset D(f)$, define the restriction map:

$$ \text{res}_{D(f), D(f')} : \mathscr{O}_{\text{Spec } A}(D(f)) \rightarrow \mathscr{O}_{\text{Spec } A}(D(f')) $$

in the obvious way; the latter ring is a further restriction of the former ring. The restriction maps obviously commute, and this is a presheaf on a distinguished base.

The data just described give a sheaf on a distinguished open set, and hence determine a sheaf on the topological space on $\text{Spec } A$. The topological space along with the structure sheaf will be called an affine scheme. The notation $\text{Spec } A$ will henceforth denote the data of the topological space along with the structure sheaf. $\mathscr{O}_{\text{Spec } A}$ contains data not just of the sheaf, but also the underlying distinguished base that provides a good way of working with $\mathscr{O}_{\text{Spec } A}$. To prove this, we need to show that base gluability and base identity axioms hold on $\text{Spec } A$. Suppose $\text{Spec } A = \bigcup_{i \in I} D({f_i})$, or equivalently, the ideal generated by $f_i$ is the entire ring $A$. By quasicompactness of $A$, $\text{Spec } A = \bigcup_{i = 1}^n D({f_i})$ for finite $n$; i.e. $(f_1, \ldots, f_n) = A$. Suppose we are given $s \in A$ such that $\text{res}_{\text{Spec } A, D(f_i)} s = 0$ for all $A_{f_i}$ for all $i$; then we wish to show that $s = 0$. This implies that there is some $m$ such that for all $i \in {1, \ldots, n}$ such that $f_i^m s = 0$. Now, $(f_1^m, \ldots, f_n^m) = A$, so there are $r_i \in A$ with $\sum_{i = 1}^n r_i f_i^m = 1$ from which
$$s = \sum_{i = 1}^n r_i (f_i^m s) = 0$$

Hence, we have shown base identity axiom. For the base gluability axiom, we have $a_i/f_i^{l_i} \in A_f$ agreeing on overlaps $A_{f_i, f_j}$. Letting $g_i = f_i^{l_i}$, using $D(g_i) = D(f_i)$, our elements are of the form $a_i/g_i \in A_{g_i}$. The fact that $a_i/g_i$ and $a_j/g_j$ agree on overlap means that for some $m_{ij}$,
$$(g_i g_j)^{m_{ij}} (g_j a_i - g_i a_j) = 0$$

Taking $m = \text{max } m_{ij}$, we get:
$$(g_i g_j)^m (g_j a_i - g_i a_j) = 0$$

for all $i, j$. Let $b_i = a_i g_i^m$ for all $i$, and $h_i = g_i^{m + 1}$, so $D(h_i) = D(g_i)$. On each $D(h_i)$, we have a function $b_i/h_i$, and the overlap condition is:
$$h_j b_i = h_i b_j$$

Now $\bigcup D(h_i) = \text{Spec } A$ implying that $1 = \sum r_i h_i$, for some $r_i \in A$. Define $r = \sum r_i b_i$. This will be an element of $A$ that restricts to each $b_i/h_j$. Indeed, from the overlap condition,
$$r_j h_j = \sum_i r_i h_i b_j = b_j$$

completing the proof.

Suppose $M$ is an $A$-module. The following construction describes sheaf $\tilde{M}$ on a distinguished base. Define $\tilde{M}(D(f))$ to be the localization of $M$ at a multiplicative set of all functions that do not vanish on $V(f)$. Define restriction maps $\text{res}_{D(f), D(g)}$ in an analogous way to $\mathscr{O}_{\text{Spec } A}$. This defines a sheaf on a distinguished base, and hence on $\text{Spec } A$. Hence, this is an $\mathscr{O}_{\text{Spec } A}$-module.

