November 29, 2016

Linear Algebraic Groups (Part 8)

Posted by John Baez

The course proceeds apace, but my notifications here have slowed down as I become over-saturated with work.

In Part 8, I began explaining a bit of algebraic geometry. Following the general pattern of this course I took a quasi-historical approach, explaining some older ideas before moving on to newer ones. I’m afraid I never got to explaining schemes. That’s a tragedy, but hey—life is full of tragedies, nobody will notice this one. Affine schemes is all I had time for, despite the fact that I was discussing a lot of projective geometry. And before explaining affine schemes, it seemed wise to mention some earlier ideas and their defects.

• Lecture 8 (Oct. 18) - Group objects in various categories. What’s the right category for linear algebraic groups? First try: algebraic sets. Over an algebraically complete field $k$, Hilbert’s Nullstellensatz says there’s an order-reversing one-to-one correspondence between algebraic sets $S \subseteq X$ in a finite-dimensional vector space $X$ over $k$ and radical ideals $J \subseteq k[X]$ of the polynomial algebra $k[X]$. The algebra $k[S]$ of polynomials restricted to $S$ is isomorphic to $k[X]/J$. Problem: we’d like an ‘intrinsic’ approach that does not make use of the ambient space $X$. Second try: affine algebras. For any algebraic set $S$, the algebra $k[S]$ is an affine algebra, meaning a finite-generated commutative algebra without nilpotents. Up to isomorphism, every affine algebra arises this way. Thus we can use affine algebras as a more intrinsic substitute for algebraic sets. Problem: all this works only over an algebraically complete field.

Posted at November 29, 2016 1:36 AM UTC

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Re: Linear Algebraic Groups (Part 8)

I think it’s very silly that schemes are taught as a big package deal. The only ones you need for this class are projective over affine, i.e. are Proj of some ring. (Go ahead, prove me wrong, but I’m pretty confident you’re not going to use the quasiaffine guy G/N for anything.) With that in mind, it’s easy to introduce Proj(R graded) as (Spec R \ Spec R_0)/scaling.

Posted by: Allen K. on November 29, 2016 7:07 PM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 8)

But already Spec R \ Spec R_0 fails to be projective over affine… And then to write /scaling opens a huge can of worms, namely the problem of taking quotients.

Really, in my experience, every way of teaching algebraic geometry is flawed.

Posted by: Xerxes on December 1, 2016 3:20 PM | Permalink | Reply to this

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