Date
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What we discussed/How we spent our time
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Jan 11
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Will be a course
on homological algebra. Grades will be
based on HW.
Definition of a differential R-module.
A differential R-module is equivalent to a module over R[d]/(d^2),
the ring of dual numbers over R.
Definitions of chains, cycles, boundaries, homology modules.
Examples: Computing the homology of a 1-simplex and of a 2-simplex.
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Jan 13
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Example: Computing the homology of the boundary of a 2-simplex
(an oriented triangle without interior).
Definition of homology of any finite simplicial complex.
Chain complexes.
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Jan 15
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Description of singular homology.
Relationship between simplicial and singular homology.
Intuition behind cycles, boundaries and homology.
Interpretation of H0(X).
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Jan 20
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Categories: definition and examples. Monomorphisms,
epimorphisms, isomorphisms, retractions, sections.
Opposite category. Functors: definition and examples
of covariant and and contravariant functors.
Interpretations of `functor': (i) homomorphism,
(ii) uniform construction, (iii) representation.
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Jan 22
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Example of a `uniform construction' that is not
a functor: the construction of the center of a group.
Description of the covariant and contravariant
power sets functors.
Full, faithful and representative functors.
Natural transformations: definition.
Comparison with homotopies between continuous maps.
Identity transformations. Composition of natural transformations
is associative. The collection
DC of functors
from C to D equipped with natural transformations
is a category, except that homsets Nat(F,G) may
be proper classes. If C is small, then DC is a category.
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Jan 25
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Natural isomorphism: definition (= invertible
natural transformation). Same as natural transformation
whose components are isomorphisms. Isomorphism, equivalence and
duality of categories. Examples. Skeleton of a category.
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Jan 27
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Characterizations of equivalence. Axiom of choice
for classes. Representable functors. Yoneda lemma.
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Jan 29
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Yoneda embedding. Natural transformations between representable functors.
Natural operations on functors.
Algebras and coalgebras in categories.
Representable functors to categories of algebras.
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Feb 1
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Examples of
representable functors to categories of algebras.
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Feb 3
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Preadditive and additive categories.
A preadditive category has binary products and
coproducts and a zero object iff it has finite
biproducts.
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Feb 5
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Kernels, cokernels, normal monomorphisms,
conormal epimorphisms.
Kernels are monic and cokernels are epic.
A monic is zero iff its domain is a zero object.
If ker(φ) exists, then φ is monic
iff ker(φ) = 0.
Abelian categories.
A group homomorphism is a monomorphism iff it is 1-1;
it is a normal monomorphism iff its image is a normal subgroup.
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Feb 8
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Examples of abelian categories:
categories of R-modules; "large" versions of categories
of R-modules; compact, T2, abelian groups
(Pontrjagin duality);
full subcategories of abelian categories which are closed under
0, ⊕ ker, coker;
products of abelian categories; functor categories
AC with C small and A abelian.
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Feb 10
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Chain
complexes and exact sequences in abelian categories.
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Feb 12
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Pullback theorems. Application to diagram chasing.
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Feb 15
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Snake Lemma. Diagram chasing by Swan's method.
Mitchell Embedding Theorem.
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Feb 17
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The category of chain complexes (= a subcategory of a functor category).
Proof that the category of chain complexes of objects from
an abelian category A is itself an abelian category. Discussion
of biproducts, kernels and cokernels in
a category of chain complexes. Chain maps are monic/epic
iff they are componentwise. An exact sequence of complexes is
exact iff it is componentwise. Projection of complexes onto their
nth components is a functor. Homology of a complex.
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Feb 19
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Hn is a functor. Exact homology sequence.
Nullhomotopic chain maps and homotopic pairs of chain maps.
Homotopic chain maps induce the same function on homology.
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Feb 22
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Topological background for chain homotopies:
We computed the simplicial chain complexes for the
circle and the cylinder, and their homology.
We compared the chain maps
resulting from the embeddings of the circle into the
cylinder at the top and the bottom. The chain homotopy
between these maps was derived from the operation of
"cylindrification" of chains.
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Feb 24
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Definition of resolution and deleted resolution.
Types of resolutions to the
left include: free, projective and flat resolutions.
Types of resolutions to the right include
injective and flabby resolutions.
Characterization of projectives:
Thm. The following are equivalent for an R-module P.
(1) P is projective.
(2) Every SES ending at P splits.
(3) P is a retract of a free module.
(4) P is a direct summand of a free module.
(5) The covariant hom functor represented by P
maps epis to surjections.
(6) The covariant hom functor represented by P is exact.
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Feb 26
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Hom functors are left exact.
Left or right exact functors are additive.
An abelian category has enough projectives if
every object is an epimorphic image of a projective object.
Using splicing of SES's,
this is seen to be equivalent to the condition that
every object has a projective resolution.
