**Yoshifumi Tsuchimoto**

is a prime ideal of $A$

For any subset of we define

Then we may topologize in such a way that the closed sets are sets of the form for some . Namely,

closed

We refer to the topology as the

- Show that for any , is an open set of .
- Show that given a point of and an open set which contains , we may always find an element such that . (In other words, forms an open base of the Zariski topology.

- For any subset
of
, we have
- For any subset
of
, let us denote by
the ideal of
generated by
. then we have

- For any ideal of , let us denote by the canonical projection. Then gives a homeomorphism between and .
- For any element
of
, let us denote by
be the canonical map. Then
gives a homeomorphism between
and
.

holds. is said to be

such that

- For any ideal of , we have .
- For two ideals , of , holds if and only if .
- For an ideal of , is irreducible if and only if is a prime ideal.

It is knwon that has a structure of ``locally ringed space''. A locally ringed space which locally lookes like an affine spectrum of a ring is called a scheme.

We define

It is known that carries a ringed space strucure on it and that it is a scheme.

- sheaves

- Benefit of being a sheaf.
- homomorphisms of (pre)sheaves
- example of presheaves and sheafification
- sheafification of a sheaf
- stalk of a presheaf
- kernels, cokernels, etc. on sheaves of modules
- About this document ...