We have already shown that $\mathbb{A}_k^n$ is irreducible; this argument behaves well under gluing yielding the fact that $\mathbb{P}_k^n$ behaves well under gluing. If there is some property $P$ of schemes that is "open", then to check if all points of the quasicompact scheme have $P$, it is sufficient to check just the closed points. In many good situations, the closed points are dense.

A scheme is said to be quasiseparated if the intersection of any two quasicompact subsets is quasicompact. We will later realize that this is a property of morphisms, not schemes.

## Reducedness and integrality

Recall that a ring is said to be reduced if it has no nonzero nilpotents. A scheme is said to be reduced if $\mathscr{O}_X(U)$ is reduced for every open set $U$. The scheme $\text{Spec } k[x, y]/(y^2, xy)$ is nonreduced, as we have already seen as a fuzz around the origin.

A scheme $X$ is said to be an integral if it is nonempty and if $\mathscr{O}_X(U)$ is an integral domain for every open subset $U$ of $X$. Be cautioned that integrality is not a stalk-local property, since $\text{Spec } A \sqcup \text{Spec } B = \text{Spec}(A \times B)$.

Irreducible varities have the convenient property that sections over different open subsets can be considered as subrings of the same ring. In particular, restriction maps are always considered inclusions, and gluing is easy: $f_i$ on a cover $U_i$ of $U$ glue iff they are the same element of $K(X)$.

## Properties of schemes that can be checked affine-locally

We have defined a scheme to be a topological space with a sheaf of rings, that can be covered by affine schemes. Hence, we have all affine open sets in the cover, and the Affine Communication Lemma will provide us a way for communicating between them.

Affine-local properties can be checked on any affine cover; i.e. a covering by affine open sets. If a scheme is quasicompact, we only need to check a finite number of affine open sets.

Suppose $\text{Spec } A$ and $\text{Spec } B$ are affine local subschemes. Then, $\text{Spec } A \cap \text{Spec } B$ is the union of open sets that are simultaneously distinguished open subschemes of $\text{Spec } A$ and $\text{Spec } B$. To prove this, notice that given any point $p \in \text{Spec } A \cap \text{Spec } B$, we can produce the open neighborhood of $\text{Spec } A \cap \text{Spec } B$ that is simultaneously distinguished in both $\text{Spec } A$ and $\text{Spec } B$. Let $\text{Spec } A_f$ be the distinguished open subset in $\text{Spec } A$ that is contained in $\text{Spec } A \cap \text{Spec } B$ and contains $p$. Let $\text{Spec } B_g$ be the distinguished open subset of $\text{Spec } B$ contained in $\text{Spec } A_f$ and containing $p$. Then, $g \in \Gamma(\text{Spec } B, \mathscr{O}_X)$ restricts to an element $g' \in \Gamma(\text{Spec } A_f, \mathscr{O}_X) = A_f$. The points of $A_f$ where $g$ vanishes are precisely the points of $A_f$ where $g'$ vanishes, so

If $g' = g''/f^n (g'' \in A)$, then $\text{Spec}(A_f)_{g'} = \text{Spec}(A_{fg''})$, and we are done.

Affine Communication Lemma: Let $P$ be some property enjoyed by some affine open subsets of scheme $X$, such that:

- If an affine open subset $\text{Spec } A \hookrightarrow X$ has property $P$, then for any $f \in A$, $\text{Spec } A_f \hookrightarrow X$ does too.
- If $(f_1, \ldots, f_n) = A$, and $\text{Spec } A_f \hookrightarrow X$ has property $P$ for all $i$, then so does $\text{Spec } A \hookrightarrow X$.

Suppose $X = \cup_{i \in I} \text{Spec } A_i$ where each $\text{Spec } A_i$ has property $P$. Then, every affine open subset of $X$ also has property $P$. We say that such a property is affine-local: indeed, we can check it on any affine open cover. For example, reducedness is an affine-local property. If $U$ is an affine open subscheme of $X$, then $U$ inherits any affine-local property of $X$. Note that any property that is stalk-local is affine-local as well. To prove this, let $\text{Spec } A$ be an affine subscheme of $X$. Cover $\text{Spec } A$ with a finite number of distinguished open sets $\text{Spec } A_{g_j}$, each of which is distinguished in some $\text{Spec } A_i$. By (i), each $\text{Spec } A_{g_j}$ has property $P$, and by (ii), each $\text{Spec } A$ does too.

Suppose $X$ is a scheme. If $X$ can be covered by affine open sets $\text{Spec } A$ where $A$ is Noetherian, we say that $X$ is a locally Noetherian scheme. If $X$ is quasicompact, or equivalently, can be covered by a finite number of such open sets, we say that $X$ is a Noetherian scheme. The underlying topological space of a Noetherian scheme is Noetherian, and all finite open subsets of a Noetherian scheme are quasicompact.

