Proving The Alternative Definition Of Rational Equivalence In Fulton's Intersection Theory
Introduction
Alright, guys, let's dive into a tricky concept in algebraic geometry: rational equivalence. This is a cornerstone of intersection theory, and understanding it is crucial for anyone delving into this fascinating field. We're going to tackle an alternative definition of rational equivalence, specifically as presented in Fulton's "Intersection Theory." This stuff can get pretty dense, so we'll break it down step by step to make it as clear as possible. Our focus will be on unraveling the proof of Proposition 1.6, which lays out this alternative definition. So, buckle up, and let's get started!
Setting the Stage: The Foundation of Rational Equivalence
Before we jump into the nitty-gritty details, let's set the stage. We're working within the realm of schemes of finite type over a field K. Think of these schemes as geometric objects defined by polynomial equations. We're interested in understanding how these objects intersect, and that's where intersection theory comes in. A key player in this theory is the concept of rational equivalence. Rational equivalence helps us define when two cycles (formal sums of subvarieties) are considered "equivalent" from an intersection-theoretic point of view. This equivalence relation allows us to create a well-behaved intersection theory. At its heart, rational equivalence is about understanding how cycles can be deformed within a variety without changing their fundamental intersection properties.
Now, let's talk about the specific setup we'll be using. We have a scheme X of finite type over a field K. We'll also be dealing with the projection map p from X to another scheme. This projection essentially "projects" the geometry of X onto a simpler space. Understanding how cycles behave under this projection is crucial for our alternative definition. We'll be focusing on how rational functions on subvarieties of X induce rational equivalences. These rational functions, which are basically fractions of polynomials, play a vital role in deforming cycles. The core idea is that the "divisors" of these rational functions (the places where they vanish or have poles) define cycles that are rationally equivalent to zero. These divisors are formal sums of subvarieties, each with a coefficient that reflects the multiplicity of the zero or pole.
Understanding the relationship between rational functions and rational equivalence is key. When a rational function vanishes along a subvariety, it means that the subvariety is part of the "zero locus" of the function. Similarly, poles indicate where the function "blows up." The formal difference between the zeros and poles, weighted by their multiplicities, gives us the divisor. This divisor captures how the rational function deforms cycles. The beauty of rational equivalence lies in its ability to capture the essential geometric information about intersections while allowing for flexibility in how cycles are represented. This flexibility is crucial for making computations and proving theorems in intersection theory. We will see how this alternative definition elegantly captures this concept, providing a powerful tool for studying intersections.
Delving into Proposition 1.6: The Alternative Definition
Proposition 1.6 in Fulton's "Intersection Theory" provides an alternative way to define rational equivalence, and it's this definition we're going to dissect. This proposition essentially states that rational equivalence can be understood in terms of the pushforward of divisors of rational functions. Let's break that down. We have a subvariety V of X, and on V, we have a rational function f. The divisor of f, denoted as div(f), is a cycle on V that captures the zeros and poles of f, as we discussed earlier. Now, we consider the pushforward of this divisor under the inclusion map from V into X. The pushforward is an operation that essentially extends the cycle on V to a cycle on X. The pushforward takes into account how the subvariety V sits inside the larger scheme X. It's a way of translating the information contained in the divisor from the smaller subvariety to the larger space.
The proposition then says that cycles that are rationally equivalent to zero can be expressed as sums of such pushforwards of divisors. In other words, a cycle is rationally equivalent to zero if it can be obtained by considering rational functions on subvarieties, taking their divisors, and then pushing these divisors forward to the ambient scheme. This definition provides a powerful tool for understanding and manipulating rational equivalence. It connects the abstract notion of equivalence to the more concrete world of rational functions and their divisors. It also highlights the importance of subvarieties and their role in defining rational equivalence. Understanding the proposition's statement is the first step; now, we need to delve into the proof to see why it holds true. The proof involves carefully constructing cycles and demonstrating how they relate to rational functions and divisors.
The beauty of this alternative definition is that it often makes it easier to work with rational equivalence in practice. Instead of directly manipulating cycles, we can work with rational functions and their divisors, which are often simpler to handle. This is especially useful when dealing with complex schemes and intersections. The proposition also provides a bridge between algebraic and geometric perspectives. The rational functions and divisors are algebraic objects, while the cycles and their equivalence are geometric. This bridge allows us to use algebraic tools to study geometric problems and vice versa. By expressing rational equivalence in terms of these simpler objects, we gain a powerful handle on the concept. This alternative definition not only provides computational advantages but also deepens our understanding of the geometric meaning of rational equivalence.
Unpacking the Proof: A Step-by-Step Guide
Okay, let's get our hands dirty and dive into the proof of Proposition 1.6. This is where things can get a bit technical, so we'll take it slow and break it down into manageable chunks. Remember, the goal is to show that the alternative definition of rational equivalence, using pushforwards of divisors, is indeed equivalent to the standard definition. The standard definition, which we haven't explicitly stated yet, involves considering cycles defined by rational functions on subvarieties of X × P1, where P1 is the projective line. We need to show that these two definitions are equivalent, which means that a cycle is rationally equivalent to zero in one definition if and only if it's rationally equivalent to zero in the other.
