Skip to main content

Section 1.5 Theorem of the Cube (Ricky)

Subsection 1.5.1 Crash Course in Line Bundles

Consider \(\RR^2\text{,}\) \(f\colon \RR \to \RR\) , \(f(x,y) = x^2 + y^2 -1\text{,}\) now \(S = \{f=0\}\subseteq \RR^2\) is a closed submanifold (in fact a circle). Question: Do all closed submanifolds arise in this way? Lets switch to \(\CC\) better analogies with AG.

Example 1.5.1.

Let \(X\in \PP^n(\CC)\text{,}\) the answer here is no! (Because \(f\colon X \to \CC^1\) is constant!) Want to define functions locally that give us level sets, but gluing such will give us a global section. Instead glue in a different way (i.e. into different “copies” of \(\CC\)) so that this doesn't happen.

Example 1.5.2.

\(X\in \PP_\CC^1\text{,}\) \(\sheaf O_X\) the structure sheaf.

\begin{equation*} X = U_0 \cup U_1 = (\aff^1,t) \cup(\aff^1,s) \end{equation*}

on \(U_0\cap U_1\text{,}\) \(t = s^{-1}\text{.}\) What is a global section of \(\sheaf O_X\text{,}\) a section of \(U_0\) and a section of \(U_1\) that glue. \(\sheaf O_X(U_0) = k[t], \sheaf O_X(U_1) = k[s]\) so given \(f(t), g(s)\) these glue to a global section iff \(f(t) = g(1/t)\) so \(f,g\) must be constant.

Definition 1.5.3. Line bundles.

A line bundle on \(X\) is a locally free \(\sheaf O_X\)-module of rank 1, i.e. \(\exists \{U_i\}\) open cover along with isomorphisms \(\phi_i\colon \sheaf L|_{U_i} \xrightarrow\sim \sheaf O_X |_{U_i}\text{.}\)

Alternative definition: A line bundle on \(X\) is equivalent to the following data:

  • An open cover of \(X\text{.}\)
  • Transition maps \(\tau_{ij} \in \GL_1(\sheaf O_X(U_i\cap U_j))\) satisfying \(\tau_{ij}\tau_{jk} =\tau_{ik}\) and \(\tau_{ii} = \id\text{.}\)
Example 1.5.5.

On \(X = \PP^n_k\text{,}\) we have line bundles \(\sheaf O(d)\) for all \(d\in \ZZ\text{.}\) Just have to give cover and transition functions, use usual open cover \(\{U_i\}\) with \(U_i\cong \aff^n\text{.}\) Then \(\tau_{ji}\) is given by multiplication by \((x_i/x_j)^d\text{.}\)

\begin{equation*} H^0(X,\sheaf O(d)) ( = \Gamma (X,\sheaf O(d))) \end{equation*}

\(= k\)vector space spanned by deg. \(d\) homogeneous polynomials in \(k[x_0,\ldots, x_n]\text{.}\)

All line bundles on \(\PP^n\) are isomorphic to some \(\sheaf O(d)\text{.}\)

We say a line bundle \(\sheaf L\) on \(X\) is trivial if \(\sheaf L \cong \sheaf O_X\text{.}\) Given \(\sheaf L_1\) and \(\sheaf L_2\) on \(X\) (line bundles) we can create a new line bundle \(\sheaf L = \sheaf L_1 \otimes \sheaf L_2\text{.}\) So isomorphism classes of line bundles on \(X\) with \(\otimes\) form a group, denoted \(\Pic(X)\) with identity \(\sheaf O_X\) and inverses \(\sheaf L^{-1} = \Hom(\sheaf L , \sheaf O_X)\text{.}\)

Example 1.5.8.

By previous exercise \(\Pic(\PP_k^n) \cong \ZZ\) since \(\sheaf O_X(d_1) \otimes \sheaf O_X(d_2) \cong \sheaf O_X(d_1+d_2)\text{.}\)

Subsection 1.5.2 Relation to (Weil) divisors

Let \(X\) be a normal variety, call \(Z\subseteq X\text{,}\) a closed subvariety of codimension 1, a prime divisor. Then a divisor on \(X\) is a formal sum

\begin{equation*} D = \sum_{Z\subseteq X} n_Z\cdot Z \end{equation*}

of prime divisors.

