==== Week 2 Graded rings, Spec R and Proj R ====

Week 1 worked with the informal notion of variety, and introduced
the notion of a space with locally defined regular functions. This
is important background motivation for Grothendieck's more formal
and much more general theory of scheme.

A graded ring is R = sum R_d taken over integers d >= 0. Here I
almost always assume in addition that R_0 is a field k, and R is
generated by finitely elements of weight d_i > 0.

Week 2 discussed one or two technical points in the general
construction of X = Spec R and its structure sheaf OX (the
discussion gets fairly gristly), and the notion of the graded
spectrum Proj R. The latter generalises the construction of
projective space
  PP^n = (k^{n+1} \ 0) / k^x
as the quotient by the multiplicative group GGm. As in the
quotient of an affine variety by a finite group, the idea is
to take GGm-invariant open sets _not contained in the vertex_
and pass to their rings of GGm-invariant functions and the
affine varieties or affine schemes given by their Spec.

====
A projective variety X in PP^n corresponds to a graded ring.
Work first with the case of irreducible variety over k = kbar.

affine scheme: Spec R and its structure sheaf.
Spec of a ring

X = Spec R = set of prime ideals  (includes maximal ideals)
Zariski topology. The closed sets if X are subsets
   V(I) = set of prime ideals P such that P > I
          P contains f means f |-> 0 mod P,
                   f vanishes in field k_P = Frac(R/P)
Sheaf of rings: open set U |-> ring Ga(U, OX)
   restriction maps to open subset U1 in U.
Particular case: Xf = principal open set X \ V(f)
                    = Spec R[1/f]
These Xf form a basis for the topology: any open is
union  U = union Xf union
           taken over f s.t. U, V(f) disjoint.

Need a few more words on definition of sheaf, and the
stalk of a sheaf at a point: for F a sheaf on X, the
stalk of F at a point P in X is the direct limit
   lim_{->} Ga(U, F)
or the set of germs of sections near P.
If the restriction maps are injective, the direct lim
means union. Otherwise, the
   set of pairs U,s_U
   with U a nbd of P and s_U a section of F over U
modulo _germ equivalence_
   which means identify s_U and s1_U1 if they restrict
   to equal elements on some smaller neighbourhood.

Roughly, a germ of function near P. An element s in FP of
the stalk can be defined as a section on an open nbd, and
any two such are equal on a smaller nbd

=====
The construction of the structure sheaf OX of an affine
scheme X = Spec R is a complicated array of \forall \forall
\exists \exists \suchthat etc. The crucial logical step is
that with this definition, Ga(X, OX) = R and for the
principal open sets Xf, also Ga(Xf, OX) = R[1/f]. In other
words, the definition is consistent with the starting point.

At the end, we will have a sheaf of rings OX on X = Spec R
with two basic properties:

(i) for every f in R, the principal open set Xf = X \ V(f)
has sections  Ga(Xf, OX) = R[1/f]

(ii) for every prime P in X, the stalk of OX at P is the
local ring OX,P = RP.

Reminder concerning the definitions:
(1) the closed sets of the Zariski topology on X are
  V(I) = {primes containing I}.
The principal open sets form a basis for the open sets. [If you have
not seen this before, you need to show that V(I) = intersect V(f)
taken over f in I, and that any open set X \ V(I) is the union of
Xf.]

(2) The localisation RP = S^-1R = directlim R[1/f].
The stalk of OX at P is
  directlim Ga(U,OX) for U open nbd of P
    = directlim Ga(Xf,OX) for f notin P
[If you have not seen this before: the Xf with f notin P form a
basis for the open nbds of P. Thus we can refine a limiting sequence
{g_U in Ga(U,OX)} for an arbitrary sequence of open nbds to
{g_{Xf} in Ga(Xf,OX)} over basic opens.]

Statement: Ga(X,OX) = R
[Harts, Ch2, Prop 2.2, or same for modules Lemma 5.3]
There is a map
  R[1/f] -> Ga(X,OX)
that localises g/f in R[1/f] to g/f in RP for every
P in Xf. First, this is INJECTIVE: In fact, if 
g/f^n <> 0 in R[1/f], its annihilator is a nontrivial ideal,
so there is a prime ideal P of R[1/f] contained in it
(existence of primes). Then P in Xf, and g/f^n maps to a
nonzero element of RP = OXp at that P. (An element is in the
kernel of localisation S^-1 means that s*g = 0.)

