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Chapter 1: Problem 20

Let \(Y\) be a variety of dimension \(\geqslant 2,\) and let \(P \in Y\) be a normal point. Let \(f\) be a regular function on \(Y-P\) (a) Show that \(f\) extends to a regular function on \(Y\) (b) Show this would be false for dim \(Y=1\) See (III. Ex. 3.5) for generalization.

Short Answer

Expert verified
(a) Because \(P\) is a normal point in the variety \(Y\), any function that is regular on \(Y - P\) can be extended over \(P\) as \(P\) belongs to a regular local ring. Therefore, the function \(f\) extends to a regular function on \(Y\). (b) This would not hold if the dimension of \(Y\) is 1, as shown by the counterexample where a function defined on \(Y - P\) is not extendable to \(Y\).

Step by step solution

01

Recall the definitions

In algebraic geometry, a variety is a separated scheme of finite type over a field. We are dealing with regular (or rational) functions, which are analogues of polynomial functions. A regular function on a variety \(Y\) is a function that is locally a quotient of polynomials. Also recall that a normal point in an algebraic variety is a point that locally corresponds to a regular local ring. A regular local ring is an integral domain whose local dimension equals its Krull dimension.
02

Extension of f to Y

Now, we need to show that the function \(f\) can be extended from \(Y - P\) to \(Y\). We need to note that \(f\) is already a regular function on \(Y - P\). So at all points except possibly \(P\), \(f\) is defined and is a regular function. Since \(P\) is a normal point, the local ring at \(P\) should be a regular local ring. Now, any function that is regular on the punctured neighborhood of \(P\) can be extended over \(P\). The process of extension is a basic property of regular local rings. This is essentially due to the fact that a regular local ring is integrally closed in its fraction field. So, the function \(f\) extends to a regular function on \(Y\).
03

Counterexample for dim Y=1

For part (b), we can construct a counterexample when the dimension of \(Y\) is 1. Consider \(Y = \mathbb{P}^1\), which is the projective line over our field, and let \(P\) be any point on this line. There are regular functions on \(Y - P\) (which is isomorphic to the affine line), that is not extendable to \(Y\). For instance, let \(f\) be the function \(1/x\) on \(Y - P\). Although \(1/x\) is a regular function on \(Y - P\), it is not extendable to \(Y\) because \(f(P)\) is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variety Dimension
Understanding the dimension of an algebraic variety is crucial when studying properties and behavior of functions within algebraic geometry. In simple terms, the dimension of a variety refers to the number of independent parameters needed to describe a point within it. Imagine it like the number of coordinates you'd need to pinpoint a location on a map, where a two-dimensional map requires two numbers (like latitude and longitude).

In our context, let's say we have a variety that has a dimension greater than or equal to 2. This effectively means that the variety is at least 'surface-like', with at least two independent directions. So, algebraic curves (like lines or circles) are one-dimensional, whereas surfaces (like planes or spheres) are two-dimensional. The concept of dimension helps in concluding the extension potential of functions in such varieties, which is a focal point of the exercise we are looking at.
Regular Function
Algebraic geometry often deals with functions that are more general than our usual polynomials. A regular function on an algebraic variety can be locally expressed as a ratio of two polynomials. The key word here is 'locally'. In essence, if you zoom into any point on the variety (except a set of points with specially defined properties), the function will look like a ratio of polynomials at that point.

For example, consider the function defined by \( f(x,y) = \frac{x^2}{y} \) on a variety except at points where \( y = 0 \). This function is not polynomial, but locally around any point where \( y eq 0 \), it behaves just like a fraction of two polynomials 鈥� which makes it a regular function in those regions. This concept is essential in understanding when and how functions can be extended across varieties.
Normal Point
A 'normal point' on a variety holds a special place in algebraic geometry. It signifies a point around which the variety behaves nicely, in a certain algebraic sense. Specifically, at a normal point, you can consider the local ring of functions - which is an algebraic structure encapsulating functions defined near that point - and it turns out to be a 'regular local ring'. This is significant because regular local rings have desirable properties that make dealing with functions around normal points more manageable.

One of these properties is that they are integrally closed, meaning that if a function behaves like an integer within this ring (is integral), it's actually part of the ring. This property is the backbone for extending functions across a variety, as it ensures that no 'gaps' or irregularities arise at these normal points.
Regular Local Ring
A regular local ring is a sturdy building block in the architecture of algebraic geometry. It's a type of algebraic structure that represents the local behavior of functions around a point on a variety, much like examining the soil composition in a handful of earth from your backyard to infer about the entire yard's makeup.

In our exercise, the normal point's local ring is a regular local ring, which implies certain high-quality attributes. One such attribute is that it matches its Krull dimension - think of this as the ring having no 'excess dimensions'. Another is being integrally closed, as mentioned before. These attributes enable us to 'fill in' the function at the missing point \( P \) by a process akin to completing a jigsaw puzzle; the surrounding pieces fit in so nicely that only one piece can complete the picture 鈥� and because of the integral-closedness, that piece is guaranteed to exist within the ring.

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Most popular questions from this chapter

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