Let \(K = K(X)\) be the function field of \(X\text{.}\) Given \(f\in K^\times\) we can define

\begin{equation*} \divisor (f) = \sum v_Z(f) \cdot Z\text{.} \end{equation*}

Given \(D \in \Div(X)\text{,}\) we can define a line bundle \(\sheaf L(D)\) on \(X\) via

\begin{equation*} \sheaf L(D)(U) = \{f\in K^\times : (D + \divisor (f))|_U\ge 0 \} \cup \{0\} \end{equation*}

where \(D|_U = \sum_{Z\cap U\ne \emptyset} n_Z\cdot (Z\cap U)\text{.}\)

Subsection 1.5.3 Onto cubes

Let \(m \colon A\times A \to A\) be multiplication (addition?) and \(p,q\) the projections \(A\times A \to A\text{.}\) Then the composites of the maps \(\phi\colon A\times A \to A \times A \times A, \phi(x,y) = (x,y,0)\) with \(p_{123}, p_{12}, p_{23}, p_{13}, p_1, p_2, p_3\) are respectively \(m,m,q, p, p, q, 0\text{.}\) Hence the restriction of \(\sheaf M\) to \(A \times A \times \{0\}\) is

\begin{equation*} m^* \sheaf L \otimes m^* \sheaf L^{-1} \otimes q^* \sheaf L^{-1} \otimes p^*\sheaf L^{-1} \otimes p^*\sheaf L \otimes q^*\sheaf L \otimes \sheaf O_{A\times A} \end{equation*}

this is trivial by tensor commuting with pullback. Similarly \(\sheaf M\) restricts to a trivial bundle on \(A\times \{0\} \times A\) and \(\{0\}\times A \times A\text{.}\) So by theorem of the cube 1.5.11 \(\sheaf M\) is trivial.

\(\sheaf M\) is the pullback of the line bundle of Corollary 1.5.12 via the map \((f,g,h) \colon V \to A\times A\times A\text{.}\)

On \(A\) we have \(n_A \colon A\to A\) be \(n_A(a) = a+\cdots + a\) (\(n\) times) for \(n\in \ZZ\text{.}\)

Use Corollary 1.5.13 with \(f= n_A, g = 1_A, h = (-1)_A\text{.}\) So the line bundle

\begin{equation*} (n)^*\sheaf L \otimes (n+1)^*\sheaf L^{-1}\otimes (n-1)^*\sheaf L^{-1}\otimes (1-1)^*\sheaf L^{-1} \otimes n^*\sheaf L \otimes 1^*\sheaf L \otimes (-1)^*\sheaf L \end{equation*}

is trivial i.e.

\begin{equation*} (n+1)^*\sheaf L = (n-1)^*\sheaf L^{-1}\otimes n^*\sheaf L^2 \otimes \sheaf L \otimes (-1)^*\sheaf L \end{equation*}

in statement \(n = 1\) is clear, so use \(n=1\) in the above to get

\begin{equation*} 2_A^*\sheaf L \cong \sheaf L^2 \otimes \sheaf L \otimes (-1)^*_A \sheaf L\cong \sheaf L^3 \otimes (-1)_A^*\sheaf L\text{.} \end{equation*}

Then induct on \(n\) in above.

Use Corollary 1.5.13 with \(f= \id\text{,}\) \(g(x) = a, h(x) = b\) to get

\begin{equation*} t_{a+b}^*\sheaf L \otimes t_{a}^*\sheaf L ^{-1}\otimes t_b^* \sheaf L ^{-1} \otimes \sheaf L \end{equation*}

is trivial.

Remark 1.5.16.

Tensor by \(\sheaf L^{-2}\) in the above equation to get

\begin{equation*} t_{a+b}^*\sheaf L \otimes\sheaf L^{-1} \cong ( t_{a}^*\sheaf L \otimes \sheaf L^{-1}) \otimes(t_b^* \sheaf L\otimes \sheaf L^{-1})\text{.} \end{equation*}

This gives a group homomorphism

\begin{equation*} A(k) \to \Pic(A) \end{equation*}

via

\begin{equation*} a\mapsto t_a^*\sheaf L \otimes \sheaf L^{-1} \end{equation*}

for any \(\sheaf L \in \Pic(A)\text{.}\)