Projectivization

Here we define the projectivization of a graded ring and the projectivization of a sheaf of graded rings, and prove some important lemmas about them. We also record some important facts about graded rings. This basically follows the discussion in Hartshorne’s Algebraic Geometry, Chapter II, Section 2.

Preliminaries

I may eventually decide to copy this into an algebra article, but I think it’s important to have here.

Definition (Graded Ring):

A graded ring is a ring $A$, together with a decomposition

of the underlying abelian group of $A$, subject to the condition $A_iA_j\subseteq A_{i+j}$ on the multiplicative structure of $A$. A homogeneous element of the ring is an element contained in $A_i$ for some $i$; the degree of that element is then $i$. A homogeneous ideal is an ideal which can be generated by homogeneous elements (note, in particular, this does not mean that each element of the ring is homogeneous. Additionally, the homogeneous elements may not all be of the same degree). A homogeneous ideal can be prime or maximal if it also satisfies the conditions of a prime or maximal ideal. The irrelevant ideal $A_+$ is the ideal

See this example.

Lemma (Primality on Homogeneous Elements):

A homogeneous ideal $\mf a$ of $A$ is prime if and only if for any homogeneous elements $a, b$ of $A$, $ab\in A$ implies $a\in A$ or $b\in A$.

A morphism of graded rings is a morphism of rings $\psi: A\to B$ where

and $\psi(A_i)\subseteq B_i$. Graded rings form a category; the standard identity morphism is a morphism of graded rings, the composition of morphisms of graded rings is a morphism of graded rings, and composition is associative. A sheaf of graded rings is a sheaf with values in the category of graded rings.

Proj of a Graded Ring

Let $A$ be a graded ring. We seek to define a scheme $\proj(A)$. The underlying set of $\proj(A)$ is the set of all homogeneous prime ideals of $A$ which are not the irrelevant ideal. If $\mf a$ is a homogeneous ideal of $A$, let $V(\mf a)$ be the set of all homogeneous prime ideals $\mf p$ such that $\mf a\subseteq \mf p$. We then wish to establish the

Lemma (Closed Sets of Proj):

The sets $V(\mf a)$ satisfy the following conditions:

1. If $\mf a$ and $\mf b$ are homogeneous ideals, then $V(\mf a\mf b) = V(\mf a) \cup V(\mf b)$
2. If $\mf a_i$ are a family of homogeneous ideals, then

Proof. Firstly, $\mf p$ is a homogeneous prime ideal that contains $\mf a$ and $\mf b$ if and only if $\mf p$ is a homogeneous prime ideal which contains $\mf a\mf b$, since we can check primality on homogeneous elements and $\mf a$ and $\mf b$ are generated by homogeneous elements. The second part holds similarly. QED.

We see from this that sets of the form $V(\mf a)$ satisfy the conditions for closed sets of a topology, and so we give $\proj(A)$ the topology where the closed sets are sets of the form $V(\mf a)$. It remains to specify a structure sheaf $\mc O_{\proj(A)}$. We say that $\mc O_{\proj(A)}(U)$ is the set of functions $s:U\to \coprod S_{(\mf p)}$ such that for each $\mf p\in U$, $s(\mf p)\in s_{(\mf p)}$ and furthermore that for every point $\mf p$, there is a neighborhood of $\mf p$ such that for each $\mf q$ in that neighborhood $s(\mf q)$ a fracton $\frac{f}{g}$ where $f$ and $g$ are independent of $\mf q$, homogeneous, and satisfy $\deg f= \deg g$. This set is a subring of the field of fractions: addition and multiplication of two such elements yield a third such element (in particular, addition of fractions where the numerator and denominator are homogeneous of the same degree yields another fraction woth homogenous numerator and denominator of the same degree). Clearly $1=\frac11$ and $0=\frac 01$ are of this form, because $0$ and $1$ are homogeneous of degree 0. Additive inverses follow from the closure of the $A_i$ under additive inversion. Moreover, this is clearly a presheaf; gluing follows from the fact that if two functions $s$ and $t$ agree on the overlaps of their domain, then we can define a function on the union of their domain which restricts to each. (Also, this definition is essentially a sheafification).

It’s perhaps useful to note that the sections of this sheaf over an open set $U$ are locally constant functions with values in the field of fractions of $A$, such that the numerator and denominator are both homogeneous, the “total degree” of the fraction (the degree of the numerator minus the degree of the denominator) is zero, and the denominator does not lie in $\mf q$ for any $\mf q\in U$. Another way to view total degree is that the grading on $A$ induces a grading on any localization of $A$, by taking $\deg\left(\frac fg\right) := \deg f -\deg g$; this is the total degree of the fraction.

