More mathematics

Math 515, Commutative Algebra, Spring 1998

**19980114:**
Ring homomorphisms.
The natural map from R to R[[x]].
Polynomials.
Polynomial degrees.
Subrings.
The polynomial ring R[x].
Freeness of \Z[x]; polynomial evaluation.
Lack of polynomial identities in x over \Z.
Lack of polynomial identities in x over any infinite domain.
Examples of rings with polynomial identities.

**19980116:**
[Homework assignment #2.]
Bound on number of polynomial roots over domain.
R-algebras.
R-algebra homomorphisms.
Freeness of R[x].
R[x][y].
Freeness of \Z[x][y].
Lack of polynomial identities in x,y over \Z.
(0,1,-,+,.) structures.
Construction of a free (0,1,-,+,.) structure.
Proof that any (0,1,-,+,.) identity in x,y over \Z is an identity over any ring.

**19980119:**
No class.

**19980121:**
[Some-constructions-of-rings handout. Homework assignment #3.]
The category of rings.
Chopping a homomorphism into quotient, isomorphism, inclusion.
Kernels.
Intuitive definition of ideal: kernel of some ring homomorphism.
Usual definition of ideal.
Kernels as ideals.
Quotients; ideals as kernels.
Intuitive definition of prime ideal:
kernel of some ring homomorphism to some domain.
Usual definition of prime ideal.
Examples.
The image of composition of maps with a surjective map; induced maps.
The isomorphism induced by a homomorphism.
Examples.
Composition with the inclusion map; ideals lying over other ideals.
How composition affects kernels in general.
Examples.

**19980123:**
[Homework assignment #4.]
Modules and actions.
Examples of modules: vector spaces, ideals, commutative groups.
Algebras as modules.
The image of an action as an algebra.
Module homomorphisms.
Submodules.
Noetherian module:
every nonempty collection of submodules has a maximal element.
Every submodule of a Noetherian module is finitely generated.
Every ascending chain of submodules of a Noetherian module has a maximal element.

**19980126:**
Noetherian induction.
Example of Noetherian induction:
every submodule of a Noetherian submodule is a finite intersection of irreducible submodules.
Construction of Noetherian modules:
preimage is Noetherian if and only if image and kernel are Noetherian.
Restatement in terms of quotient modules.
Restatement in terms of exact sequences.
Finitely generated \Z-modules are Noetherian.

**19980128:**
[Homework assignment #5.]
Noetherian rings.
Examples of Noetherian rings.
Finitely generated free modules over a Noetherian ring are Noetherian.
Finitely generated modules over a Noetherian ring are Noetherian.
Quotients of a Noetherian ring are Noetherian.
Hilbert's basis theorem.
Examples of Hilbert's basis theorem.
Ideals of R[x] in terms of leading coefficients and low-degree elements.
Proof of Hilbert's basis theorem.
Examples of localization.
Universal mapping property of localization.

**19980130:**
[Homework assignment #6.]
Warnings about terminology: epimorphism; generated by; maximal.
Manufacturing all R-algebras as quotients of polynomial rings over R.
Manufacturing \Q as a quotient of a polynomial ring over \Z.
Manufacturing localizations in general.
Rabinowitch's localization.
Usual construction of localization of modules.
Localization of modules as modules.
Localization of ring as ring.
Localization of module as module over localization of ring.
Recovering ideal of localization from preimage in ring.
Localizations of a Noetherian ring are Noetherian.

**19980202:**
[Homework assignment #7.]
Kernel of localization.
Prime ideals of localizations.
Examples.
Localizing at a prime.
Local rings.
Comments on proper ideals being contained in maximal ideals.
Sample local-global theorem:
ideal is trivial if and only if it is locally trivial.
Spec and Maxspec.
Some maximal ideals of K[x,y].

**19980204:**
[Homework assignment #8.]
Some prime ideals of K[x,y].
What Hilbert's Nullstellensatz says.
Pictures of localizing K[x,y] at a point.
Another local-global theorem: module is zero if it is zero locally.
Principles for proving more local-global theorems.
Generalization of principles: ``flat base change,'' special case of tensor product.
Many examples of tensor products.
Universal property of tensor products.