Since we will be interested in working with schemes over a particular ring $A$, we define an $A$-scheme, where $A$ is a ring, as a scheme where all rings of sections over the structure sheaf are $A$-algebras, and all restriction maps are $A$-algebras. Now suppose $X$ is an $A$-scheme. If $X$ can be covered by $\text{Spec } B_i$ where each $B_i$ is an $A$-algebra, we say that $X$ is a locally finite type $A$-scheme. If, furthermore, $X$ is quasicompact, then it is a finite type $A$-scheme. Note that a scheme of locally finite type over $k$ or $\mathbb{Z}$ is locally Noetherian, and similarly, a scheme of finite type over a Noetherian ring is Noetherian. To give a couple of examples, $\text{Spec } \mathbb{C}[x_0, \ldots, x_n]/I$ and $\mathbb{P}_\mathbb{C}^n$ are finite type $\mathbb{C}$ schemes.

An affine scheme that is reduced and of affine type over $k$ is said to be an affine $k$-variety. A reduced (quasi-)projective $k$-scheme is a (quasi-)projective $k$-variety. The degree of a locally finite type $k$-scheme (eg. a $k$-variety) is the degree of the field extension $\kappa(p)/\kappa$. For example, in $\mathbb{A}_k^1 = \text{Spec } k[t]$, the point $[(p(t))]$, $p \in k[t]$ irreducible, is $\text{deg } p(t)$. If $k$ is algebraically closed, then the degree of all closed points is $1$.

Suppose $A$ is a ring, and $(f_1, \ldots, f_n) = A$.

- If $A$ is Noetherian, then so is $\text{Spec } A$. If each $A_{f_i}$ is Noetherian, then so is $A$.
- If $A$ be a finitely-generated $B$-algebra, then so is $A_{f_i}$. If each $A_{f_i}$ is a finitely-generated $B$-algebra, then so is $A$.

We divide the proof into parts (i) and (ii) in accordance with the Affine Communication Lemma. To prove (a)(i), consider the strictly increasing chain of ideals $I_1 \subsetneq I_2 \subsetneq \ldots$ of $A_f$. Then, we can verify that $J_1 \subsetneq J_2 \subsetneq \ldots$ is a strictly increasing chain of ideals of $A$, where:

$$J_j = \{r \in A : r \in I_j\}$$

where $r \in I_j$ means image of $r$ in $A_f$ lies in $I_j$. Clearly $J_j$ is an ideal of $A$. If $x/f^n \in I_{j + 1} \backslash I_j$, where $x \in A$, then $x \in J_{j + 1}$ and $x \notin J_j$. For part (ii), consider the strictly increasing chain of ideals $I_1 \subsetneq I_2 \subsetneq \ldots$ in $A$. Then, for each $1 \leq i \leq n$:

$$I_{i, 1} \subset I_{i, 2} \subset I_{i, 3} \subset \ldots$$

is an increasing chain of ideals of $A_f$. Then, $I_{i, j} = I_j \otimes_A A_{f_i}$. It remains to be shown that for each $j$, $I_{i, j} \subsetneq I_{i, j + 1}$, for some $i$; then the result will follow.

Now, (b)(i) is clear: if $A$ is generated over $B$ as $r_1, \ldots, r_n$, then $A_f$ is generated over $B$ as $r_1, \ldots, r_n, 1/f$. For part (ii), since $f_i$s generate $A$, we can write $1 = \sum c_i f_i$ for $c_i \in A$. We have generators of $A_f : r_{ij}/f_i^{k_j}$, where $r_{ij} \in A$. Now, claim that $\{c_i\} \cup \{f_i\}_i \cup \{r_{ij}\}_{ij}$ generate $A$ as a $B$-algebra. Suppose we have $r \in A$. Then in $A_{f_i}$, we can write $r$ as some polynomial in $r_{ij}$s and $f_i$, divided by some huge power of $f_i$. So, in each $f_i$, we have described $r$ in the obvious way, except for the huge denominator. Now use the partition of unity argument to combine all this expressions into a single expression killing the denominator. The resulting expression we build agrees with $r$ in each $A_{f_i}$. Thus, it is indeed $r$ by identity axiom for the structure sheaf.

