The n-Category Café

• 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.

Supplementary reading:

• J. S. Milne, Algebraic Geometry.