Cech Covers I wrote this song with my old beloved room mate Christian Gorski in my last year of grad school while I was wrapping up my thesis.

For weeks, I was doing nothing but computing etale sheaf cohomologies of ramified covers of the projective plane. I would decompose sheaves on these curves by cutting around the neighborhood of ramification point (the stomach), and capturing the properties of the ramification group from the ramification point (the heart). The gluing back datum was the punctured neighborhood of the sheaf.

Let $X$ be a curve over a complete local ring, and let us decompose it into two affine pieces $X = A \cup B.$

Given a sheaf $\mathcal{F}$ over $X$, we have the following Mayer-Vietoris sequence.

\[0 \to H^0(X, \mathcal{F}) \hookrightarrow \mathcal{F}(A) \times \mathcal{F}(B) \to \mathcal{F}(A) \cap \mathcal{F}(B) \to H^1(X, \mathcal{F} \to 0.\]

In particular, if take the decomposition of $X$ into the point at infinity and its complement $X = U \cup {\infty}$, we get the following sequence, let $T$ be a divisor at $\infty$.

\[0 \to H^0(X, \mathcal{F}) \hookrightarrow \mathcal{F}(U) \times \mathcal{F}^\wedge_\infty \to \mathcal{F}^\wedge_\infty[\frac{1}{T}] \to H^0(X, \mathcal{F}) \to 0.\]

A global section $H^0(X, \Omega^1_X)$ can be specified by either a basis in terms of $\mathcal{F}(U)$, in terms of $\mathcal{F}^\wedge_\infty.$ This is because we are considering $H^0(X, \mathcal{F})$ as a subset of $\mathcal{F}(U) \times \mathcal{F}^\wedge_\infty$.

Lyrics:

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

It’s an easy thing to fall in love with a hand
When its lifeless and the fingers are missing
I can trace the truth As it sits motionless
beneath my glass, slowly twitching

And when I stitch it back together
It jumps and motions toward the sky
There’s a hanging sequence of all of your pieces
And your other hand is waving hi

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

Those intersections between your skin
Reveal raw flesh that I’m kissing
There’s so many ways to stitch you back
I GIVE YOU MORE EYES MORE EYES MORE EYES MORE EYES YOUR DICK IS MISSING, i needed it to make more eyes

I love to cut you apart to find the ways you combine
How many times can I do this before you fall apart forever?