Purity (algebraic geometry)
Okay kiddo, let me try to explain what "purity" means in algebraic geometry!
Purity is a way to measure how "nice" a mathematical object is. In algebraic geometry, we often work with something called a "scheme" which is like a geometric object made up of points and lines and curves, but it's described using algebraic equations instead of measurements.
Now, some schemes are easier to work with than others. Some are very complicated and messy, while others are simpler and more organized. And purity is a way to tell which schemes are the "good" ones and which ones are the "bad" ones.
Basically, if a scheme is "pure," it means that it's nice and organized and easy to work with. It has some special properties that make it easier to understand and manipulate. For example, if we know that a scheme is pure, we can use certain techniques to study it and learn more about it.
On the other hand, if a scheme is "impure," it means that it's messier and more complicated. It might not have those nice properties that make it easy to work with. And that can make it harder to study and understand.
So when we talk about "purity" in algebraic geometry, we're really just talking about how nice and organized a scheme is. If it's pure, it's good. And if it's impure, it's bad. Simple as that!
Purity is a way to measure how "nice" a mathematical object is. In algebraic geometry, we often work with something called a "scheme" which is like a geometric object made up of points and lines and curves, but it's described using algebraic equations instead of measurements.
Now, some schemes are easier to work with than others. Some are very complicated and messy, while others are simpler and more organized. And purity is a way to tell which schemes are the "good" ones and which ones are the "bad" ones.
Basically, if a scheme is "pure," it means that it's nice and organized and easy to work with. It has some special properties that make it easier to understand and manipulate. For example, if we know that a scheme is pure, we can use certain techniques to study it and learn more about it.
On the other hand, if a scheme is "impure," it means that it's messier and more complicated. It might not have those nice properties that make it easy to work with. And that can make it harder to study and understand.
So when we talk about "purity" in algebraic geometry, we're really just talking about how nice and organized a scheme is. If it's pure, it's good. And if it's impure, it's bad. Simple as that!