Topological methods in Commutative Ring Theory - Summary of
lectures and exercises
Classes: Monday, 9-11 (Kopernikusgasse 24, seminar room NT02008) and
14-16 (Kopernikusgasse 24, seminar room NT04064).
October 3, 2016 (1 hour). Introduction to the course. Algebraic
subsets of the $n$-dimensional affine space $\mathbb A^n_K$ over a
field $K$. The ideal of a subset of $\mathbb A^n_K$. Statement of
Hilbert's Nullstellensatz.
October 10, 2016 (2 hours, morning). Proof of the equivalence of
several forms of Hilbert's Nullstellensatz. G-ideals and G-domains.
The radical of an ideal $\mathfrak a$ of a ring $A$ is the
intersection of all the G-ideals of $A$ containing $\mathfrak a$.
For an integral domain $D$ with quotient field $K$, the following
conditions are equivalent: (i) $D$ is a G-domain; (ii) there is an
element $d\in D$ such that $K=D[d^{-1}]$; (iii) $K$ is of finite
type over $D$. Hilbert rings. Examples and exercises.
October 10, 2016 (2 hours, afternoon). A ring $A$ is a Hilbert ring
if and only if any radical ideal of $A$ is intersection of maximal
ideals of $A$. For a ring $A$ and a prime ideal $\mathfrak p$ of
$A$, the following conditions are equivalent: (i) $\mathfrak p$ is a
G-ideal of $A$; (ii) there exists a maximal ideal $\mathfrak m$ of
the polynomial ring $A[T]$ such that $\mathfrak p=\mathfrak m\cap
A$; (iii) there is a G-ideal $\mathfrak q$ of the polynomial ring
$A[T]$ such that $\mathfrak p=\mathfrak q\cap A$. Main theorem on
Hilbert rings: A ring $A$ is a Hilbert ring if and only if
$A[T]$ is a Hilbert ring. Furthermore, if $A$ is a Hilbert ring,
$\mathfrak q$ is any maximal ideal of $A[T]$ and $\mathfrak m:=A\cap
\mathfrak q$, then $\mathfrak q$ is generated by $\mathfrak m$
and by a polynomial $f\in A[T]$ whose canonical image in
$(A/\mathfrak m)[T]$ is irreducible. Corollary: for any field $K$,
the polynomial ring $K[T_1,\ldots,T_n]$ is a Hilbert ring. Proof of
Hilbert's Nullstellensatz.
October 17, 2016 (2 hours, morning). Some consequence of Hilbert's
Nullstellensatz. The Zariski topology on $\mathbb A^n_K$. Noetherian
spaces and their properties. The space $\mathbb A^n_K$, endowed with
the Zariski topology, is Noetherian. Irreducible spaces and
examples. A closed subset $X$ of $\mathbb A^n_K$ is irreducible if
and only if the ideal $I(X)$ of $X$ is prime. If $K$ is an infinite
field, then $\mathbb A^n_K$ is irreducible. Irreducible components.
Any space $X$ is the union of its irreducible components. If $X$ is
Noetherian, then it has only finitely many irreducible components.
Any irreducible component is closed. Irreducible components of
$Z(\{f\})\subseteq \mathbb A^n_K$, where $f\in K[T_1,\ldots, T_n]$
and $K$ is algebraically closed. Examples and exercises.