Note that $\mathscr{O}_{\text{Spec } A}(U)$ is not a localization of $A$ at the multiplicative set of functions that do not vanish at any point on $U$. For instance, let $\text{Spec } A$ be two copies of $\mathbb{A}_k^2$ glued together at their origins, and $U$ be a complement of the origins. Then, the function which is $1$ on the first copy of $\mathbb{A}_k^2/\{(0, 0)\}$ is $0$ on the second copy of $\mathbb{A}_k^2/\{(0, 0)\}$. Let us collect five counter-examples which we will see in more detail later:

  1. The cone over the quadratic surface, $\text{Spec } A$ and $\text{Proj } A$, where $A = k[w, x, y, z]/(wz - xy)$.
  2. Two planes meeting at a point.
  3. Affine plane minus the origin, $\mathbb{A}^n - 0$; its inclusion with $\mathbb{A}^n$; and $\mathbb{A}^n$ with its doubled origin; each with $n = 1, 2, \infty$.
  4. Infinite disjoint union of schemes, especially, $\bigsqcup \text{Spec } k[x]/(x^n)$.
  5. $\text{Spec } \bar{\mathbb{Q}} \rightarrow \text{Spec } \mathbb{Q}$.

Visualizing schems: nilpotents

In the affine scheme $(\text{Spec } A, \mathscr{O}_{\text{Spec } A})$, the information about nilpotents is invisible at the level of points. We have the following inclusion-reversing bijections:

  1. Maximal ideals of $A$ $\leftrightarrow$ closed points of $\text{Spec } A$
  2. Prime ideals of $A$ $\leftrightarrow$ irreducible closed subsets of $\text{Spec } A$
  3. Nilradicals of $A$ $\leftrightarrow$ closed subsets of $\text{Spec } A$

Our goal now is to figure out how to picture ideals that are not radical. We immediately picture $\text{Spec } \mathbb{C}[x]/x(x - 1)(x - 2)$ as closed subsets of $\mathbb{A}_\mathbb{C}^2$, $\{0, 1, 2\}$. In particular, the map $\mathbb{A}_\mathbb{C}^2 \rightarrow \mathbb{A}_\mathbb{C}^2/x(x - 1)(x - 2)$ can be shown to be a restriction of $\mathbb{A}_\mathbb{C}^2$ to $\{0, 1, 2\}$. Similarly, $\mathbb{C}[x]/(x^2)$ can be thought of as a point at the origin with some "fuzz" remembering information about the first derivative.

$$ \mathbb{C}[x] \twoheadrightarrow \mathbb{C}[x]/(x^2) \twoheadrightarrow \mathbb{C}[x]/(x) $$

$\mathbb{C}[x, y]/(x, y)^3$ can be pictured as a "circular fuzz" around origin remembering first and second derivatives, while $\mathbb{C}[x, y]/(x^2, y^2)$ can be thought of as a square fuzz around origin that is circumscribed by $\mathbb{C}[x, y]/(x, y)^3$, and inscribed by $\mathbb{C}[x, y]/(x, y)^2$. In fact, given two ideals $I, J$ of $A$, $\text{Spec } A/IJ$ should be pictured as the intersection of pictures $\text{Spec } A/I$ and $\text{Spec } A/J$ in $\text{Spec } A$. As another example, consider a polynomial in $\mathbb{C}[x, y]/(y^2, xy)$; knowing what it is is equivalent to knowing its values on the $x$-axis, as well as its first-derivative information around the origin.

Next, consider the intersection of the parabola $y = x^2$ and the $x$-axis:

$$ \begin{align*} \text{Spec } \mathbb{C}[x, y]/(y - x^2) \cap \text{Spec } \mathbb{C}[x, y]/(y) &= \text{Spec } \mathbb{C}[x, y]/(y - x^2, y) \\ &= \text{Spec } \mathbb{C}[x, y]/(x^2, y) \end{align*} $$

We interpret this intersection as having multiplicity $2$ on the $x$-axis.

Definition of schemes

First, define isomorphism of ringed spaces $(X, \mathscr{O}_X), (Y, \mathscr{O}_Y)$ as:

  1. A homeomorphism from $X$ to $Y$.
  2. An isomorphism of sheaves $\mathscr{O}_X, \mathscr{O}_Y$, considered to be on the same space via $\pi$. In other words, it is the isomorphism $\mathscr{O}_Y \rightarrow \pi_* \mathscr{O}_X$, or equivalently using adjoints, $\pi^{-1} \mathscr{O}_Y \rightarrow \mathscr{O}_X$ of sheaves on $X$.

