Suppose we want to define "X is Noetherian" to mean "X has an affine cover by Noetherian rings." We need to verify:

1. If one affine cover consists of Noetherian rings, do all affine covers?
2. If we take a finer affine cover, is the property preserved?

Without the Affine Communication Lemma, we would need to verify these questions separately for each property we care about.

The property "$A$ is a Noetherian ring" satisfies both conditions of the Affine Communication Lemma:

Localization: If $A$ is Noetherian and $f \in A$, then $A_f$ is Noetherian. This is a standard result: every ideal of $A_f$ is of the form $I \cdot A_f$ where $I$ is an ideal of $A$.

Cover: If $f_1, \ldots, f_n$ generate the unit ideal and each $A_{f_i}$ is Noetherian, then $A$ is Noetherian. To see this, let $I_1 \subseteq I_2 \subseteq \cdots$ be an ascending chain of ideals in $A$. Then $I_1 \cdot A_{f_i} \subseteq I_2 \cdot A_{f_i} \subseteq \cdots$ is an ascending chain in each $A_{f_i}$, which stabilizes. By the gluing property of ideals, the original chain stabilizes in $A$.

Therefore, we can define: A scheme $X$ is Noetherian if it admits a cover by affine opens $\mathrm{Spec}(A_i)$ where each $A_i$ is Noetherian. The Affine Communication Lemma guarantees this is well-defined.

For a scheme $X$, the following are equivalent:

1. $X$ is Noetherian (admits a cover by Noetherian affine schemes)
2. Every open affine subscheme $\mathrm{Spec}(A) \subseteq X$ has $A$ Noetherian
3. $X$ is quasi-compact and every open subset is quasi-compact
4. $X$ satisfies the descending chain condition on open subsets

The Affine Communication Lemma proves the equivalence of (1) and (2).

The property "$A$ is reduced" (no nonzero nilpotent elements) is affine-local:

Localization: If $A$ is reduced and $f \in A$, then $A_f$ is reduced. Indeed, if $(a/f^n)^m = 0$ in $A_f$, then $f^k a^m = 0$ in $A$ for some $k$. Since $A$ is reduced, $f^k a^m = 0$ implies $a^m = 0$ (as $f^k$ is not a zero divisor in a reduced ring... actually, we need to be more careful).

More precisely: if $a \in A$ is nilpotent in $A_f$, then $a^n/1 = 0/1$ in $A_f$ for some $n$, so $f^m a^n = 0$ in $A$ for some $m$. This means $(fa)^{\min(m,n)} = 0$, so $fa$ is nilpotent in $A$, hence $fa = 0$ since $A$ is reduced. But then $a$ maps to zero in $A_f$, so $a/1 = 0/1$.

Actually, the cleanest proof: $A_f = A[1/f] \subseteq A[f^{-1}]$ embeds in $\mathrm{Frac}(A)$ when $A$ is reduced, so $A_f$ is reduced.

Cover: If $f_1, \ldots, f_n$ generate the unit ideal and each $A_{f_i}$ is reduced, then $A$ is reduced. If $a \in A$ is nilpotent with $a^m = 0$, then $a/1 = 0$ in each $A_{f_i}$, so $f_i^{k_i} a = 0$ for each $i$. Taking $k = \max k_i$, we have $f_i^k a = 0$ for all $i$. Since $(f_1, \ldots, f_n) = A$, we have $(f_1^k, \ldots, f_n^k) = A$, so $\sum r_i f_i^k = 1$ for some $r_i$. Then $a = \sum r_i f_i^k a = 0$.

Therefore, reducedness is affine-local.

A scheme $X$ is reduced if and only if the structure sheaf $\mathcal{O}_X$ has no nilpotent sections on any open set. Equivalently, the nilradical $\mathcal{N} \subseteq \mathcal{O}_X$ is zero.

This is the scheme-theoretic version of saying that the space has no "embedded fuzz" or "fattened points."

Integrality: A scheme $X$ is integral if it is both reduced and irreducible. However, this is NOT purely an affine-local property in the sense of the Communication Lemma, because irreducibility involves global topology. Nevertheless, for an affine scheme $X = \mathrm{Spec}(A)$:

• $X$ is integral $\iff$ $A$ is an integral domain

Normality: A scheme $X$ is normal if every local ring $\mathcal{O}_{X,x}$ is an integrally closed domain. For affine schemes, $\mathrm{Spec}(A)$ is normal $\iff$ $A$ is a normal ring (integrally closed in its fraction field).