The injective part only involved finding one P at which the
localisation of g/f^n is nonzero, so that was easy.

The surjective part involves the full definition of OX. For
an open subset U in X = Spec R a section s in Ga(U,OX) means
  I have sP in AP for every P in X satisfying
  (local property) there is an open cover of U by principal
  opens Xh and for each Xh an element g/h^n in R[1/h] that
  represents sP at P for all P in Xh.
Suppose that for a given f in R, some set {h in R} give rise
to basic opens Xh that between them cover Xf. This means that
for every P in Xf, there is h so that P in Xf. The local
property in the definition of OX means I can choose {h in R}
so that
 (1) Xh cover Xf, and
 (2) sP in AP is represented by g/h^n in R[1/h] for every P.
Now (1) together with the familiar (existence of primes)
result implies 
  f is in the radical of I = ideal {h}.
In fact (1) with the definition  Xf = X \ V(f) means that
f \not P for any P in Xh so that f in every prime of V(I).

Hard point: Let Xf = Spec R \ V(f).
If s in Ga(Xf, OX) in the definition of OX, then s in R[1/f]
-- this means that Ga(Xf, OX) = R[1/f] as in the starting
point description of OX.

This is quite similar to "uniformly continuous" in ep-de
calculus. We require s to be "continuous" (in the local
sense of being covered by open) over the whole of Xf, and
conclude it is actually defined once and for all by an
element of R[1/f] with the single denominator f.

Digression. The proof below and in [Harts, Ch2, Prop 2.2, or
for modules and their associated sheaf of OX-modules,
Lemma 5.3] is slightly complicated logic that takes care to
get correct, but remembering the classical case based on the
NSS should help.

If we work with the old notion of irreducible variety over
k = kbar, the statement is that if s in k(X) is regular at
every point of a principal open set Xf in X then s is in the
affine coordinate ring k[X][1/f] of Xf. Recall the argument:
define the ideal of denominators of s. Then the assumption
is that f vanishes on V(Denom(s)). So the NSS implies that
f is in the radical, or f^r is a denominator of s, so
s in k[X][1/f].

Proof of surjective assertion.
Saying s in Ga(Xf,OX) means that for every P in Xf, there
is expression s = gi/hi in R_P. The contrapositive of this
is that any P that contains one possible denominators hi of
an expression s = gi/hi also contains f. Therefore
  f is contained in every prime that contains one of the hi.
Thus f in rad( ideal generated by all the hi ). Because

rad(I) = intersection of all primes containing I.
(the "scheme version of Nullstellensatz", the first result on
existence of primes in commutative algebra).

Now f^r in (ideal of hi) so it is in the ideal of finitely
many hi, and to treat s we can assume
  there are finitely many hi with Xhi in Xf,
  Xf = union Xhi, and s = gi/hi^Ni.
The inclusions mean hi in rad(f) and f in rad({hi}). We can
replace hi -> hi^Ni or f -> f^N, so the rad. Now
gi/hi ~ gj/hj on the overlaps X_{hi*hj}, so
  (hi*hj)^N*(hj*gi-hi*gj) = 0  and  f^N*(hj*gi-hi*gj) = 0.
Since there are only finitely many i, we can suppose these
holds for fixed N and all i,j. Then
  g = f^N*hi*gj in R, so g in R[1/f] as required.   QED

From Spec to Proj

Proj R = Spec R \ V( ideal R_+ )
       = union Spec (R[1/g])^0
taken over all homogeneous g of degree > 0.
This means take gi of degree di > 0  (all of them if your prefer,
or just finitely many, or just the xi if you prefer).
The construction (R[1/g])^0 means the ring  { f / g^n | all f in R_{n*di} }
 -- allow only powers of g in denominator, and adjust f to be homogeneous
 of deg n*di so that f/g^n is homogeneous of degree 0, so GGm-invariant.

In scheme theoretic terms Proj R is the set of homogeneous prime ideals,
(not irrelevant, meaning p not R_+ -primary, so that V(P) in Spec R
contains nontrivial GGm orbit). Its Zariski topology and structure
sheaf are constructed exactly as for Spec, using GGm-invariant open
sets and GGm-invariant fractions as above.