October 17, 2016 (2 hours, afternoon). Irreducible ideals. Primary
decompositions of an ideals. Any proper ideal of a Noetherian ring
has a primary decomposition. By using a primary decomposition of an
ideal $\mathfrak a$ of $K[T_1,\ldots, T_n]$ ($K$ algebraically
closed) it is possible to find the irreducible components of the
closed set $Z(\mathfrak a)\subseteq \mathbb A^n_K$. If $\mathfrak a$
is a decomposable ideal of a ring $A$ and $\mathcal P:=\{\mathfrak
q_1,\ldots, \mathfrak q_n\}$ is an irredundant primary decomposition
of $\mathfrak a$, then the set of prime ideals ${\rm Ass}(\mathfrak
a):=\{\sqrt{\mathfrak q_1}, \ldots, \sqrt{\mathfrak q_n}\}$ of $A$
depends only on $\mathfrak a$ and not on $\mathcal P$. If $\mathfrak
a$ is a decomposable ideal of a ring $A$, then there are only
finitely many prime ideals over it: they are precisely the minimal
elements of ${\rm Ass}(\mathfrak a)$. Let $X$ be an infinite
Tychonoff space and let $C(X)$ be the ring of all continuous
functions $X\longrightarrow \mathbb R$ (where $\mathbb R$ is endowed
with the euclidean topology). Then the ideal $(0)$ of $C(X)$ is not
decomposable. In particular, $C(X)$ is non Noetherian.
October 24, 2016 (2 hours, morning). Basic properties of the Zariski
topology on the prime spectrum $X:={\rm Spec}(A)$ of a ring $A$.
Description of the closure of a subset of $X$. Closed points.
Principal open subsets. Irreducible closed sets of $X$ and
irreducible components. $X$ is a compact space. Functorial
properties. The closed embedding induced by a ring surjection. The
topological embedding induced by a localization map. Compactness of
any principal open subset. Canonical homeomorphism of a closed
subset $Y$ of $\mathbb A^n_K$ (where $K$ is algebraically closed)
and the maximal spectrum of the coordinate ring $\Gamma(Y)$ of $Y$
(with the subspace Zariski topology induced by that of ${\rm
Spec}(\Gamma(Y))$. Any prime spectrum $X$ satisfies T$_0$ axiom.
Moreover $X$ is Hausdorff if and only if $X$ is T$_1$. If $A$ is a
Noetherian ring, then ${\rm Spec}(A)$ is a Noetherian space.
October 24, 2016 (2 hours, afternoon). Examples of non Noetherian
rings with Noetherian spectrum. Nagata idealization of a module. For
any ring $A$ there exists a non Noetherian ring $B$ such that ${\rm
Spec}(A)$ is homeomorphic to ${\rm Spec}(B)$. Fiber products of
rings. Description of the prime spectrum of any fiber product (under
mild assumptions), from a set-theoretic point of view. Examples.
October 31, 2016 (2 hours, morning). Disjoint union of topological
spaces. Let $f:A\longrightarrow C, g:B\longrightarrow C$ be ring
homomorphisms and let $D:=\{(a,b)\in A\times B: f(a)=g(b)\}$ be the
fiber product; identify ${\rm Spec}(C)$ with a closed subspace of
${\rm Spec}(B)$, via the closed embedding $g^\star:{\rm
Spec}(C)\longrightarrow {\rm Spec}(B)$. Then, ${\rm Spec}(D)$ is
canonically homeomorphic to the quotient space of the disjoint union
$T$ of ${\rm Spec}(A)$ and ${\rm Spec}(B)$, with respect to the
equivalence relation on $T$ generated by identifying a prime ideal
of ${\rm Spec}(C)$ with its image via $f^\star$.
October 31, 2016 (2 hours, afternoon). Review on Noetherian and
Artin modules. Faithful modules. If $A$ is a ring and there exists a
Noetherian faithful $A$-module, then $A$ is a Noetherian ring.
Formanek's Theorem: let $A$ be a ring and let $M$ be a finitely
generated faithful $A$-module such that the collection $\{\mathfrak
aM:\mathfrak a \mbox{ ideal of }A\}$ satisfies a.c.c.. Then $M$ is a
Noetherian $A$-module (and $A$ is a Noetherian ring). Eakin-Nagata's
Theorem: if $A\subseteq B$ is a finite ring extension and $B$ is a
Noetherian ring, then $A$ is a Noetherian ring. Exercises on rings
of formal power series and on the prime spectrum of fiber products.
November 7, 2016 (2 hours, morning). Necessary and sufficient
condition for a fiber product to be Noetherian. Example: given a
ring extension $A\subseteq B$ and an indeterminate $T$ over $B$,
then the ring $A+TB[T]$ is Noetherian if and only if $A$ is
Noetherian and the ring extension $A\subseteq B$ is finite. The
Krull Intersection Theorem and its corollaries.
