Separatedness, properness, and valuative criteria
Use diagonals and valuation-ring extension problems to distinguish unique limits, existing limits, and proper algebraic families.
On this page
The question for today
What you’ll understand
- Define separatedness and properness as properties of a morphism.
- State an all-valuation-ring properness criterion with its finiteness hypotheses.
- Compute a projective limit over a discrete valuation ring and distinguish existence from uniqueness.
Local affine calculations do not guarantee that limits fit together globally. Separatedness prevents two different extensions of one generic family, while properness also ensures existence of extensions under the appropriate hypotheses. Both notions concern a morphism to a specified base. They generalize useful features of familiar compact geometric objects, but their definitions are algebraic and stable under base change. A valuation ring is a particularly effective test object because its fraction field supplies generic data and its closed behavior tests whether that data can acquire a compatible limit.
The diagonal tests whether two descriptions can separate
A morphism f:X→S is separated when its diagonal X→X×_S X is a closed immersion. The diagonal represents pairs of equal maps over the base. A closed diagonal implies that equality cannot disappear under specialization in the way it does for a doubled origin. Quasi-separatedness is weaker: the diagonal is quasi-compact. For schemes the diagonal is automatically an immersion, so having closed image is an equivalent test in this particular case. The structural definition remains a closed immersion, while quasi-compactness alone gives only quasi-separatedness.
The line with a doubled origin exhibits the failure concretely. A map from the spectrum of a discrete valuation ring can follow the same nonzero coordinate on the generic point and choose either origin on the closed point. The two maps agree generically but not globally. Ordinary affine schemes are separated over an affine base, so separatedness must be checked across the overlaps in this example rather than inferred from the fact that every point has a separated affine neighborhood.
A precise valuative statement includes all valuation rings
Let f:X→Y be of finite type and quasi-separated. Then f is proper if and only if every commutative square consisting of Spec K→X and Spec R→Y has a unique compatible lift Spec R→X, for every valuation ring R with fraction field K. The maps from Spec K to Spec R and from X to Y are fixed in the square. Uniqueness reflects separatedness; existence supplies the missing limit. This is a theorem with hypotheses, not a definition that can be applied to an arbitrary infinite-dimensional morphism.
Discrete valuation rings provide especially computable examples, but the general statement tests all valuation rings, including those that are not discrete or Noetherian. Versions that only require discrete valuation rings need extra assumptions on the base and morphism; we do not silently substitute that narrower test here. Likewise, the existence part and uniqueness part have their own precise valuative formulations. The combined finite-type, quasi-separated criterion stated above is the version used throughout this lesson.
Projective coordinates can retain a limit outside one affine chart
Over a discrete valuation ring, a point of projective space over the fraction field can be represented by coordinates in the valuation ring with at least one coordinate a unit. Multiply a homogeneous coordinate tuple by a suitable power of a uniformizer so its minimum valuation is zero. Reducing the normalized tuple then gives the special point. An affine coordinate may have a pole even though the projective point extends perfectly well into another chart.
Projective space is proper over its base, and a closed subscheme of projective space is also proper over that base. Therefore the normalization calculation illustrates a general extension theorem; it is not the proof of every properness assertion for arbitrary schemes. Proper need not mean projective, and a proper family need not have smooth fibers or a constant number of underlying points. Those additional geometric questions require their own hypotheses and invariants.
Work it through
Follow a pole into the projective chart at infinity
Let R=k[[t]] and K=k((t)). Consider the K-point x=1/t of A¹_k. Does it extend to an R-point of A¹_k? Find its extension in P¹_k and its special point.
An R-point of A¹_k is a k-algebra homomorphism k[x]→R. To extend the given K-point it would have to send x to 1/t, which is not in R. Thus the affine extension does not exist.
Embed the affine chart by x↦[x:1]. The generic projective point [1/t:1] equals [1:t], a tuple in R² with a unit first coordinate. It defines an R-point of P¹, using the chart where the first coordinate is nonzero.
Reducing modulo t gives [1:0], the point outside the original affine chart. The extension is unique because P¹ is separated; its existence agrees with properness.
The pole obstructs an affine extension but not a projective one. A missing chart point and a nonunique limit are different failures.
The living notebook
Explore the model
Interactive lab · vectors
See how a coordinate ratio can leave one chart
Pause and Reset; set Angle=45 degrees,Magnitude=1 and predict equal nonzero components, so the ratio x/y is 1. For a fresh run Reset and set Angle=0 degrees,Magnitude=1; predict (1,0), where that affine ratio is undefined but the nonzero homogeneous pair [1:0] remains meaningful. The instrument draws real vectors only, not valuation rings, scheme families, or a properness test. It illustrates the chart change used in the worked extension.
- x component
- 1.393
- y component
- 0.975
- Magnitude
- 1.7
Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forA homogeneous point can remain well-defined while a particular affine coordinate develops a pole.
Pause & reason
Check your understanding
Your turn
Try it yourself
Over K=k((t)), normalize the P²-point [t^(-3):t^(-1):1] to coordinates over R=k[[t]] with a unit entry. Compute its special point and explain why multiplying every normalized coordinate by t would spoil the chosen representation over R.
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Subtract the minimum of the three coordinate valuations, which are −3,−1,0.
Compare with the solution
Multiplication by t³ gives [1:t²:t³], whose reduction is [1:0:0]. The first coordinate is a unit, so the tuple defines an R-point. Multiplying it by t gives (t,t³,t⁴), whose entries all lie in the maximal ideal and do not generate the unit ideal. That tuple represents the same K-point but is not an admissible global homogeneous coordinate tuple for this R-point until the common nonunit factor is removed.
Keep the idea
Take it with you
- Separatedness is a closed-diagonal condition on a specified morphism.
- The general properness criterion requires all valuation rings and explicit finiteness assumptions.
- Projective extension can repair a pole without asserting anything about smoothness or fiber counts.
Read further
Original sources and deeper explanations to follow at your own pace.
Optional progress, saved in this browser when available. Revisit any lesson at any time.
Lesson not marked complete.