We have a correspondence between sets, topologies, and structure sheaves. An affine scheme is a ringed space isomorphic to $(\text{Spec } A, \mathscr{O}_{\text{Spec } A})$ for some $A$. A scheme $(X, \mathscr{O}_X)$ is a ringed space such that every point $X$ has open neighborhood $U$, and $(X, \mathscr{O}_X|_U)$ is an affine scheme. The topology on a scheme is the Zariski topology. The isomorphism between two schemes is an isomorphism as ringed spaces. If $U \subset X$ is an open set, $\Gamma(U, \mathscr{O}_X)$ is said to be a function on $U$; this generalizes in the obvious way to functions on an affine scheme.

From the definition of a structure sheaf on an affine scheme, if $(X, \mathscr{O}_X)$ is an affine scheme, we can recover its ring $A$ where $X = \text{Spec } A$ by taking global sections as $X = D(1)$:
$$\Gamma(X, \mathscr{O}_X) = \Gamma(D(1), \mathscr{O}_X) = A$$

We can recognize $X$ as the scheme on $\text{Spec } A$: we get the isomorphism $\pi : (\text{Spec } \Gamma(X, \mathscr{O}_X), \mathscr{O}_{\text{Spec } \Gamma(X, \mathscr{O}_X)}) \rightarrow (X, \mathscr{O}_X)$. For example, if $\mathfrak{m}$ is a maximal ideal of $\Gamma(X, \mathscr{O}_X)$, $\{\pi(\mathfrak{m})\} = V(\mathfrak{m})$.

Given $f \in A$, $\Gamma(D(f), \mathscr{O}_{\text{Spec } A}) = A_f$. Under natural inclusion $A_f \hookrightarrow A$, the Zariski topology on $\text{Spec } A$ restricts to that in $\text{Spec } A_f$. Moreover, the structure sheaf on $\text{Spec } A$ restricts to the structure sheaf on $\text{Spec } A_f$.

We say that $(U, \mathscr{O}_X|_U)$ is an open subscheme of $X$. If $U$ is also an affine scheme, $U$ is called affine open subscheme, or simply affine open. For example, $D(f)$ is an affine open subscheme.

Open subsets of schemes come with a natural scheme structure. For comparison, closed subschemes on $X$ are, informally, a particular kind of scheme structure on closed subsets of $X$. For example, if $I \subset A$ is an ideal, then $\text{Spec } A/I$ endows $V(I) \subset \text{Spec } A$ with a scheme structure, but note that there can be different ideals with the same vanishing set; for example, $(x), (x^2)$ in $k[x]$.

The stalk of sheaf $\tilde{M}$ at point $[\mathfrak{p}]$ is $M_\mathfrak{p}$. A section is determined by its germs meaning that $M \rightarrow \prod_p M_p$ is an inclusion. For example, an $A$-module is zero iff all its localizations at prime ideals are zero.

We say that a ringed space is a locally ringed space if all its stalks are local rings. Schemes and manifolds are examples of locally ringed spaces. Taking quotient at the maximal ideal can be interpreted as evaluation at the point. The maximal ideal of a local ring $\mathscr{O}_{X, p}$ is denoted by $m_p$ and the residue field $\mathscr{O}_{X, p}/m_p$ is denoted by $\kappa[p]$. Functions on an open subset $U$ of a locally ringed space have values at each point of $U$. The value $p$ of such a function lies in $\kappa[p]$. We say that a function vanishes at point $p$ if its value at $p$ is $0$. Notice that we can't even make sense of a "function vanishing" on a ringed space in general.