The proof typically starts by showing that cycles that are rationally equivalent to zero in the standard sense can be expressed as sums of pushforwards of divisors. This direction is often the trickier one. The idea is to take a cycle that is rationally equivalent to zero in the standard sense, which means it's the difference of the cycle at 0 and the cycle at ∞ of a subvariety W of X × P1, and then relate it to divisors of rational functions. This involves carefully choosing a rational function on W and relating its divisor to the cycle. We need to show that the cycle can be written as a sum of terms of the form p(div(f)), where p is the inclusion map and div(f) is the divisor of a rational function. This often involves using properties of rational functions, divisors, and pushforwards to manipulate the cycle and express it in the desired form.
Conversely, the proof also needs to show that sums of pushforwards of divisors are rationally equivalent to zero in the standard sense. This direction is usually more straightforward. If we have a divisor div(f) on a subvariety V of X, we need to show that p(div(f))* is rationally equivalent to zero in the standard sense. This typically involves constructing a subvariety of X × P1 and a rational function on it such that the cycle at 0 minus the cycle at ∞ is equal to p(div(f))*. This construction often relies on geometric intuition and careful manipulation of the rational function and the subvariety. By proving both directions, the proof establishes the equivalence of the two definitions. This equivalence is a powerful result, as it provides us with two different ways to think about rational equivalence, each with its own advantages.
Key Techniques and Concepts in the Proof
To really nail this proof, there are some key techniques and concepts you'll need to have in your toolkit. First and foremost, you need a solid understanding of divisors. As we've mentioned, divisors are formal sums of subvarieties that capture the zeros and poles of rational functions. Being comfortable with how divisors are constructed and manipulated is essential. You should know how to compute the divisor of a given rational function and how divisors behave under various operations.
Next, you'll want to be familiar with the concept of pushforwards. The pushforward is a way of translating cycles from a subvariety to the ambient variety. It's a fundamental operation in intersection theory, and understanding how it works is crucial for this proof. You should know how to compute the pushforward of a cycle and how it interacts with other operations, such as taking divisors. The pushforward is a key tool for relating divisors on subvarieties to cycles on the larger scheme.
Another important concept is the projective line, P1. The projective line is a fundamental object in algebraic geometry, and it plays a central role in the standard definition of rational equivalence. You should be familiar with its properties and how cycles on X × P1 are used to define rational equivalence. The projective line provides the deformation space for cycles, allowing us to smoothly move them from one position to another.
Finally, a good grasp of rational functions and their properties is essential. You should know how to construct rational functions, how to compute their divisors, and how they behave under various operations. Rational functions are the workhorses of this proof, and understanding them is key to unraveling the argument. Rational functions provide the algebraic tools for deforming cycles and defining rational equivalence. By mastering these techniques and concepts, you'll be well-equipped to tackle the proof of Proposition 1.6 and gain a deeper understanding of rational equivalence.
Implications and Applications of the Alternative Definition
So, we've dissected the proof, but what's the big deal? Why is this alternative definition of rational equivalence so important? Well, it turns out this definition has some significant implications and applications in intersection theory and beyond. One of the main advantages is that it often simplifies computations. Working directly with divisors of rational functions can be much easier than working with cycles defined on products with the projective line. This simplification is especially valuable when dealing with complex schemes and intersections. The alternative definition provides a more direct and efficient way to calculate intersection products.
Furthermore, this definition provides a more geometric intuition for rational equivalence. It connects the abstract notion of equivalence to the concrete world of rational functions and their zeros and poles. This connection can make it easier to visualize and understand what rational equivalence means geometrically. The divisors of rational functions provide a tangible representation of how cycles are deformed, making the concept more accessible.
This alternative definition is also crucial for extending intersection theory to more general settings. It allows us to define rational equivalence and intersection products in situations where the standard definition might be difficult to apply. This is particularly important when dealing with singular varieties or other non-smooth situations. The flexibility of the alternative definition makes it a valuable tool for pushing the boundaries of intersection theory.
Beyond intersection theory, the concepts and techniques used in this proof have applications in other areas of algebraic geometry and even in number theory. The interplay between rational functions, divisors, and cycles is a fundamental theme in many areas of mathematics. Understanding this interplay can provide insights into a wide range of problems. For instance, the ideas behind rational equivalence are closely related to concepts in algebraic K-theory, which is a powerful tool for studying the structure of algebraic varieties. By mastering the alternative definition of rational equivalence, you're not just learning about intersection theory; you're also gaining valuable tools and insights that can be applied to other areas of mathematics.
Conclusion: Mastering Rational Equivalence
Alright, we've reached the end of our journey into the alternative definition of rational equivalence. We've explored the setting, dissected Proposition 1.6, unpacked its proof, and discussed its implications and applications. Hopefully, you now have a much clearer understanding of this crucial concept in intersection theory. Remember, rational equivalence is a cornerstone of intersection theory, and mastering it opens the door to a deeper understanding of how geometric objects intersect. It's a challenging concept, but with persistence and a solid grasp of the key ideas, you can conquer it.
This alternative definition, as presented in Fulton's "Intersection Theory," provides a powerful tool for working with rational equivalence. It connects the abstract notion of equivalence to the concrete world of rational functions and their divisors. This connection not only simplifies computations but also provides valuable geometric intuition. So, keep practicing, keep exploring, and keep pushing your understanding of algebraic geometry. You've got this!