November 7, 2016 (2 hours, afternoon). Any Noetherian ring with a
unique prime ideal is an Artin ring. The Principal Ideal Theorem: if
$A$ is a Noetherian ring and $\mathfrak p$ is a prime ideal of $A$
which is minimal over a principal ideal, then ${\rm ht}(\mathfrak
p)\leq 1$. Important corollary: let $A$ be a Noetherian ring and let
$\mathfrak p \subsetneq \mathfrak q$ be prime ideals of $A$ which
are not adjacent (i.e., there is a prime ideal $\mathfrak k$ of $A$
such that $\mathfrak p \subsetneq \mathfrak k\subsetneq \mathfrak
q$). Then there are infinitely many prime ideals $\mathfrak h$ of
$A$ such that $\mathfrak p \subsetneq \mathfrak h\subsetneq
\mathfrak q$. The principal ideal Theorem and its corollary can fail
when $A$ is not a Noetherian ring. Example of a Noetherian prime
spectrum which is not homeomorphic to the prime spectrum of any
Noetherian ring. Examples of valuation domains arising as fiber
products. Dimension of certain types of fiber products.
November 14, 2016 (2 hours, morning). The product of any nonempty
collection of fields is a zero-dimensional ring. Filters and
ultrafilters on sets. Trivial (or principal) ultrafilters. Tarski's
Lemma: any collection of subsets of a set $X$ with the finite
intersection property can be extended to an ultrafilter on $X$. In
particular, any filter can be extended to an ultrafilter. Examples
and exercises.
November 14, 2016 (2 hours, afternoon). The Stone-Cech topology on
the space $\beta X$ of all the ultrafilters on $X$. The space $\beta
X$, together with the natural topological embedding
$\iota:X\longrightarrow \beta X$, where $\iota(x)$ is the trivial
ultrafilter generated by $x$, provides a compactification of the
discrete space $X$, called the Stone-Cech compactification of $X$.
Universal property of $\beta X$: given any function
$f:X\longrightarrow K$, where $K$ is a compact and Hausdorff space,
there is a unique continuous function $\widehat f:\beta
X\longrightarrow K$ such that $\widehat f\circ\iota=f$. Consequence:
let $\{K_x:x\in X\}$ be a nonempty collection of fields and let
$A:=\prod_{x\in X}K_x$. Then, the canonical map
$\lambda:X\longrightarrow {\rm Spec}(A)$, $x\mapsto\mathfrak
m_x:=\{f\in A:f(x)=0\}$, induces (via the universal property of
$\beta X$) a unique homeomorphism $\widehat \lambda: \beta
X\longrightarrow {\rm Spec}(A)$ such that $\widehat
\lambda\circ\iota =\lambda$. Furthermore $\widehat\lambda(\mathscr
U)=\{f\in A:\{x\in X:f(x)=0\}\in\mathscr U\} $, for any ultrafilter
$\mathscr U$ on $X$.
November 21 (2 hours, morning). The constructible topology on the
prime spectrum of a ring. Basic properties: the constructible
topology is Hausdorff, totally disconnected and admits a basis
consisting of clopen sets. Ultrafilter limit points. The ultrafilter
topology on the prime spectrum of a ring. The ultrafilter topology
is compact and it is equal to the constructible topology.
November 21 (2 hours, afternoon). The constructible topology on
${\rm Spec}(A)$ is equal to the Zariski topology if and only if $A$
is zero-dimensional. The canonical map $f^\star:{\rm
Spec}(B)\longrightarrow {\rm Spec}(A)$ induced by a ring
homomorphism $f:A\longrightarrow B$ is continuous and closed, when
${\rm Spec}(B), {\rm Spec}(A)$ are endowed with the constructible
topology. Description of the closure of a set, with respect to the
constructible topology, in terms of ultrafilter limit points. A
subset $Y$ of ${\rm Spec}(A)$ is closed, with respect to the
constructible topology, if and only if $Y=f^\star({\rm Spec}(B))$,
for some ring homomorphism $f:A\longrightarrow B$ and some ring $B$.
