Understanding Prevarieties and Affine Varieties in Projective Space

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In summary, the book "Linear Algebraic Groups" by Humphreys defines a prevariety X in projective space P^n as a noetherian topological space with a sheaf of functions, composed of finitely many open subsets that are isomorphic to affine varieties. This may be confusing because affine varieties, which are closed sets in the Zariski topology, are being mapped isomorphically onto open subsets in P^n. This is possible due to the relative properties of openness and closedness, as shown by examples such as the projection of a hyperbola onto the x-axis. The distinction between a prevariety and a variety is usually the lack of a separation axiom, which is always satisfied for objects in
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dg0666
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The book I am reading, Linear Algebraic Groups by Humphreys defines a prevariety X in projective space P^n to be a noetherian topological space endowed with a sheaf of functions such that X is the union of finitely many open subsets, each isomorphic to an affine variety.

This confuses me because I do not understand how affine varieties, which are closed sets in the Zariski topology, can be isomorphic to open sets in P^n.

Please help!

Daniel
 
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  • #2
Ooh, is it the closed vs open bit that's confusing you?

Don't forget that openness and closedness are properties of a subspace relative to the whole, and not properties of the subspace itself.

For example, (0, 1) is a closed subset of (0, 1), despite the fact it's an open subset of [0, 1].
 
  • #3
In fact I understand that distinction. What is confusing me is that the affine varieties are closed in one topology and they are mapped isomorphically onto open subsets in P^n. It is just this bijection between open and closed subsets which is confusing me.

Should I be confused by this or is this just something counter-intuitive? I always thought that a continuous map sends closed sets to closed sets. Do you have to hand a simple elementary example to clarify how this situation is possible.

Dan
 
  • #4
A continuous map reflects closed sets -- that is if f(T) is closed, then T is closed.

The other way is not guaranteed. For example, consider the map RR:x→arctan x. This is an isomorphism of a closed set onto an open set.

Another popular example is to take the curve in R² given by xy = 1, and project it onto the x-axis, giving an isomorphism of a closed subset of R² onto an open subset of R.

But, continuous maps do preserve compact sets. One of the oddities of Zariski topologies is that compact sets are not necessarily closed.

For example, C - {0} is an open, compact subset of C with the Zariski topology.


Incidentally, the embedding (0, 1)→[0, 1] is representative of what's going on here. If we identify the endpoints of [0, 1], then it is exactly analogous to embedding the real line into the projective real line.
 
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  • #5
Great. I think that has answered my question very well. My confusion arosed because of a misunderstanding between isomorphisms and continuous maps but thanks very much for your help.

Dan
 
  • #6
It strikes me that I should be more explicit about which topology I was using, and in which category I was claiming isomorphism in my examples. (Though I presume you figured out my meaning, I still feel the need to elaborate)

x→arctan x was using the Euclidean topology, since it's clearly not a continuous map under the Zariski topology. It is an isomorphism onto its image, that is an isomorphism of R and (-π/2, π/2), as a map of topological spaces. (i.e. a homeomorphism)

The second example, the projection of the hyperbola onto the punctured x-axis is continuous in both the Zariski and the Euclidean topologies. The domain is closed in both topologies, and the image is open in both topologies. This one is actually a regular map of quasiaffine varieties, and thus an isomorphism in that category, as well as being a homeomorphism of topological spaces.
 
  • #7
I am puzzled about your saying that the prevariety is "in projective space" as then you give a completely abstract definition.the distinction between a "pre"variety and a variety, is usually the lack of a separation axiom, which is always satisfied for such things that occur in projective space. are you sure you read him correctly?
 
  • #8
I am puzzled about your saying that the prevariety is "in projective space" as then you give a completely abstract definition.


the distinction between a "pre"variety and a variety, is usually the lack of a separation axiom, which is always satisfied for such things that occur in projective space. are you sure you read him correctly?
 

FAQ: Understanding Prevarieties and Affine Varieties in Projective Space

What is a prevariety?

A prevariety is a mathematical concept that is used in algebraic geometry to study algebraic varieties. It is a generalization of the notion of a variety, allowing for more flexibility in defining and studying geometric objects.

How is a prevariety different from a variety?

A prevariety is more general than a variety in that it allows for objects that may not satisfy all the defining properties of a variety. For example, a prevariety may not be irreducible or reduced, while a variety must be both. However, every variety is also a prevariety.

What are the defining properties of a prevariety?

A prevariety must satisfy the following properties: it must be a locally ringed space, meaning it has a sheaf of rings associated to it; it must be locally irreducible, meaning every open subset contains an irreducible open subset; and it must be locally reduced, meaning every open subset contains a reduced open subset.

How are prevarieties used in mathematics?

Prevarieties are used in algebraic geometry to study algebraic varieties. They provide a more flexible framework for defining and studying geometric objects, and allow for a wider range of objects to be studied. They also play an important role in the theory of schemes, a generalization of varieties.

What are some examples of prevarieties?

Some examples of prevarieties include affine and projective spaces, Grassmannians, and flag varieties. These are all examples of algebraic varieties, but they can also be studied as prevarieties, allowing for more generalizations and flexibility in their definition and properties.

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