In order to define any algebraic geometry structures (a *sheaf*, for example), one has to first understand what an algebraic variety is. And thus:

**Observation I.** It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.

Instead, most authors tend to define an *affine algebraic variety* – first as the common zero set of a collection of complex polynomials in and later as a “variety that can be embedded in affine space as a Zariski-closed set” (Smith et. al., *An Invitation to Algebraic Variety*). Then, half a book later or more (it’s on page 144 of the aforementioned book), it’s said that an *(abstract) algebraic variety* is a topological space with an open cover consisting of sets homeomorphic to affine algebraic varieties which are glued together by so-called *transition functions* that are morphisms in the category of affine algebraic varieties.

This of course requires knowledge of category theory, the Zariski topology, etc. etc.

As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning -modules.

*le sigh*

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