**19980206:**
Definition of bilinear maps.
Examples of bilinear maps.
Bilinear maps in terms of Hom.
M-by-N R-modules.
M and \Z^2 versus M^2.
M-by-N R-module homomorphisms.
M^2 as a tensor product.
General definition of tensor product.
Construction of tensor product as R-module.
Bilinear map to the tensor product.
Maps from the tensor product.
Tensor product of R-module homomorphisms.
Tensor products as base change.
Universal property of base change.
Examples of base change.
Tensoring a commutative diagram with an algebra.

**19980209:**
[Homework assignment #9.]
IA as image of A tensor I in A.
Tensor product of algebras.
Universal property of tensor product of algebras.
Localization of modules as base change.
Exactness of localization.
Exactness of flat tensor products in general.
Moving between local-global principles with tensor products.
Lengths.
Exactness of length.
Length over \Z of finite module.

**19980211:**
Composition series.
Computing lengths with composition series.
Finite length implies Noetherian.
Endomorphisms of finite-length modules: injective implies surjective.
Finite-length algebras: domain implies field.
Finite-length modules from maximal ideals.
Finite-length modules from powers of finitely generated maximal ideals.
Maximal ideal annihilating a length-1 module.
Local lengths.
The Jordan-Holder theorem.
Artin rings.

**19980213:**
Class cancelled.

**19980216:**
[Homework assignment #10.]
Constructing Artin rings.
Examples of Artin rings.
Coprime ideals.
The Chinese Remainder Theorem.
The spectrum of an Artin ring is finite.
An Artin ring has finite length.
An Artin ring is the product of local Artin rings.
The determinant trick.
A map from M to IM is integral over I when M is finitely generated.

**19980218:**
[Homework assignment #11.]
Spectrum of a matrix.
Nakayama's lemma for M=IM.
Counterexample for infinitely generated module.
Nakayama's lemma for M=PM, R local.
Nakayama's lemma for M=L+PM, R local.
Counterexample for non-local ring.
Endomorphisms of finitely generated modules: surjective implies injective.
Rank of free finitely generated modules.
Alternate proof for Artin rings using length.
Extending proof to any ring with a maximal ideal.
Ford's proof for characteristic 0 using trace.
Proof for any ring using determinant.
Integral elements.
Integral algebras.
Examples of integral algebras.
Algebra finitely generated as a module is integral.

**19980220:**
Algebra generated by finitely many integral elements is
finitely generated as a module.
Arithmetic example of integral algebra.
Geometric example of integral algebra.
Integral closure.
Integral closure is an algebra.
Integral closure is idempotent.
Arithmetic example of integral closure.
Geometric example of integral closure.
Normal domains.
Various definitions of normal rings.

**19980223:**
Statement of the Goldman-Krull theorem.
Algebraically closed fields.
Hilbert's Nullstellensatz for maximal ideals.
What the Nullstellensatz says about roots of proper ideals.
Classical statement of the Nullstellensatz.
Hilbert's Nullstellensatz, strong form.
Proof of strong form with Rabinowitch's trick.
Domain under integral field is field.
Element integral over field generates field.
Element not integral over field generates polynomial ring.

**19980225:**
[Goldman-Krull handout. Homework assignment #12.]
Goldman domains.
Polynomial rings are not Goldman domains.
A one-variable Goldman extension of a field is integral.
Goldman ideals.
Jacobson rings.
The inductive step of the Goldman-Krull theorem.
Proof of the Goldman-Krull theorem.

**19980227:**
The 10-adic integers \Z_{10}.
Natural numbers in \Z_{10}.
-1 in \Z_{10}.
-1/9 and -1/7 in \Z_{10}.
The square root of 41 in \Z_{10}.
The natural logarithm of 101 in \Z_{10}.
Nontrivial idempotents in \Z_{10}.
Definition of \Z_{10}.
Universal property of \Z_{10}.
Metric on \Z_{10}.
Measure on \Z_{10}.
Ideals of \Z_{10}.
\Z_{10} as product of \Z_2 and \Z_5.
Ideals of \Z_2.
Proving that \Z_{10} is product with universal property.
Lifting idempotents from \Z/10 to \Z_{10}.

