Thoughts on mathematics
June 10, 2011
Soft sheaves
Posted by Akhil Mathew under topology | Tags: sheaves, soft sheaves |
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The statement of Verdier duality was fairly fancy–it used the jazzed up language of derived categories,
and was phrased in terms of the existence of an adjoint to a suitable derived functor. Though by the end
of it hopefully it’ll become clear that, at least for manifolds, the mysterious
adjoint functor has a fairly
concrete interpretation that will turn out to be plain old Poincare duality.
But before we get there, we need to discuss some sheaf theory. I have usually thought of sheaves in the
context of algebraic geometry, and I wanted to review (at least for my own benefit) some of the classical
theory of sheaves on ordinary non-pathological topological spaces, like locally compact Hausdorff ones.
In addition, it will at any rate be necessary to introduce the compactly supported cohomology functors to
get Verdier duality. So I’ll do a couple of preparatory posts.
1. Soft sheaves
Let
be a locally compact space. If
for
and
is closed, we define
the inclusion. This allows us to make sense of a section
“over a closed subset.” The interpretation via the espace étale is helpful here: if
étale of
, then a section over
is the same thing as a section of the projection
follows from the latter interpretation that if we have sections
, one obtains uniquely a closed subset on
on closed subsets
is the espace
over
. It
that agree on
. Now we want to show that we can recover
this notion from the familiar idea of sections over an open set.
Lemma 1 If
where
is compact, we have
ranges over open sets containing
. (more…)
June 4, 2011
The nPOV, sheaves, and derived categories
Posted by Akhil Mathew under category theory, math education | Tags: derived category,
derived functors, higher category theory, nPOV, sheaves |
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I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent
complex, partially because I’ve been busy doing other things–namely, trying to learn about the
foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent
complex thing will get finished as a short expository note on my website.
One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact
shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many
such concepts, to replace them with the newer ones.
Sheaves
An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working
out the exercises there. But as I have more recently discovered, many of the methods there are not the
appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines
sheafification (and many other things) on a topological space using stalks. However, on a site this is
meaningless because there is no analogous notion in general.
The stalk of a sheaf (or presheaf) on a space
the inclusion
at a point corresponds to the inverse image functor via
. The analogy in the theory of sites would be the inverse image via a morphism
from the site with one point (or something equivalent to this). It turns out, fortunately, in etale
cohomology this more general notion does make sense, if
closed field. So, if
is taken to be the spectrum of a separably
is a scheme, it is not topological points
etale cohomology, but the morphisms
for
that lead to the stalk functors in
a separably closed field (e.g. the separable
closure of the residue fields of the topological points).
It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point
is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of
sets and the given topos. The direct and inverse image functors obtained from maps
show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there
is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves.
Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far
from understanding the story here though. (more…)
March 10, 2011
The site of G-sets and the associated category of sheaves
Posted by Akhil Mathew under algebraic geometry, category theory | Tags: Grothendieck
topologies, representable functors, sheaves, sites |
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In the past, I said a few words about Grothendieck topologies and fpqc descent. Well, strictly speaking, I
didn’t get very far into the descent bit. I described a topology on the category of schemes (the fpqc
topology) and showed that it was a subcanonical topology, that is, any representable presheaf was a
sheaf in this topology.
This amounted to saying that if
the same thing as homming out of
was a fpqc morphism of schemes, then to hom out of
such that the two pull-backs to
was
were the same. If I
had gotten further, I would have shown that to give a quasi-coherent sheaf on
the same as giving “descent data” of a quasi-coherent sheaf on
between the two pull-backs to
(among other things) is
together with an isomorphism
satisfying the cocycle condition. Maybe I’ll do that later. But
there is a more basic “toy” example that I now want to describe of a site (that is, category with a
Grothendieck topology) and the associated category of sheaves on it.
1.
Our category
is going to be the category of left
equivariant morphisms of
-sets
-sets for a fixed group
; morphisms will be
-sets. We are now going to define a Grothendieck topology on this
category. For this, we need to axiomatize the notion of “cover.” We can do this very simply: a collection
of maps
is called a cover if the images cover
. Now, fiber products of
-sets are
calculated in the category of sets, or in other words the forgetful functor
commutes with limits (as it has an adjoint, the functor
). Thus, taking pull-backs
preserve the notion of covering, and it is easy to see the other axioms are satisfied too: if we have a cover
of each of the
(which cover
), then collecting them gives a cover of
. Similarly, an isomorphism is
a cover. This is obvious from the definitions.