Proposition (Proj is a Scheme) (INCOMPLETE):

For any graded ring $A$, $\proj(A)$ is a scheme.

THIS PROOF IS INCOMPLETE

Proof. We must show that $\proj (A)$ is a locally ringed space which is locally isomorphic to $\spec R$ for some $R$. In fact, we will attempt the second part first; if we can show that $\proj(A)$ is locally isomorphic to $\spec R$ for some $R$ without considering the locally ringed structure, we obtain that $\proj (A)$ is locally ringed and that isomorphism preserves the locally ringed structure. Indeed, that $\proj(A)$ is locally isomorphic to $\spec(A)$ is mostly a fact of commutative algebra; it suffices to show that for any homogeneous element $f\in A$ of degree at least 1, that there is an isomorphism between the degree zero peice of $A[f^{-1}]$ and $A/(f-1)$, and moreover that this isomorphism induces an isomorphism of schemes between $U(f)$ and $\spec A/(f-1)$. We refer to these isomorphisms as the dehomogenization isomorphisms, and their inverses homogenization isomorphisms.

To construct these isomorphisms, we will begin by constructing homogenization and dehomogenization maps between the collection of homogeneous elements of the ring $A$ which do not contain $f$ and the elements of $A/(f-1)$, and then show that these induce the appropriate maps.

Note that there is an inclusion of rings $\iota:A/(f-1)\to A$, given by lifting a coset $g$ in the quotient to the unique representative in which $f$ does not appear; i.e., we can write an element $h\in A$ as $h_1 + fh_2$, which in the quotient becomes $h_1+fh_2 + (f-1) = h_1 + h_2 +(f-1)$ (because

and so they are equal); we lift $h_1+f_h2+(f-1) = h_1+h_2 + (f-1)$ to $h_1+h_2$. This is unique for any choice of $f$ and $h$, as the way of writing $h= h_1+fh_2$ is unique. Interestingly, this means the short exact sequence

splits; there is a group-theoretic section of the second map, turning this short exact sequence into a product diagram and insuring that $A \cong (f-1)\oplus A/(f-1)$ as (additive) abelian groups. (This isn’t particularly relevent, but I’m writing these notes for me and I find it interesting). We define the (set-theoretic) map $\phi:A/(f-1)\to A$ by

where we write

for $g_i$ homogeneous (we can do this by definition; note that this property is the reason we want a direct sum in the definition, not a direct product.) Note that $\phi(g)$ is homogeneous of degree $n$. Thus this map (the homogenization map) maps $A/(f-1)$ into the set of homogeneous elements of $A$. We also note that the quotient map $\psi: A\to A/(f-1)$ induces a map on the set of homogeneous elements of $A$. If $g$ is homogeneous of degree $n$, we can write $g$ as

where $g_i$ is of degree $i$. Then the quotient map restricted to homogeneous elements (which we call the dehomogenization map) followed by $\iota$ sends this to $\sum_{i=0}^ng_i$ and homogenizing gives $\sum_{i=0}^ng_if^{n-i}.$ conversely, $\iota$ acting on any element of the quotient can be written as $\sum_{i=0}^ng_i;$ homogenizing gives $\sum_{i=0}^ng_if^{n-i},$ and dehomogenizing via $\psi$ yields the same equivalence class as $\sum_{i=0}^ng_i$ in the quotient. Thus the maps are mutual inverses.

Let $\mf p$ be a homogeneous prime ideal in $\proj (A)$ which does not contain $f$. Note that $\psi(\mf p)$ be $\mf p / (\mf p \cap (f-1))$. Since $f\not \in \mf p$, we know that $\psi(\mf p)$ cannot be the unit ideal; we seek to show that $\psi(\mf p)$ is a prime ideal of $A/(f-1)$, that is, a point in $\spec A/(f-1)$. Note that

By the ring isomorphism theorems); from this we conclude that $\psi$ induces a map $\proj(A)\to \spec A/(f-1)$. Conversely, $\phi=\psi^{-1}$ on homogeneous elements; we define $\phi:\spec A/(f-1)\to \proj(A)$ by the rule $\phi(\mf p) = (\phi(\mf p))$, the ideal generated by $\phi(\mf p)$. Note first that this ideal is generated by homogeneous elements, because the image of $\phi$ consists only of homogeneous elements. Further note that for two elements $g, h$ of $A$, we have $gh\in \phi(\mf p)$

Proj of a Sheaf of Graded Rings