=== Graded ring R and Proj R ===
R graded ring
Definition ring R = sum R_d  taken over d in NN = {0,1,2,..}
multiplication R_d1 x R_d2 -> R_{d1+d2}
  additional assumptions (to make a projective scheme over k)
R_0 = field k, R generated by finitely many x_i in R_di
  => R = k[x0,..xn]/IR  where IR is homogeneous ideal.

GGm = multiplicative group scheme = GL(1) acts on R, with
  la \in GGm(S) acting on R  by multiplication la^d: R_d -> R_d
GGm is a reductive group scheme. Its irreducibles are all
1-dimensional, given by la^d, and every representation is a
direct sum of representations.

Definition of Proj R
Proj R = homogeneous prime spectrum
Homogeneous ideal means I = sum In -- the condition is that
every element f in I can be written as a sum of parts fn in Rn
that are homogeneous of degree n in R, and are still in I.

If we already know the affine cone Spec R = CX.
The irrelevant ideal is M = sum R_d over d > 0. Every
homogeneous element of degree > 0 is in M, and
  M = Ideal(x0,..xn).
(Any ideal I containing M^N is also irrelevant.)
Then Proj R = (CX \ V(M)) / GGm.

Here GGm is the multiplicative group, or the group of invertible
1x1 matrices. For a ring R, GGm(R) = R^x (the units of R), but I
consider GGm as a group scheme defined over all rings. You can
write k^x or even CC^x if you prefer. (Lots of people write k^*
to mean the _set_ of units of k, but the multiplicative x is
better. I write GGm or k^x even when I say CC-star in colloquial
speech.)

Taking the quotient means: take GGm invariant affine open subset
CXf = CX \ V(f) for f in R_d (homogeneous of degree > 0).
It has affine coordinate ring R[1/f]. The GGm invariants of R[1/f]
are (R[1/f])^0 homogeneous elements of degree 0.

==== Weighted projective space
I want to get to the wPPs version of nonsingularity, called
quasi-smooth.

First, an example like PP^2(1,b,c) with b,c > 0 and coprime.
PP(1,2,3) or PP(1,3,5) if you want to be specific.

Draw picture of (say) PP(1,3,5). It has 2 "horns" P_y and P_z
that are quotient points
  Py = 1/3(1,5) iso 1/3(1,2)
  and P_z = 1/5(1,3) 

GGm-action AA^3<x,y,z> / divided by action
          ( x |-> la*x, y |-> la^3*y, z |-> la^5*z ).

At a point P = (x,y,z) with x <> 0, we can choose x = 1 as a
"slice" of the equivalence relation given by the GGm-action.
Namely, I can pass to (1, y/x^3, z/x^5) as a preferred
representative.

That does not work near the y-axis or z-axis.

Geometrically, if I try to take a preferred representative of the
orbit of GGm by setting y = 1, that plane cuts the y-axis in 1
point, but it cuts all the neighbouring orbits with (x,z) <> (0,0)
is 3 points, the cube roots of x or z.

Algebraically, the ring of fractions  R[1/y]  allows denominator
y or y^n accompanied by x^i*z^k of weight 3*n.

I could take x^3/y and x*z/y^2 and z^3/y^5 as generators of the
ring (R[1/y])^0, but there is nothing simpler.

The affine piece Spec(R[1/y])^0 has a singularity of type 1/3(1,5)
at the origin.

A way of seeing this that is convenient for computation is:
first take a cube root eta = y^{1/3}. Now eta has weight 1, so
in a slightly bigger ring x/eta and z/eta^5 are both homogeneous
of degree 0. So there is some kind of slice giving preferred
representatives of the orbits. But by taking the cube root we
have passed to a Galois extension of R with Galois group
ZZ/3 (sometimes better viewed as mu_3). The polynomials that
are GGm invariant fractions of R are obtained as the invariants
of ZZ/3 action.

In summary, where the GGm group action has fixed points, the
variety is best thought of as locally a quotient variety. The
quasismooth condition is that locally, Proj is of the form
1/r(a1,.. ak) where r = deg y, the "slice" coordinate to be
set = 1, and the transverse coordinates xi have degree
== a1,.. ak mod r. This is called orbifold. The idea that you
have to take the quotient, then remember the coordinates xi
of the space before taking the quotient is the beginning of
the theory of stacks.

There are lots of exercises concerned with constructing
interesting quasi-smooth weighted hypersurfaces (or with mild
singularities). See separate Addendum on some orbifold curves.