Consider the point $[\mathfrak{p}]$ of an affine scheme $\text{Spec } A$. The residue field $A_\mathfrak{p}/\mathfrak{p} A_\mathfrak{p}$ is isomorphic to $K(A/\mathfrak{p})$, the fraction field of the quotient. Note that the following diagram commutes:

$$ \begin{xy} \xymatrix{ & A_\mathfrak{p}\ar[dr]^{\text{quotient}} & \\ A\ar[ur]^{\text{localize}}\ar[dr]_{\text{quotient}} & & A_\mathfrak{p}/\mathfrak{p} A_\mathfrak{p} = K(A/\mathfrak{p}) \\ & A/\mathfrak{p}\ar[ur]_{\text{localize}} & } \end{xy} $$

Consider an example $(x^2 + y^2)/x(y^2 - x^5)$ in $\mathbb{A}_k^2 = k[x, y]$, in which $k$ is a field of characteristic not $2$. Then its value at $[(x - 2, y - 4)]$ is $(2^2 + 4^2)/2(4^2 - 2^5)$. Its value at the generic point on the $x$-axis is $x^2/(-x^6) = -1/x^4$, which we see by setting $y = 0$. Note that $A_\mathfrak{p}/\mathfrak{p} A_\mathfrak{p}$ is not algebraically closed, even when $A$ is a finitely-generated $\mathbb{C}$-algebra.

As another example, consider $27/4$ in $\text{Spec } \mathbb{Z}$. Its value at $[(5)]$ is $2/-1 \equiv -2 \text{ mod } 5$. Its value at generic point $[(0)]$ is $27/4$, and it vanishes at $[(3)]$.

If $\mathscr{F}$ is an $\mathscr{O}_X$-module on scheme $X$, or more generally, a locally ringed space, define the fiber of $\mathscr{F}$ at a point $p \in X$ by:
$$\mathscr{F}|_p := \mathscr{F}_p \otimes_{\mathscr{O}_{X, p}} \kappa(p)$$

For example, $\mathscr{O}_X|_p = \kappa(p)$.

Three examples

We can really work with a structure sheaf, and can compute the ring of sections of open sets that aren't just distinguished open sets of affine schemes.

First example: The plane minus the origin. Our goal here is to show that a distinguished base is something we can work with. Let $A = k[x, y]$ so that $\text{Spec } A = \mathbb{A}_k^2$. Let us work with the space of functions in $\mathbb{A}^2 - \{(0, 0)\} = \mathbb{A}^2 - \{[(x, y)]\}$. It is not immediately obvious if this is a distinguished open set. In any case, we can describe the union of two distinguished open sets $U = D(x) \cup D(y)$. We will find functions in $U$ by gluing together functions in $D(x)$ and $D(y)$. Functions on $D(x)$ are, by definition, $A_x = [x, y, 1/x]$, and those on $D(y)$ are $A_y = [x, y, 1/y]$. Notice that $A$ injects into its localizations, as it is an integral domain, so it injects onto both $A_x$ and $A_y$, and both inject into $A_{xy}$; indeed $k[x, y] = K(A)$. So we are looking for functions on $D(x) \cap D(y) = D(xy)$; i.e. we are interpreting $A_x \cap A_y$ in $A_{xy}$ or in $k(x, y)$. Clearly, those rational functions with powers in $x$ in the denominator and those with powers in $y$ in the denominator, are the polynomials. In other words, $A_x \cap A_y = A$. Thus we get:
$$\Gamma(U, \mathscr{O}_{A^2}) \equiv k[x, y]$$

We get no extra functions by removing the origin. Notice how any function on $\mathbb{A}^2 - \{(0, 0)\}$ extends over all of $\mathbb{A}^2$. This is an analog of Hartogs's lemma in complex geometry: any holomorphic function defined on the complement of a set of codimension atleast two can be extended to a complex manifold over the missing set. This works in a general algebraic setting: we can extend points in codimension atleast two, not only if they are smooth, but also if they are mildly singular (or normal).