**19980302:**
[Homework assignment #13.]
Completion of rings.
The P-adic filtration of the completion.
Metric on the completion.
Convergence of series in the completion.
Power series evaluated in the completion.
Uniqueness of the R-algebra homomorphism from power series to the completion.
If b is in completion of P then 1-b is a unit.
Completion of P is maximal if P is maximal.
Newton's method.
Hensel's lemma for single-variable roots.
Newton's method for idempotents.

**19980304:**
Multivariate Newton's method.
Bairstow's method: Newton's method for polynomial factorization.
More forms of Hensel's lemma.
The graded R-algebra associated to P.
Noetherianness of associated graded algebras.
The head of an element of the completion.

**19980306:**
[Homework assignment #14.]
Noetherianness of completion.
How to analyze any ring by localization and completion.
The Cohen structure theorem.
What the Cohen structure theorem says for finite rings: nothing.
Implications among basic homological properties.

**19980309:**
Free modules.
Maps from free modules.
A free presentation of any module.
Base change on free modules.
Free modules as direct limits of finitely generated free modules.
Proof for finitely generated modules that projective over local is free.
Torsion-free modules.
Flat modules.
What flatness means for 1x2 matrices.
IA intersect JA equals (I intersect J)A when A is flat.

**19980311:**
Example of non-free projective module.
Example of non-flat torsion-free module: a nontrivial normalization.
Flatness of localization.
Restatement of flatness for maps between free modules.
Proof that flat tensor products are left exact.
Definition of the blowup algebra.
Relation between the blowup algebra and the associated graded algebra.

**19980313:**
Noetherianness of the blowup algebra.
What the Artin-Rees lemma says.
Proof of the Artin-Rees lemma.
Closure of ideals of the completion.
Flatness of the completion.
The traditional proof of flatness of the completion.

**19980316:**
No class.

**19980318:**
No class.

**19980320:**
No class.

**19980323:**
Example of computing a degree-6 characteristic polynomial.
Review of induced maps and well-defined-ness arguments.
``Defining'' or ``choosing'' an object that exists;
the axiom of generalization.
Defining a sequence of objects by recursion.
Choosing a sequence of objects by the axiom of choice.
Using the axiom of choice to prove s(R[[x]]) = (sR)[[x]].
Avoiding indices in defining a collection.

**19980325:**
[Homework assignment #15.]
The height of a prime ideal.
The dimension of a ring.
Examples: \Z, \Z localized at 2, \Z/20, \C[x,y].
How to recover Krull dimension from the lattice of all ideals.
Other uses of ``dimension'': ideals, modules.
Uses of ``codimension'': prime ideals, ideals, modules.
Dimension is local.
Dimension ignores nilpotents.
Dimension of a field is 0.
Dimension of completion.
Dimension of polynomial rings.
Dimension of finite extensions.
What Krull's principal ideal theorem says about dimension.
What Noether normalization says about dimension.
Hilbert functions and dimension.

**19980327:**
Lying over for integral extensions, local case.
Lying over, global case.
Lying over, algebras in general.
Examples.
Going up for integral extensions.
What going up says about Krull dimension.

**19980330:**
Incomparability for the 0 prime ideal in integral extensions.
Incomparability for the 0 prime ideal in integral algebras.
Incomparability for any prime ideal in integral extensions.
What incomparability says about Krull dimension.
Krull's principal ideal theorem, local case.

**19980401:**
[Homework assignment #16.]
Krull's principal ideal theorem, global case.
What Krull's principal ideal theorem says about equations in the plane.
Krull's principal ideal theorem, relative case.
Krull's principal ideal theorem, more generators.
Proof of local case for 2 generators.

**19980403:**
Systems of parameters for primes.
Prime avoidance.
Converse of the principal ideal theorem.
Systems of parameters for local rings.
Regular systems of parameters.
Regular local rings.
Nonsingular primes and singular primes.
The singular point of K[x,y]/(x^2-y^2).
A nonsingular prime of \Z.
A singular prime of \Z[x]/(x^2-5).

**19980406:**
Constructing primes of polynomial rings.
dim R[x] is at least 1 + dim R when R is nonzero.
Characterizing primes of polynomial rings lying over 0.
dim R[x] is at most 1 + dim R when R is Noetherian.
Proof of upper bound for height in terms of base and fiber.
Comments on what going down says for flat extensions.