2. Representable presheaves
So we indeed do have a perfectly good site. Now, we want a characterization of all the sheaves of sets on
it. To start with, let us show that any representable functor forms a sheaf; that is, the topology is
subcanonical. (In fact, this topology is the canonical topology, in that it is the finest possible that makes
representable functors into sheaves.) (more…)
September 9, 2010
The fpqc topology
Posted by Akhil Mathew under algebraic geometry, category theory | Tags: descent theory, fpqc
morphisms, Grothendieck topologies, sheaves |
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We continue in the quest towards descent theory. Today, we discuss the fpqc topology and prove the
fundamental fact that representable functors are sheaves.
We now describe another topology on the category of schemes. First, we need the notion of an fpqc
morphism.
Definition 1 A morphism of schemes
satisfied:
is called fpqc if the following conditions are
1.
is faithfully flat (i.e., flat and surjective)
2.
is quasi-compact.
Indeed, “fpqc” is an abbreviation for “fidelement plat et quasi-compact.” It is possible to carry out
faithfully flat descent with a weaker notion of fpqc morphism, for which I refer you to Vistoli’s part of
FGA explained.
As with many interesting classes of morphisms of schemes, we have a standard list of properties.
Proposition 2
1.
Fpqc morphisms are closed under base-change and composition.
2.
If
are fpqc morphisms of
-schemes, then
is fpqc.
Proof: We shall omit the proof, since the properties of flatness, quasi-compactness, and surjectivity are
all (as is well-known) preserved under base-change, composition, and products. This can be looked up in
EGA 1 (except for flatness, for which you need to go to EGA 4 or Hartshorne III).
So we have the notion of fpqc morphism. Next, we use this to define a topology.
Definition 3 Consider the category
on
of
-schemes, for
is defined as follows: A collection of arrows
a fixed base-scheme. The fpqc topology
is said to be a cover of
if the map
is an fpqc morphism.
This implies in particular that each
is a flat morphism. We need now to check that this is
indeed a topology.
1.
An isomorphism is obviously an fpqc morphism, so an isomorphism is indeed a cover.
2.
If
is a fpqc cover and
, then the morphism
equal to the base-change
3.
Suppose
is
, hence is fpqc.
is a cover for each and
is a cover. Indeed, we have that
is a cover, I claim that
factors through
and we know that each morphism in the composition is flat (since
the coproduct of flat morphisms is flat) and quasi-compact (since the coproduct of quasicompact morphisms is quasi-compact). Similarly for surjectivity. It follows that
is an fpqc cover.
So we have another topology on the category of schemes, which is very fine in that it is finer than many
other topologies of interest (e.g. the fppf and etale topologies, which I will discuss at some other point).
(more…)
September 7, 2010
Sheaves in Grothendieck topologies
Posted by Akhil Mathew under algebraic geometry, category theory | Tags: descent theory,
Grothendieck topologies, sheaves |
1 Comment
It is possible to define sheaves on a Grothendieck topology. Before doing so, let us recall the definition of
a sheaf of sets on a topological space
Definition 1 A sheaf of sets
sections over
.
assigns to each open set
a set
(called the set of
) together with “restriction” maps
for inclusions
such that the following conditions are satisfied:


and for a tower
If
is a cover of
, the composite
equals
, then the map
image consists of those families
.
is injective, and the
such that the restrictions to the intersections are
equal
In particular, this says that if we have a family of elements
condition, then there is a unique
that satisfy the above gluing
which restricts to each of them.
(more…)
July 25, 2010
Notes on sheaves
Posted by Akhil Mathew under algebraic geometry | Tags: sheaves |
[2] Comments
While the material on the nonabelian Livsic theorem remains on hold, I have been writing up some
notes on algebraic geometry. Here they are. So far, there is no actual algebraic geometry in it, however;
they cover elementary facts about sheaves and ringed spaces. At some point in the future, I will certainly
post an updated and presumably much longer version.
December 24, 2009
Leray’s theorem
Posted by Akhil Mathew under algebra, algebraic geometry, blegs | Tags: cohomology, Leray's
theorem, sheaves |
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Today’s main goal is the Leray theorem (though at the end I have to ask a question):
Theorem 1 Let
be a sheaf on
, and
an open cover of
. Suppose
for all
-tuples
, and all
is an isomorphism for all
. Then the canonical morphism
.
This seems rather useless, because the theorem presupposes the vanishing of (regular) cohomology on
the covering. However, in many cases it turns out to be helpful. If
affine cover of
, and
is a separated scheme,
an open
quasi-coherent, it applies. The reason is that each of the intersections
are all affine by separatedness, so
has no cohomology on them by a basic property
of quasi-coherent sheaves. This gives a practical way of computing sheaf cohomology in algebraic
geometry. Hartshorne uses it to compute the cohomology of line bundles on projective space.
Another instance arises when
In this case
subset of
(more…)
is the sheaf of holomorphic functions over some Riemann surface
.
is a covering of charts. It is a theorem (which I will eventually prove) that for any open
(which any intersection of the
‘s is isomorphic to), the sheaf
has trivial cohomology.