(Perrin AG) Chapter 2 Highlights: What's the difference in the projective case?

Though we cannot evaluate polynomials \(F\) at points of \(\mathbb{P}^n\), it makes sense to talk about the zeroes of \(F\). Observe that for \(|k| = \infty \), if \(F = \sum F_i \) is a decomposition of \(F\) into homogeneous polynomials of degree \(i\) then for all \( \underline{\alpha} \in \mathbb{P}^n\) we have
$$F(\underline{\alpha}) = 0 \iff F_i (\underline{\alpha}) = 0 \forall i.$$ 

Proof. We must have \(F(\lambda \underline{\alpha}) = 0 \forall \lambda \in k\). Now then RHS, viewed as a polynomial in \(\lambda\) has infinitely many roots. 

This shows in particular, that the following two definitions of a homogeneous ideal \(I\) of \(k[x_1, \ldots, x_n ]\) are equivalent:

• \(I\) is generated by homogeneous polynomials
• if \(F \in I\) then so are all of its homogeneous components \(F_i\).

As in the affine case, we still have a correspondence between nonempty projective algebraic sets in \(\mathbb{P}^n\) with homogeneous, radical, relevant ideals of \(k[x_1, \ldots, x_n ]\). Under this correspondence, irreducible projective subsets still correspond to prime ideals; however, points no longer correspond to max ideals. 

In the case of a projective \(X \subset \mathbb{P}^n\), we again have a correspondence between its projective algebraic subsets and the ideals of its homogeneous coordinate rings. However, unlike the affine case, this homogeneous coordinate ring is dependent on the specific embedding \( X \hookrightarrow \mathbb{P}^n\) and even for the same embedding, there might be several graded rings structure associated to it. Furthermore, again, unlike the affine case, its elements aren't functions on \(X\). 

(Perrin) Algebraic Geometry. Chapter 1 Highlights: The Algebra-Geometry Dictionary

Some applications of Ideal-Variety Correspondence

The Nullstellensatz establishes an algebra-geometry dictionary. This allows us to prove geometric statement by passing to algebra. For example, can you show this: 

$$\text{If \(k\) is an infinite field, then \(k^n\) is irreducible.}$$

Proof. Via the algebra-geometry dictionary, it suffices to show \( \mathbb{I}(k^n) = 0\). 

Remark: Why did we need the "infinite" condition? Consider \( k = \mathbb{F}_q \) then \(\mathbb{I}(k) = (x^q - x) \neq 0 \).

Conversely, we could use geometry to prove algebraic identity. The following useful little lemma (Extension of Algebraic Identity) is a consequence of the above topological fact that \(k^n\) is irreducible. 

Assume \( |k| = \infty\). Let \( X \)  be a proper affine algebraic subvariety of \(k^n\). Let \( f \in k[x_1, \ldots, x_n]\)  such that \( f = 0\) outside \(X\). Then \( f \equiv 0\).

 Proof. Note that \(f = 0\) on a non-empty open set \(U := k^n - X\) so \(\mathbb{V}(f)\) must contains the closure of \(U\), which is \(k^n\) by irreducibility of \(k^n\). 

The most common application of this is in "Density Arguments". To prove that an identity holds for all matrices for example, it suffices to prove it for invertible ones, as they form a dense open subset of the variety of \(n \times n \) matrices. 

(Yuval Harari) Sapiens. Part I: The Cognitive Revolution, Chapter 1. An Animal of no Significance

 

Yuval Harari's Sapiens tell the story of the three important revolutions that shaped history: the Cognitive Revolution, the Agricutural Revolution and the Scientific Revolution. 

We start the tale with Part I. The Cognitive Revolution. 

Chapter 1. An animal of no Significance.

Our Forgotten Siblings: the other Human Species.

Did you know that there are multiple human species? Well, what counts as a species? Are cats and lions, for example, in the same species? They would be, if they could mate and reproduce fertile offsprings. 

From 2 million to about 10,000 years ago, the world was home to multiple human species. Wait, what makes a species a "human" species? Species that evolved from a common ancestors are grouped into genera. We, Homo sapiens, is the sapiens (wise) species of the genus Homo. By "other human species", we mean other species in the same genus. 

Common Traits among Human Species. 

All human species share several traits. First, we have extraordinarily large brains compared to other animals. These large brains require a lot of energy. Consequently, archaic humans had to spend more time searching for food and at the same time, their muscle atrophy to divert energy to neurons. 

Second, we walk upright on two legs. This has two important benefit:

• The hands are freed for other purposes. They then evolved to handle more intricate tasks, including producing and using sophisticated tools. 
• Childbirth becomes more difficult: upright gait required narrower hip, while babies heads were getting bigger. 

Because of the the difficulty of childbirth and childcare, as well as due to their weaker physique compared with other animals, humans formed social ties and structures. In addition, unlike say elephants who were born fully developed, human infants are moldable and can thus easily adapt to socialize in any communities. 

The Leap towards the Top

Despite the large brain, the sophisticated tools and the strong bonds, for millions of years, humans were still just in the middle of the food chain. Their leap towards the top is due to the invention of fire. Fire kills germs and makes food more digestible. Unlike chimpanzees which spend five hours a day chewing raw food, with cooking, human had to devote less time and energy to eating, thus shortening their intestinal track.  This in turn allows more energy to be directed towards the brain instead of the digestive system. 

The taming of this natural resource allows human to gain power that, unlike other animals, is not restricted to their physical body.  A single child could burn down the whole forest. This explains why suddenly men jump to the top of the food chain. However, this quick evolution did not give enough time for the ecosystem to develop checks and balances to prevent us from wreaking havocs. 

So What Happened to Our Siblings?

Scholar claims that from 70,000 years ago, Sapiens from East Africa spread to Euroasia, which was then already occupied by other humans. What happened to those humans? There are two theories:

 The Interbreeding Theory claims that Sapiens bred with Neanderthals (so none of us are pure Sapiens). The Replacement Theory claims, on the other hand, that the genetic makeup of these two populations make it impossible for them too mate. Rather, the Neanderthals were wiped up because of conflicts. We are all pure Sapiens.