Comparison Theorem Part 1: If φ A → B is a hom,
P → A is a complex with all Pi projective,
and
Q → B is a resolution of B, then
φ can be lifted to a chain map from
P → A to Q → B.
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Mar 1
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Comparison Theorem Part 2: Any 2 liftings of φ from
Part 1 are homotopic. Horseshoe Lemma. Injective resolutions.
Characterizations of injectivity, including Baer's Criterion.
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Mar 3
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Derived functors: definition, independent of
resolution, long exact sequence. Derived functors
of exact functors are trivial. Derived functors
of half exact functors repair exactness. Object is acyclic
with respect to T if LnT(A) = 0 for all
n>0. nth derived functor of _⊗ B is called
TornR(_,B).
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Mar 5
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We discussed a source of left exact functors: adjoints.
Definition and examples of adjoint functors.
Lm. A sequence in an abelian category
which becomes exact after applying any
covariant hom functor to it is itself exact.
(This proof contained a gap, which is filled here.)
Thm. A functor between abelian categories
that has a left adjoint is left exact.
Adjoint Associativity (= Hom and tensor are adjoints).
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Mar 8
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δ-functors and their morphisms.
Universal δ-functors. The left derived
functors of a right exact functor form a universal
δ-functor.
Definitions of
Torn(A,B), torn(A,B),
Extn(A,B) and
extn(A,B).
Thm. TFAE for a right R-module B.
(1) B is flat.
(2) TornR(A,B)=0 for all n>0 and all left R-modules
A.
(3) Tor1R(A,B)=0 for all A.
We gave a presentation for the torsion product of two abelian groups.
We started on the proof of:
Thm. (Tor for abelian groups)
(1) Torn(A,B)=0 for all n>1.
(2) Tor1(A,B) is the torsion product of A and B.
(3) Tor1(Q/Z,B)=t(B).
(4) Tor1(A,B)=Tor1(t(A),t(B)).
(5) B is flat iff B is torsion free.
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Mar 10
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We finished the proof of (1), (3), (4), (5) of:
Thm. (Tor for abelian groups)
(1) Torn(A,B)=0 for all n>1.
(2) Tor1(A,B) is the torsion product of A and B.
(3) Tor1(Q/Z,B)=t(B).
(4) Tor1(A,B)=Tor1(t(A),t(B)).
(5) B is flat iff B is torsion free.
We started on the proof that
TornR(A,B) =
tornR(A,B).
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Mar 12
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We finished the proof that
TornR(A,B) =
tornR(A,B).
In the course of the proof we established
the Dimension Shifting Lemma.
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Mar 15
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We started a discussion of the behavior of derived
functors and adjoints on limits and colimits.
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Mar 17
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More on the behavior of derived functors and
adjoints on limits and colimits.
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Mar 19
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ColimJ is a right exact functor from
AJ to A. It need not be fully exact:
we gave an example showing that pushouts are colimits
that are not always exact. We proved
that if J is filtered, then
colimJ is exact. Since homology commutes
with exact functors, it commutes with filtered
colimits. One consequence is that any directed union of flat
modules is flat, in particular Q is a flat Z-module.
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Mar 29
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Definitions of Extn(A,B) and
extn(A,B). Characterizing properties:
to each SES is an associated LES;
Ext0(A,_)≅ Hom(A,_); and
Extn(A,_) vanishes in injectives when n>0. (Dual
properties for ext.)
The evaluation of Extn(A,B) is independent
of the resolution of B. Covariant Ext preserves products,
contravariant ext converts coproducts to products (not vice versa).
Dimension shifting works the same way as for Tor.
(Dimension shifting takes place in the second variable
of Ext and the first variable of ext.)
Extn(A,B)≅
extn(A,B).
Ext for abelian groups:
Extn(A,B)=0 if n > 1.
We showed that Ext1(Zn,B)=B/nB and
Ext1(A,Zn) = A*/nA*,
where A* =
Hom(A,Q/Z) is the Pontrjagin dual of A.
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Mar 31
|
We discussed homology and cohomology with coefficients in an abelian group.
We began the proof of the Universal Coefficient Theorem
for Homology:
Thm. Let A be a left R-module and C be a
complex of flat right R-modules. Assume the boundary subcomplex
consists of flat modules. There is an exact
sequence
0→
Hn(C)⊗A→
Hn(C⊗A)→
Tor1(
Hn-1(C),A)→0.
If R=Z, then this exact sequence splits.
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Apr 2
|
We completed the proof of the Universal Coefficient Theorem
for Homology and stated the dual theorem for cohomology:
Thm. Let A be a left R-module and C be a
complex of projective left R-modules. Assume the boundary subcomplex
consists of projective modules. There is an exact
sequence
0→
Ext1(Hn-1(C),A)
→
Hn(Hom(C,A))
→
Hom(Hn(C),A)
→0.