The property "$A$ is a normal ring" is affine-local:

Localization: If $A$ is normal and $f \in A$, then $A_f$ is normal. This follows because $(A_f)_\mathfrak{p} = A_\mathfrak{q}$ where $\mathfrak{q}$ is the preimage of $\mathfrak{p}$, and integrally closed is preserved by localization.

Cover: If each $A_{f_i}$ is normal and $(f_1, \ldots, f_n) = A$, then $A$ is normal. This requires showing $A$ is integrally closed in its fraction field $K$. If $x \in K$ is integral over $A$, then $x$ is integral over each $A_{f_i}$, so $x \in A_{f_i}$ for all $i$. By gluing, $x \in A$.

Given any reduced scheme $X$, there exists a finite morphism $\nu: X' \to X$ where $X'$ is normal, called the normalization of $X$. For affine $X = \mathrm{Spec}(A)$, this is $X' = \mathrm{Spec}(\tilde{A})$ where $\tilde{A}$ is the integral closure of $A$ in its total ring of fractions.

The affine-locality of normality ensures that normalizations glue: if $X = \bigcup U_i$ with $U_i$ affine, then the normalizations $\nu_i: U_i' \to U_i$ glue to give $\nu: X' \to X$.

A scheme $X$ is Cohen-Macaulay if every local ring $\mathcal{O}_{X,x}$ is a Cohen-Macaulay ring (depth equals Krull dimension).

For affine schemes, this translates to: $\mathrm{Spec}(A)$ is Cohen-Macaulay $\iff$ $A_\mathfrak{p}$ is Cohen-Macaulay for all primes $\mathfrak{p}$.

The property "$A$ is Cohen-Macaulay" satisfies the Communication Lemma:

Localization: If $A$ is Cohen-Macaulay (meaning all localizations at primes are CM), then $A_f$ is Cohen-Macaulay, since $(A_f)_\mathfrak{p} = A_\mathfrak{q}$ for appropriate primes.

Cover: If each $A_{f_i}$ is Cohen-Macaulay, then $A$ is Cohen-Macaulay. This follows because every prime of $A$ is contained in the non-vanishing locus of some $f_i$, so we can check the CM property at that localization.

Therefore, Cohen-Macaulay is an affine-local property.

We have the following implications for properties of schemes: $\text{smooth} \implies \text{regular} \implies \text{Cohen-Macaulay} \implies \text{reduced}$

Each of these properties is affine-local (can be verified using the Communication Lemma). The regularity condition ($\mathcal{O}_{X,x}$ is a regular local ring) is particularly important in algebraic geometry.

Let $X$ be a scheme and $\mathcal{F}$ a sheaf of $\mathcal{O}_X$-modules. We say $\mathcal{F}$ is quasi-coherent if for every affine open $U = \mathrm{Spec}(A) \subseteq X$, we have $\mathcal{F}|_U \cong \widetilde{M}$ for some $A$-module $M$.

The key question: does this definition depend on the choice of affine cover?

Answer: No, by a sheaf-theoretic version of the Affine Communication Lemma. The point is that:

1. If $\mathcal{F}|_U \cong \widetilde{M}$ and $D(f) \subseteq U$, then $\mathcal{F}|_{D(f)} \cong \widetilde{M_f}$
2. If $U = \bigcup D(f_i)$ with $(f_i) = A$ and $\mathcal{F}|_{D(f_i)} \cong \widetilde{M_i}$ where the $M_i$ glue, then $\mathcal{F}|_U \cong \widetilde{M}$

This shows quasi-coherence is affine-local.

Many properties of quasi-coherent sheaves are also affine-local:

• $\mathcal{F}$ is coherent if it's quasi-coherent and on each affine open $\mathrm{Spec}(A)$, corresponds to a finitely generated $A$-module
• $\mathcal{F}$ is locally free of rank $n$ if on each point has a neighborhood where $\mathcal{F} \cong \mathcal{O}_X^n$
• $\mathcal{F}$ is flat over $Y$ if all stalks are flat modules

Each of these can be verified using Communication-Lemma-type arguments.

The Communication Lemma as stated works for arbitrary schemes because we can check properties on any affine cover, and affine schemes are quasi-compact.

However, when we want to work with modules and coherent sheaves, QCQS becomes crucial:

Gluing coherent sheaves: On a QCQS scheme, coherent sheaves can be glued from local data. On non-QCQS schemes, this can fail.

Descent: Many descent results (gluing morphisms, gluing objects) require QCQS to ensure that patching data on a cover extends uniquely to a global object.

For example, the theorem "quasi-coherent sheaves on $\mathrm{Spec}(A)$ correspond to $A$-modules" extends to: "quasi-coherent sheaves on a QCQS scheme $X$ form an abelian category with nice properties."

