math 216: foundations of algebraic geometry - Stanford University

36 Math 216: Foundations of Algebraic Geometry
2.5.2. Examples from other fields. For those familiar with representation theory:
Frobenius reciprocity may be understood in terms of adjoints. Suppose V is a
finite-dimensional representation of a finite group G, and W is a representation of
a subgroup H < G. Then induction and restriction are an adjoint pair (Ind G
H, Res G H)
between the category of G-modules and the category of H-modules.
Topologists’ favorite adjoint pair may be the suspension functor and the loop
space functor.
2.5.3. Example: groupification of abelian semigroups. Here is another motivating
example: getting an abelian group from an abelian semigroup. (An abelian
semigroup is just like an abelian group, except you don’t require an identity or
an inverse. Morphisms of abelian semigroups are maps of sets preserving the
binary operation. One example is the non-negative integers 0, 1, 2, . . . under addition.
Another is the positive integers 1, 2, . . . under multiplication. You may enjoy
groupifying both.) From an abelian semigroup, you can create an abelian group.
Here is a formalization of that notion. A groupification of a semigroup S is a map
of abelian semigroups π : S → G such that G is an abelian group, and any map of
abelian semigroups from S to an abelian group G ′ factors uniquely through G:
π
S
G
G ′
(Perhaps “abelian groupification” is better than “groupification”.)
2.5.F. EXERCISE (A GROUP IS GROUPIFIED BY ITSELF). Show that if a semigroup
is already a group then the identity morphism is the groupification. (More correct:
the identity morphism is a groupification.) Note that you don’t need to construct
groupification (or even know that it exists in general) to solve this exercise.
2.5.G. EXERCISE. Construct groupification H from the category of nonempty
abelian semigroups to the category of abelian groups. (One possible construction:
given an abelian semigroup S, the elements of its groupification H(S) are ordered
pairs (a, b) ∈ S × S, which you may think of as a − b, with the equivalence that
(a, b) ∼ (c, d) if a + d + e = b + c + e for some e ∈ S. Describe addition in
this group, and show that it satisfies the properties of an abelian group. Describe
the semigroup map S → H(S).) Let F be the forgetful functor from the category
of abelian groups Ab to the category of abelian semigroups. Show that H is leftadjoint
to F.
(Here is the general idea for experts: We have a full subcategory of a category.
We want to “project” from the category to the subcategory. We have
Morcategory(S, H) = Morsubcategory(G, H)
automatically; thus we are describing the left adjoint to the forgetful functor. How
the argument worked: we constructed something which was in the smaller category,
which automatically satisfies the universal property.)
2.5.H. EXERCISE (CF. EXERCISE 2.5.E). The purpose of this exercise is to give
you more practice with “adjoints of forgetful functors”, the means by which we
get groups from semigroups, and sheaves from presheaves. Suppose A is a ring,
∃!