If R=Z, then this exact sequence splits.
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Apr 5
|
Definition of module extensions, morphisms of extensions,
equivalence of extensions and ext(C,A).
Example:
There are only two isomorphism types of groups G for which
0 →
Zp →
G →
Zp →
0
is exact, but there are p equivalence classes of extensions.
Description of a map φ from e(C,A) to Ext1(C,A).
Proof that φ is well defined.
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Apr 7
|
Construction of the inverse map ψ from Ext1(C,A)
to ext(C,A). Proof that ψ is well defined and the inverse of
φ. Proof that the zero class of Ext corresponds to the class
of split extensions.
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Apr 9
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Construction of the Baer sum. Abelian group operations on ext(C,A).
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Apr 12
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Summary of Yoneda Ext group extn(C,A)
(similarity and equivalence of n-extensions, groups
operations on extn(C,A)).
Discussion of the evolution of
group (co)homology,
Lie (co)homology,
and Hochschild (co)homology of associative algebras.
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Apr 14
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Multilinear algebras. Ideals, abelianness, exact sequences,
actions of multilinear algebras on abelian ideals.
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Apr 16
|
Presentation of enveloping algebra of a multilinear algebra.
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Apr 19
|
Standard definitions of G-module and enveloping ring
Z[G] of a group G, the adjunction
HomRing(Z[G],R)=
HomGrp(G,R*),
module of invariants UG consisting of
all u∈U such that gu=u for all g∈G,
invariants functor U →
UG,
invariants functor
U→HomZ[G](Z,U),
example showing invariants functor need not be exact,
coinvariants functor U→UG,
coinvariants functor
U→Z ⊗Z[G] U,
coinvariants functor need not be exact.
Definitions:
♦ Hn(G,U) =
Tor1Z[G](Z,U)
♦ Hn(G,U) =
Ext1Z[G](Z,U)
Standard definitions of (A,A)-bimodule and enveloping ring
A ⊗ Aop of an associative
k-algebra A, the adjunction
Homk(A ⊗ Aop,R)=
HomD(A,R),
module of invariants UA consists
of all u∈U such that au=ua for all a∈A,
invariants functor U→UA,
invariants functor
U→Hom(A,A)(A,U),
coinvariants functor U→UA,
coinvariants functor U→A ⊗ U.
Definitions:
♦ Hn(A,U) =
Tor1
A⊗ Aop
(A,U)
♦ Hn(A,U) =
Ext1A⊗ Aop
(A,U)
Notes.
Lie definitions.
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Apr 21
|
Described the topological motivation of the different
bar resolutions for group cohomology.
Discussed homogeneous versus nonhomogeneous
and normalized versus nonnormalized
cochains. Wrote out the cocycle identity. Defined
derivations and principal derivations. Mentioned without
proof the fact that
cohomology groups can be calculated with normalized
cochains.
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Apr 23
|
We showed that elements of
Z2(Q,K) correspond to extensions
0 → K
→ G
→ Q
→ 0, an extension corresponds to an element
of B2(Q,K) iff it is split, and so
H2(Q,K) may be interpreted as the group of
equivalence classes of extensions.
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Apr 26
|
I reviewed some of the previous lecture and filled in one gap.
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Apr 28
|
We showed that elements of
Z1(Q,K) correspond to equivalences of
0 → K
→ G
→ Q
→ 0, elements of
B1(Q,K) correspond to `inner' equivalences
(those given by conjugation by an element of K, and so
H1(Q,K) may be interpreted as the group of
equivalences mod inner equivalences.
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Apr 30
|
Z1(Q,K) acts regularly on the set of complements to K
in the semidirect product K×αQ.
B1(Q,K) is the subgroup of consisting of
conjugations by elements of K. Hence, when
H1(Q,K) = 0, all complements of
K in K×αQ are conjugate
to one another.
We proved the following restrictions on
Hn(Q,K):
(1) Any exponent for K is an exponent for Hn(Q,K) for any n.
(2) The order of Q is an exponent for
Hn(Q,K) for n>0.
(3) If the order of Q and the exponent of K are relatively prime,
then Hn(Q,K)=0 for n > 0.
(4) If Q is finite and K is finitely generated as a Q-module, then
Hn(Q,K) is finite for n>0.
Call a divisor d of n a Hall divisor if gcd(d,n/d)=1.
We defined a subgroup H of a finite group G to be a
Hall subgroup of G if |H| is a Hall divisor of |G|.
Hall proved that if G is a finite solvable group
and d is a Hall divisor of |G|, then G has
a subgroup of order d and any two subgroups of order d are conjugate.
Schur-Zassenhaus Theorem: If G is finite and
K is a normal Hall subgroup, then K has a complement
and all complements of K are conjugate.
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