$(U, \mathscr{O}_{A^2}|_U)$ is a scheme but not an affine scheme. Otherwise, $(U, \mathscr{O}_{A^2}|_U) = (\text{Spec } A, \mathscr{O}_{\text{Spec } A})$, then we can recover $A$ by taking global sections:
$$A = \Gamma(U, \mathscr{O}_{A^2}|_U)$$

So if $U$ is affine, then $U \cong \mathbb{A}_k^2$. But this bijection between prime ideals in a ring and points in the spectrum of the ring is more constructive than that: given prime ideal $I$, you can recover the point as a generic point of the closed subset cut out by $I$, $V(I)$, and given point $p$, you can recover the ideal as those functions vanishing at $p$, $I(p)$. In particular, the prime ideal $(x, y)$ should cut out a point of $\text{Spec } A$. But on $U$, $V(x) \cap V(y) = \phi$. In conclusion, $(U$ is not an affine scheme.

We have now seen two examples of non-affine schemes: infinite disjoint union of nonempty schemes, and $\mathbb{A}^2 - \{(0, 0)\}$. Before reviewing more examples, let us see how to glue topological spaces together along isomorphic open sets. Given two topological spaces $X, Y$ and open subsets $U \subset X, V \subset Y$, along with the homeomorphism $U \overset{\sim}{\longleftrightarrow} V$, we can create a new topological space $W$, that we obtain by gluing together $X, Y$ along $U \overset{\sim}{\longleftrightarrow} V$. It is the quotient of the disjoint union $X \sqcup Y$ by equivalence relation $U \sim V$, where the quotient is given by quotient topology. Then $X, Y$ are naturally identified with open subsets of $W$, and indeed cover $W$. Now, let us glue schemes $(X, \mathscr{O}_X), (Y, \mathscr{O}_Y)$ together. Let $U \subset X, V \subset Y$ be open subsets, along with a homeomorphism $U \xrightarrow{\sim} V$, and isomorphism of structure sheaves $\mathscr{O}_V \xrightarrow{\sim} \pi_* \mathscr{O}_U$; i.e. an isomorphism of sheaves $(U, \mathscr{O}_X|_U) \cong (V, \mathscr{O}_X|_V)$. Then, we can glue these together to get a single scheme; let $W$ be $X, Y$ glued together using $U \cong V$. Then the structure sheaves can be glued together to get a sheaf of rings. This is indeed a scheme: any point has an the open neighborhood that is an affine scheme.

Now we will look at examples of non-affine schemes by gluing together $X = \text{Spec } k[t], Y = \text{Spec } k[u]$. Let $U = D(t) = k[t, 1/t] \subset X, V = D(u) = k[u, 1/u] \subset Y$. In both examples, we will glue $X, Y$ along $U, V$; the difference is in how we glue.

Second example: Affine line with doubled origin. Consider the isomorphism $U \simeq V$ via $k[t, 1/t] \simeq k[u, 1/u]$ given by $t \leftrightarrow u$. This can be intuitively thought of as an analog of failure of Hausdorffness, although $\mathbb{A}^1$ itself is not Hausdorff. We will define separatedness later, which will be the right condition for Hausdorffness. In a separated scheme, the affine base of a Zariski topology is nice: the intersection of two open affine sets will be affine.

Third example: The projective line. Consider isomorphism $U \cong V$ via the isomorphism $k[t, 1/t] \cong k[u, 1/u]$ given by $t \leftrightarrow 1/u$. The resulting scheme is called the projective line over field $k$, and is denoted by $\mathbb{P}_k^1$. Traditional point $a$ on the $t$-line is glued to point $1/a$ on the $u$-line assuming $a \neq 0$, and generic point $(0)$ on the $t$-line maps to the generic point $(0)$ on the $u$-line, under the isomorphism $t \rightarrow 1/u$. If $k$ is algebraically closed, we can interpret the closed points of $\mathbb{P}_k^1$ in the following way: the points are $[a, b]$ where both $a, b$ are nonzero, and $[a, b]$ is identified with $[ac, bc], c \in k^\times$. If $b \neq 0$, this is identified with $a/b$ on the $t$-line, and if $a \neq 0$, this is identified with $b/a$ on the $u$-line.