## Normality and factoriality

We define a property of schemes that are "not too far from smooth", normality. Locally Noetherian normal schemes satisfy Hartogs's lemma: functions defined away from a set of codimension $2$ extend over that set. As a consequence, rational functions that have no poles are defined everywhere. An integral domain $A$ is integrally closed iff the zeros in $K(A)$ to any monic polynomial in $A[x]$ must lie in $A$ itself. We say that scheme $X$ is normal if all of its stalks $\mathscr{O}_{X, p}$ are normal; i.e. are integral domains, and integrally closed in their fraction fields. By definition, normality is a stalk-local property, as is reducedness. If $A$ is an integrally closed domain, then $\text{Spec } A$ is normal. For quasicompact schemes, normality can be checked at closed points. For such schemes, any point is a generization of a closed point. Normal schemes, however, need not be integral. Thus, $\text{Spec } k \sqcup \text{Spec } k \cong \text{Spec}(k \times k) \cong \text{Spec } k[x]/x(x - 1)$ is normal, but its ring of global sections is not an integral domain.

To quote a result from commutative algebra, if $A$ is an integral domain, then the following statements are equivalent:

- $A$ is integrally closed.
- $A_\mathfrak{p}$ is integrally closed for prime ideal $\mathfrak{p} \subset A$.
- $A_\mathfrak{m}$ is integrally closed for maximal ideal $\mathfrak{m} \subset A$.

If all stalks of a scheme $X$ are UFDs, we say that $X$ is factorial. Thus, if $A$ is a UFD, $\text{Spec } A$ is factorial, but the converse need not hold. We will later see that elliptic curves are factorial, yet no affine open set is the Spec of a UFD. We can show factoriality by finding an appropriate affine open cover, but there need not be such a cover of a factorial scheme. There are few means of checking if a Noetherian integral domain is a UFD:

- Localization of a UFD is also a UFD.
- Height $1$ prime ideals are principal.
- Normal and $CI = 0$.
- Nagata's lemma.

Moreover, factoriality implies normality.

## Associated points and primes

Associated points of schemes capture some crucial information about the sheaf of functions over them. To start with a geometric intuition, we will think of them as "fuzzy attractors". In $\text{Spec } k[x, y]/(y^2, xy)$, the fuzz about origin captured nonreduced behavior there. This can be justified as: origin is the only point where the structure sheaf is nonreduced. Since irreducible closed subsets are in bijection with points, we may say that two keys points with respect to nonreducedness were the generic point $[(y)]$ and origin $[(x, y)]$; these will be the associated points in the scheme.

Another way to think about them: associated points are generic points of irreducible components of the support of sections. Recall that support of a function on a scheme is a closed subset. The fact that the same closed subsets arise in two different ways is no coincidence: their generic points are associated points of $\text{Spec } k[x, y]/(y^2, xy)$. When discussing associated points of $M$, we consider $M$ to be a finitely generated $A$-module, and $A$ to be Noetherian. When talking about rings rather than schemes, we discuss associated primes rather than associated points. We list three properties now:

- The associated primes/points of $M$ are precisely the generic points of generic points of irreducible components of the support of some element of $M$.
- $M$ has a finite number of associated points/primes. For example, there are only a finite number of irreducible closed subsets of $\text{Spec } A$, such that the only possible functions of $\text{Spec } A$ are unions of these. We can interpret this as Noetherianness forcing some kind of finiteness. This gives some meaning to the statement that generic points are the few crucial points of the scheme. We can completely describe which supports of $\text{Spec } A$ are the support of an element of $M$: precisely those subsets which are the closure of the subsets of the associated points.

We immediately see from (a) that, if $M = A$, the generic points of the irreducible components of $\text{Spec } A$ are the associated primes of $M = A$, by considering the function $1$. Other associated points of $\text{Spec } A$ are called embedded points. Thus, in the case of $\text{Spec } k[x, y]/(y^2, xy)$, the only embedded point is the origin. Furthermore the map:

$$M \rightarrow \prod_{\text{associated } p} M_p$$

is injective (Once again, we see that the associated points are where all the action happens). We show this by showing that the kernel is $0$. Suppose the function $f$ has a germ $0$ at each associated point, so its support contains no associated points. It is supported on a closed subset, which by (a), must be the union of closures of associated points. Thus, it must be supported nowhere, and be the zero function.

- An element $A$ is a zerodivisor of $M$ iff it vanishes on some associated point in $M$; i.e. is contained within some associated prime of $M$.

Associated points/primes can be determined locally. For example, associated points of $A$ can be checked by looking at the stalks of the structure sheaf. As another example, associated prime ideals of $M$ may be determined by working on a distinguished open cover of $\text{Spec } A$. We define the associated points of a locally Noetherian scheme $X$ to be those points $p \in X$ such that, on any affine open set in $\text{Spec } A$ containing $p$, $p$ corresponds to the associated prime of $A$. This notion is independent of the choice of open neighborhood of $\text{Spec } A$: if $p$ has two open neighborhoods $\text{Spec } A$ and $\text{Spec } B$, $p$ corresponds to the associated prime of $A$ (say, $p$ corresponds to a prime ideal in $A$ and $q$ corresponds to a prime ideal in $B$) iff it corresponds to an associated prime $A_p = \mathscr{O}_{X, p} = B_q$ iff it corresponds to an associated prime of $B$.

