Properties of Schemes
Most important properties of schemes are local in nature, i.e. a scheme X has a certain property P if and only if for any cover of X by open subschemes Xi, i.e. X= Xi, every Xi has the property P. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is Zariski-local, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology.
Consider a scheme X and a cover by affine open subschemes Spec Ai. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings Ai. A property P is local in the above sense, iff the corresponding property of rings is stable under localization.
For example, we can speak of locally Noetherian schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a Noetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations.
An example for a non-local property is separatedness (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme.
The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let X = Spec Ai be a covering of a scheme by open affine subschemes. For definiteness, let k denote a field in the following. Most of the examples also work with the integers Z as a base, though, or even more general bases.
| notion | definition | example | non-example |
|---|---|---|---|
| related to scheme structure | |||
| connected | The scheme is connected as a topological space. Since the connected components refine the irreducible components any irreducible scheme is connected but not vice versa. An affine scheme Spec(R) is connected iff the ring R possesses no idempotents other than 0 and 1; such a ring is also called a connected ring. | affine space, projective space | Spec(k×k) |
| irreducible | A scheme X is said to be irreducible when (as a topological space) it is not the union of two closed subsets except if one is equal to X. Using the correspondence of prime ideals and points in an affine scheme, this means X is irreducible iff X is connected and the rings Ai all have exactly one minimal prime ideal. (Rings possessing exactly one minimal prime ideal are therefore also called irreducible.) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. | affine space, projective space | Spec k/(xy) = |
| reduced | The Ai are reduced rings. Equivalently, none of its rings of sections (U any open subset of X) has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations from varieties to schemes. | varieties (by definition) | Spec k/(x2) |
| integral | A scheme that is both reduced and irreducible is called integral. For locally Noetherian schemes, to be integral is equivalent to being a connected scheme that is covered by the spectra of integral domains. (Strictly speaking, this is not a local property, because the disjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property.) | Spec k/f, f irreducible polynomial | Spec A×B. (A, B ≠ 0) |
| normal | An integral scheme is called normal, if the Ai are integrally closed domains. | regular schemes | singular curves |
| related to regularity | |||
| regular | The Ai are regular. | smooth varieties over a field | Spec k/(x2+x3-y2)= |
| Cohen-Macaulay | All local rings are Cohen-Macaulay. | regular schemes, Spec k/(xy) | |
| related to "size" | |||
| locally noetherian | The Ai are Noetherian rings. If in addition a finite number of such affine spectra covers X, the scheme is called noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. | (Virtually everything in algebraic geometry). | |
| dimension | The dimension, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. It can be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See also Global dimension. | equidimensional schemes in dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces. | |
| catenary | A scheme is catenary, if all chains between two irreducible closed subschemes have the same length. | (Virtually everything, e.g. varieties over a field) | |
Read more about this topic: Glossary Of Scheme Theory
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)
“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)