The affine line with doubled origin is a scheme that is not quasi-separated: $X = \mathrm{Spec}(k[t]) \sqcup_{\mathrm{Spec}(k[t,t^{-1}])} \mathrm{Spec}(k[t])$

This is the gluing of two copies of $\mathbb{A}^1$ along $\mathbb{A}^1 \setminus \{0\}$.

On this scheme:

• Quasi-coherent sheaves can have pathological behavior
• Some affine-local properties become ambiguous
• Descent fails in certain cases

This example shows why quasi-separatedness is a reasonable hypothesis for many theorems.

The property "X is an affine scheme" is NOT affine-local. For example, $\mathbb{P}^1$ can be covered by two affine opens $U_0 \cong \mathbb{A}^1$ and $U_1 \cong \mathbb{A}^1$, but $\mathbb{P}^1$ itself is not affine.

The issue is that being affine is a global property related to the algebra of global sections: $X \cong \mathrm{Spec}(\Gamma(X, \mathcal{O}_X))$ if and only if $X$ is affine. This cannot be detected by looking at an affine cover alone.

Topological properties like connectedness are generally not affine-local in the Communication Lemma sense, though they have their own local characterizations.

For instance, $X = \mathrm{Spec}(A_1 \times A_2)$ is disconnected (two connected components), but this can be covered by affine opens in various ways. The property depends on the global topology, not just local ring-theoretic properties.

Similarly, irreducibility is a global topological property: $\mathbb{A}^1 \setminus \{0\} = D(t) \cup D(1-t)$ is irreducible but covered by two principal opens.

A morphism $f: X \to Y$ being of finite type means it's locally of finite type and quasi-compact. The "locally of finite type" part is affine-local (on an affine cover, the coordinate rings are finitely generated algebras), but the quasi-compactness is a global condition.

Similarly, being proper, finite, or étale are properties of morphisms that involve both local (affine-local) conditions and global (topological) conditions.

A sheaf $\mathcal{F}$ on $X$ is globally generated if the global sections map surjectively to every stalk: $\Gamma(X, \mathcal{F}) \otimes k(x) \to \mathcal{F}_x$ is surjective for all $x \in X$.

This is not affine-local because it depends on the global sections $\Gamma(X, \mathcal{F})$, which is not determined by local data. For example, on $\mathbb{P}^n$, the sheaf $\mathcal{O}(d)$ is globally generated for $d \geq 0$ but not for $d < 0$, even though locally it's always isomorphic to $\mathcal{O}$.

The Affine Communication Lemma can be viewed as descent for the Zariski topology:

Given a Zariski open cover $X = \bigcup U_i$, we have a "covering family" of morphisms $\{U_i \to X\}$. The Communication Lemma says that a property descends along this cover if:

1. It pulls back along localizations (open immersions)
2. It descends from a cover

This is precisely the pattern of descent theory: ascent (pulling back) and descent (gluing from a cover).

Many important properties satisfy fpqc (faithfully flat and quasi-compact) descent:

• Reduced
• Normal
• Regular
• Cohen-Macaulay
• Locally Noetherian

The Communication Lemma handles the special case of descent for Zariski covers (where the covering maps are open immersions), which is sufficient for proving affine-locality.

Not all properties satisfy descent:

Affineness: As noted, being affine does not satisfy Zariski descent (though it does satisfy fpqc descent!).

Geometric connectivity: Connectedness does not satisfy descent in general, though there are refined versions that do.

Ampleness: For a line bundle $\mathcal{L}$ on $X$, being ample is not Zariski-local. The sections of $\mathcal{L}^{\otimes n}$ for large $n$ must separate points and give a projective embedding globally.

Let's verify that "having finite Krull dimension" is affine-local:

Affine version: $\mathrm{Spec}(A)$ has finite Krull dimension if $\dim A < \infty$.

Localization: If $\dim A < \infty$, then $\dim A_f \leq \dim A < \infty$. (Every prime of $A_f$ contracts to a prime of $A$.)

Cover: If $(f_1, \ldots, f_n) = A$ and $\dim A_{f_i} < \infty$ for all $i$, then $\dim A < \infty$.

To see this, let $\mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_r$ be a chain of primes in $A$. Some $f_i \notin \mathfrak{p}_0$ (since $(f_1, \ldots, f_n) = A$). Then all primes in the chain localize to primes in $A_{f_i}$, giving a chain of length $r$ in $A_{f_i}$. So $r \leq \dim A_{f_i} < \infty$.

Therefore, having finite Krull dimension is affine-local.