**19980408:**
[Homework assignment #17.]
Length series.
Length series for 2\Z, 2(\Z/8), point of \C[x1,x2], singular point of \Z[x]/(x^2-5).
Hilbert series for a sequence of finitely generated modules over an Artin ring.
Proof that Hilbert series for a small graded module is a rational function.
Expressing length series in terms of Hilbert series.
Length polynomials.

**19980410:**
The Hilbert dimension of a Noetherian local ring.
Hilbert dimension is at most Krull dimension.
Hilbert dimension is at least the Hilbert dimension of a quotient.
Hilbert dimension of a domain exceeds the Hilbert dimension
of the quotient by a nonzero element of the maximal ideal.
Generalization to any serious quotient.
Hilbert dimension is at least Krull dimension.
More comments on regular local rings.

**19980413:**
Affine rings.
Nagata's trick for two variables.
Example.
Nagata's trick for more variables.
Non-Krull proof of the dimension of K[x1,x2,...,xd].
Noether normalization.
Transcendence degree.
What Noether normalization says about
the transcendence degree of an affine domain.
Comments on generalizations: the dimension formula; affine domains are catenary.

**19980415:**
Number fields.
Examples of number fields.
Number rings.
Function rings.
What the Krull-Akizuki theorem says about number rings and function rings.
The Krull-Akizuki situation.
Lengths of nontrivial quotients of finitely generated modules.
The Krull-Akizuki theorem.

**19980417:**
[Homework assignment #18.]
The ring of integers of a number field.
Dedekind domains.
Prime factorization of ideals in Dedekind domains.
The classical e, f, sum of ef.
Example: \Z[x]/(x^2+5).
Practical problem: computing the ring of integers is difficult.
Solution: work with an approximation to the ring of integers.
Defining e and f in more generality.
Finding the primes of \Z[x]/phi.
The sum of ef in general.

**19980420:**
Working with elements of \Q[x].
Working with ideals of \Q[x].
The ideal membership question.
Proving that 12x^4+7x^2+1 is in (4x^2+1)\Q[x].
Proving that 8x^3+1 is not in (4x^2+1)\Q[x].
What goes wrong for multiple generators.
Non-unique remainders for an element of (4x^2+1)\Q[x]+(8x^3+1)\Q[x].
A unique non-zero remainder for another element.
Groebner bases for ideals of \Q[x].
Reduced Groebner bases for ideals of \Q[x].
Computing reduced Groebner bases for ideals of \Q[x] with Euclid's algorithm.
Heavy dependence of Groebner bases on generators.
What goes wrong for non-principal ideals;
a Groebner basis for (x^2+1)\Z[x]+2x\Z[x].

**19980422:**
Lexicographic order of terms in \Q[x,y].
The head of a polynomial in \Q[x,y].
Groebner bases and obvious combinations of polynomials.
Example: x^2, xy+y^2 is not a Groebner basis.
Characterization of Groebner bases in terms of head divisibility.
Example of expressing element of ideal as an obvious combination
of elements of a Groebner basis for the ideal.
Remainders.
Expressing any polynomial as
an obvious combination of generators plus a remainder.

**19980424:**
Remainders produced by division.
Example of division.
Uniqueness of remainders modulo Groebner bases.
S-polynomials.
The Buchberger criterion.
Using Buchberger's criterion to prove that x^2,xy+y^2,y^3 is a Groebner basis.
Outline of Buchberger's proof.
How the proof works given x^2y^2=x(xy+y^2)-y(x^2) in terms of x^2,xy+y^2,y^3.
Summary of how to test ideal membership.

**19980427:**
Properties of extensions: incomparability, going up, going down.
Context of normal going down: domain, integral over a normal domain.
Minimal polynomials of elements integral over ideals.
Applying minimal polynomials to prime ideals.
Proof of normal going down.

**19980429:**
Speed of multiplying elements of R[x] of degree up to n.
Time for the straightforward method: n^2 multiplications in R.
Karatsuba's trick.
Speed of Karatsuba's trick.
The ring homomorphisms behind Karatsuba's trick.
General philosophy of multiplication methods.
Toom's trick.
Speed of Toom's trick.
Gauss's trick: the Fast Fourier Transform.
Speed of the FFT.

**19980501:**
The FFT in general.
What the FFT does without divisions by 2.
Nussbaumer's trick.
Radix-3 FFTs.
The Cantor-Kaltofen theorem.