To prove that $\mathbb{P}_k^1$ is not affine, we compute the ring of global sections. Global sections correspond to sections on $X$ and sections on $Y$ that agree on overlap. Restricting to overlap, we get polynomials $f(t)$ in $X$, and $g(u)$ in $Y$. We want $f(t) = g(1/t)$, but the only polynomials that are both polynomials in $t$ and $1/t$ are the constants $k$. Thus, if $\mathbb{P}_k^1$ we affine, $\Gamma(\mathbb{P}^1, \mathscr{O}_{\mathbb{P}^1}) = k$, and hence, $\text{Spec } \Gamma(\mathbb{P}^1, \mathscr{O}_{\mathbb{P}^1}) = \text{Spec } k$, i.e. one point. But it isn't: it has lots of points. We have proved an analog of an important theorem: the only holomorphic functions on $\mathbb{CP}^1$ are constants.

We will now define projective $n$-space over field $k$, $\mathbb{P}_k^n$, by gluing $n + 1$ open sets, each isomorphic to $\mathbb{A}_n^k$. We think of points $(x_0, x_1)$ in the projective space as being equivalent upto scalars; $(\lambda x_0, \lambda x_1)$ is equivalent to $(x_0, x_1)$. The first patch can be thought of as points $[1, t]$ where $t = x_1/x_0, x_0 \neq 0$, and the second patch can be thought of as points $[u, 1]$ where $u = x_0/x_1, x_1 \neq 0$. It will be useful, to instead use the notation $x_{0/1}$ for $u$ and $x_{1/0}$ for $t$. Then, the $i^{th}$ open set will have coordinates $x_{0/i}, \ldots, x_{i/i}, \ldots, x_{n/i}$. It will be convenient to write this as:
$$\text{Spec } k[x_{1/i}, \ldots, x_{n/i}]/(x_{i/i} - 1)$$

so we have introduced dummy variable $x_{i/i}$ which we immediately set to $1$. We glue distinguished open sets $D(x_{i/j})$ of $U_j$ to distinguished open sets $D(x_{j/i})$ of $U_i$, by identifying these two schemes by describing the identification of rings

$$ \text{Spec } k[x_{1/i}, \ldots, x_{n/i}]/(x_{i/i} - 1) \cong \\ \text{Spec } k[x_{1/j}, \ldots, x_{n/j}]/(x_{j/j} - 1) $$

via $x_{k/i} = x_{k/j}/x_{i/j}$ and $x_{k/j} = x_{k/i}/x_{j/i}$, which implies that $x_{i/j} x_{j/i} = 1$.

The Chinese remainder theorem is embedded in our argument; it can be thought of as a geometric fact. The prime ideals of $\mathbb{Z}/60$ are $(2), (3), (5)$, and its stalks are $\mathbb{Z}/4, \mathbb{Z}/3, \mathbb{Z}/5$. Nilpotents at $(2)$ can be thought of as a fuzz at $(2)$ in the corresponding picture. The global sections on this scheme are sections on the open sets $(2), (3)$, and $(5)$. Indeed, there is a natural isomorphism of rings:
$$\mathbb{Z}/60 \rightarrow \mathbb{Z}/4 \times \mathbb{Z}/3 \times \mathbb{Z}/5$$

Projective schemes

Schemes that were of interest in classical geometry are projective schemes or open subsets thereof, quasiprojective schemes. There are very few examples of schemes that are not provably quasiprojective. The notion of a projective $k$-scheme is a good approximation of the algebro-geometric version of compactness, or properness. Although projective schemes can be obtained by gluing together affine schemes, it can be annoying to keep track of gluing information. Just as there is a rough correspondence between rings and affine schemes, there is also a correspondence between graded rings and projective schemes. $\mathbb{P}^n$ can be interpreted as lines through origin in $\mathbb{R}^{n + 1}$, and subsets as unions of lines thereof, and closed subsets as such unions that are closed. We can picture $\mathbb{P}^n$ as being points at infinite distance in $\mathbb{R}^{n + 1}$, with points at infinity in one direction being associated with points at infinitely in the other direction. To make this precise, let us define the decomposition:
$$\mathbb{P}^{n + 1} = \mathbb{R}^{n + 1} \bigsqcup \mathbb{P}^n$$

by which we mean that the open subset of $\mathbb{P}^{n + 1}$ is identified with $\mathbb{R}^{n + 1}$, the points with the last projective coordinate nonzero, and the complementary closed subset $\mathbb{P}^n$ with the last coordinate zero.