Combining the properties above, we get the following list of facts:

- A Noetherian scheme has finitely many associated points.
- Each of the irreducible components of the support of any function of a locally Noetherian scheme is the union of closures of some subset of the associated points.
- The generic points of the irreducible components of a locally Noetherian scheme are the associated points.
- A reduced locally Noetherian scheme has no embedded points.
- A nonreduced locus of a locally Noetherian scheme is the closure of those associated points that have nonreduced stalks.

Furthermore, note that for any integral scheme $X$ (for instance, an irreducible affine variety), that is not shared by all schemes, is that for any nonempty open subset $U \subset X$, the natural map $\Gamma(U, \mathscr{O}_X) \rightarrow K(X)$ is an inclusion. Thus, sections over any nonempty open subset, and elements of all stalks, can be thought of as lying in $K(X)$, the stalk at the generic point. Associated primes allow us to generalize this idea.

A rational funtion on a locally Noetherian scheme is an element of the image of $\Gamma(U, \mathscr{O}_U)$ for some $U$ containing the associated primes. Equivalently, the set of rational functions is the colimit of $\mathscr{O}_X(U)$ over the open sets containing associated points. Another equivalent definition: a rational function is a function defined on the open set containing all associated points; i.e. an ordered pair $(U, f)$ with $U$ an open set containing all associated points, and $f \in \Gamma(U, \mathscr{O}_X)$. Two such data $(U, f), (U', f')$ define the same rational function iff the restrictions of $f, f'$ on $U \cap U'$ are the same. If $X$ is reduced, it is the same as requiring that they be defined on an open set of each of the irreducible components. For example, in $\text{Spec } k[x, y]/(y^2, xy)$, $\frac{x - 2}{(x - 1)(x - 3)}$ is a rational function but $\frac{x - 2}{x(x - 1)}$ is not.

A rational function has a maximal domain of definition because any two actual functions that agree as rational functions must be the same. We say that a rational function $f$ must be regular at point $p$ if $p$ is contained in this maximal domain of definition. For example, in $\text{Spec } k[x, y]/(y^2, xy)$, $\frac{x - 2}{(x - 1)(x - 3)}$ has domain of definition on everything except at $1, 3$. A rational function is regular if it is regular at all points.

A rational function forms a ring, called total fraction ring or total quotient ring of $X$. If $X = \text{Spec } A$ is affine, then this ring is called the total fraction ring of $A$. If $X$ is integral, the total fraction ring is the total function field $K(X)$: the stalk at the generic point.

To augment our earlier definitions, let $M$ be an $A$-module for arbitarary ring $A$. Then:

- A prime $\mathfrak{p} \subset A$ is said to be associated to $M$ if $\mathfrak{p}$ is an annihilator of element $m \in M$. Equivalently, $\mathfrak{p}$ is associated to $M$ iff $M$ has a submodule associated to $A/\mathfrak{p}$. The set of prime ideals associated to $M$ is denoted by $\text{Ass } M$. Awkwardly, if $I$ is an ideal, the associated prime ideals of the module $A/I$ are said to be the associated prime ideals of $I$.

The properties of associated prime ideals can be listed as follows. Given Noetherian ring $A$ and $M \neq 0$ finitely generated:

- The set $\text{Ass } M$ is finite and nonempty.
- The natural map $M \rightarrow \prod_{\mathfrak{p} \in \text{Ass } M} M_\mathfrak{p}$ is an injection.
- The set of zerodivisors of $M$ is $\bigcup_{\mathfrak{p} \in \text{Ass } M} \mathfrak{p}$.
- Association commutes with localization. If $S$ is a multiplicative set then $\text{Ass}_{S^{-1} A} S^{-1} M = \text{Ass}_A M \cap \text{Spec } S^{-1} A$.

If $A$ is a Noetherian ring, then any element of any minimal prime $\mathfrak{p}$ is a zerodivisor. Suppose $s \in \mathfrak{p}$. Then, by minimality of $\mathfrak{p}$, $\mathfrak{p} A_\mathfrak{p}$ is a unique prime ideal in $A_\mathfrak{p}$, so the element $s/1$ of $A_\mathfrak{p}$ is nilpotent, because it is contained in all prime ideals of $A_\mathfrak{p}$. Thus, for $t \in A \backslash \mathfrak{p}$, $ts^n = 0$, so $s$ is a zerodivisor.

The associated prime ideals of an ideal turn out to be precisely the radicals of ideals in any primary decomposition.