For example, an equation cutting out a set of points $V$ in $\mathbb{P}^n$ also cut out a set of points in $\mathbb{R}^{n + 1}$ that will be a closed union of lines. We will call this the affine cone of $V$. This equation will cut out some union of $\mathbb{P}^1$s in $\mathbb{P}^{n + 1}$, and we will call this the projective cone of $V$. The projective cone is the disjoint union of the affine cone and $V$. For example, the affine cone defined by $x^2 + y^2 = z^2$ in $\mathbb{P}^2$ can be pictured as a classical cone in $\mathbb{R}^3$. For our $\text{Proj}$ construction, we switch coordinates from $x, y, z$ to $x_0, x_1, x_2$.

We informally observe that degree $d$ polynomials in $n + 1$ variables over a field form a vector space of dimension $\begin{pmatrix} n + d \\ d \end{pmatrix}$. It is not true that any two polynomials cut out the same subset of $\mathbb{P}_k^n$ if one is a nonzero multiple of the other; $x^2 y = 0$ and $x y^2 = 0$ show this. Instead, two polynomials cut out the same closed subscheme if one is a nonzero multiple of the other.

The $\text{Proj}$ construction produces a scheme out of a graded ring. We will now discuss graded rings. A $\mathbb{Z}$-graded ring is $S_\bullet = \oplus_{n \in \mathbb{Z}} S_n$, where the subscript is called grading; i.e. it sends $S_m \times S_n$ to $S_{m + n}$. Clearly $S_0$ is a subring, each $S_n$ is an $S_0$-module, and $S_\bullet$ is an $S_0$-algebra. Suppose that $S_\bullet$ is a $\mathbb{Z}$-graded ring. Those elements of some $S_n$ are called homogenous elements of $S_\bullet$; nonzero homogenous elements have an obvious degree. An ideal $I$ of $S_\bullet$ is a homogenous ideal if it is generated by homogenous elements. If $T$ is a multiplicative subset of $S_\bullet$ containing only homogenous elements, then $T^{-1} S_\bullet$ has a natural structure as a $\mathbb{Z}$-graded ring.

A $\mathbb{Z}^{\geq 0}$ ring is a $\mathbb{Z}$-graded ring with no elements of negative degree. Henceforth, a graded ring will refer to $\mathbb{Z}^{\geq 0}$ ring. $S_\bullet$ is assumed to be a graded ring. Fix ring $A$ as the base ring. If $S_0 = A$, we say that $S_\bullet$ is a graded ring over $A$. A key example is $A[x_0, \ldots, x_n]$ or even $A[x_0, \ldots, x_n]/I$ where $I$ is a homogenous ideal with $I \cap S_0 = 0$. Here, we take conventional grading on $A[x_0, \ldots, x_n]$ where each $x_i$ has weight $1$. The subset $S_+ := \oplus_i S_i \subset S_\bullet$ is an ideal, called the irrelevant ideal. If the irrelevant ideal $S_+$ is finitely generated, we say that $S_\bullet$ is a finitely generated graded ring over $A$. If $S_\bullet$ is generated by $S_1$ as an $A$-algebra, we say that $S_\bullet$ is generated in degree $1$.

We now define the scheme $\text{Proj } S_\bullet$, where $S_\bullet$ is a graded ring. Let us start with two instructive examples. If $S_\bullet = A[x_0, \ldots, x_n]$, we will recover $\mathbb{P}_A^n$; and if $S_\bullet = A[x_0, \ldots, x_n]/(f_0, \ldots, f_n)$, where $f$ is homogenous, we construct something cut out in $\mathbb{P}_A^n$ by $f = 0$.

As we did with the spectrum of a ring, we will construct $\text{Proj } S_\bullet$ first as a set, then as a topological space, and finally as a ringed space. As in our preliminary discussion, we will glue together $n + 1$ well-chosen affine spaces, but we do it by considering all possible affine open sets. Our affine building blocks will be as follows. For each $f \in S_+$, note that the localization $(S_\bullet)_f$ is naturally a $\mathbb{Z}$-graded ring, where $\text{deg}(1/f) = -\text{deg } f$. Consider $\text{Spec }(((S_\bullet)_f)_0)$, a zero-graded piece of the ring $(S_\bullet)_f$; the first and third subscripts refer to grading, and the second to localization. Applying this ring to $S_\bullet = k[x_0, \ldots, x_n]$ where $f = x_i$, we obtain the ring:
$$k[x_0/x_i, \ldots, x_n/x_i]/(x_i/x_i - 1)$$

The points of $\text{Proj } S_\bullet$ are the set of homogenous prime ideals of $S_\bullet$ not containing the irrelevat ideal $S_+$.

The correspondence of $\text{Proj } S_\bullet$ with homogenous prime ideals helps us picture $\text{Proj } S_\bullet$. For example, if $S_\bullet = k[x, y, z]$ with the usual grading, then we picture the homogenous prime ideal $(z^2 - x^2 - y^2)$ first as the subset of $\text{Spec } S_\bullet$; it is a cone. We picture $\mathbb{P}_k^2$ as the plane at infinity. Thus, we picture this equation as cutting out a conic at infinity in $\text{Proj } S_\bullet$. If $T$ is the set of homogenous elements of $S_\bullet$, define the projective vanishing set $V(T) \subset \text{Proj } S_\bullet$ to be those homogenous prime ideals containing $T$, but not $S_+$. Define $V(f)$ if $f$ is a homogenous element of positive degree, and $V(I)$ if $I$ is a homogenous ideal contained in $S_+$. Define $D(f) := \text{Proj } S_\bullet \backslash V(f)$, the projective distinguished open set, to be the complement of $V(f)$. Once we define a scheme structure on $\text{Proj } S_\bullet$, we will define $D(f)$ to be, not just the open subset, but the open subscheme. As in the affine case, $V(I)$s satisfy the axioms for a closed set in topology, and we call this the Zariski topology on $\text{Proj } S_\bullet$. Many results about Zariski topology in the spectrum of a ring carry over with a little extra work. Clearly $D(fg) = D(f) \cap D(g)$ by the same immediate argument as in the affine case.

We now redefine the projective space over a ring $A$ by $\mathbb{P}_A^n = \text{Proj } A[x_0, \ldots, x_n]$. This definition involves no messing gluing or choice of patches. Note that projective coordinates $x_0, \ldots, x_n$ is part of the definition.

We call a scheme that is isomorphic to $\text{Proj } S_\bullet$, where $S_\bullet$ is a finitely generated graded ring over $A$, a projective scheme over $A$, or a projective $A$-scheme. A quasiprojective $A$-scheme is a quasicompact open subscheme of a projecive $A$-scheme.

Consider the example of projectivization of $\mathbb{P}V$. Let $V$ be an $(n + 1)$-dimensional vector space over $k$. Define:
$$\text{Sym}^\bullet V^\vee := k \oplus V^\vee \oplus \text{Sym}^2 V^\vee \oplus \ldots$$

If, for example, $V$ is the dual of the vector space with basis associated to $x_0, \ldots, x_n$, we would have $\text{Sym}^\bullet V^\vee = k[x_0, \ldots, x_n]$. Then we can define $\mathbb{P}V = \text{Proj}(\text{Sym}^\bullet V^\vee)$. In this language, $x_0, \ldots x_n$ can be interpreted as linear functionals on the underlying vector space $V$. We can also describe a natural bijection between the points of $V$ and closed points of $\text{Proj}(\text{Sym}^\bullet V^\vee)$. This construction respects the affine/projective cone picture.