Constructible subsets of a topological space. If ${\rm Spec}(A)$ is
Noetherian, then the constructible subsets of ${\rm Spec}(A)$ are
precisely the clopen subsets of ${\rm Spec}(A)$, endowed with the
constructible topology. Basic properties of the G-topology on
the prime spectrum of a ring and exercises. Introduction to spectral
spaces.
November 28 (2 hours, morning). Statement of Hochster's
characterization of spectral spaces. Applications; a Hausdorff space
is spectral if and only if it is compact and admists a basis
consisting of clopen sets. Any compact, Hausdorff and totally
disconnected space $X$ is spectral; precisely, $X$ is homeomorphic
to the prime spectrum of the ring $A(X)$ of continuous functions
$X\longrightarrow \mathbb F_2$, where $\mathbb F_2$ is equipped with
the discrete topology. The patch space of any topological space. If
$X$ is a spectral space and ${\rm Spec}(A)$ is homeomorphic to $X$,
for some ring $A$, then the patch space of $X$ is homeomorphic to
${\rm Spec}(A)$, endowed with the constructible topology.
November 28 (2 hours, afternoon). A topological space is spectral if
and only if it is T$_0$, has a basis of open and compact subspaces
and its patch space is compact. The ultrafilter criterion for
spectrality of a topological space. Example: if $A$ is a subring of
ring $B$ and $X$ is the set of all the subrings $C$ of $B$ such that
$A$ is a subring of $C$, equip $X$ with the topology whose subbasic
open sets are the sets of the form $U(x):=\{C\in X: x\in C\}$. Then
$X$ is spectral. Introduction to Riemann-Zariski spaces.
December 5 (2 hours, morning). Let $K$ be a field and $D$ be a
subring of $K$; then the Riemann-Zariski space ${\rm Zar}(K|D)$ of
the valuation domains of $K$ containing $D$ (as a subring) is
spectral, in view of the ultrafilter criterion. Order of domination
for local rings. If $\mathcal L$ is the set of all the local
subrings of a fixed field $L$ having quotient field $L$, and
$\mathcal L$ is ordered by domination, then: (a) maximal elements of
$\mathcal L$, under domination, are precisely the valuation domains
of $L$; (b) any element of $\mathcal L$ is dominated by some
valuation domain of $L$. Properties of the domination map
$\delta:{\rm Zar}(K|A)\longrightarrow {\rm Spec}(A)$, where $A$ is a
domain with quotient field $K$. The map $\delta$ is always
continuous and surjective; moreover, if $A$ is a Prufer domain, then
$\delta$ is a homeomorphism.
December 5 (2 hours, afternoon). $K$-Halter-Koch rings of a field
$K$. The Gaussian extension of any valuation domain of a field
$K$ to $K(T)$ is a $K$-Halter-Koch ring. Any $K$-Halter-Koch
ring $H$ is a Bézout domain with quotient field $K(T)$. Moreover, if
$f:=f_0+\ldots+ f_nT^n\in K[T]$, then $fH=f_0H+\ldots+f_nH$. The
Riemann-Zariski space of $K(T)$ over a $K$-Halter-Koch ring consists
of Gaussian extensions of valuation domains of $K$.
December 12 (2 hours, morning). Structure theorem of $K$-Halter-Koch
rings: a subring $H$ of $K(T)$ is a $K$-Halter-Koch ring if and only
if $H$ is integrally closed and ${\rm Zar}(K(T)|H)$ consists of
Gaussian extensions of valuation domains of $K$, if and only if $H$
is the intersection of a collection of Gaussian extensions of
valuation domains of $K$. Spectral representation of ${\rm
Zar}(K|D)$, where $K$ is a field and $D$ is a subring of $K$: if
$H:=\bigcap_{V\in {\rm Zar}(K|D)}V(T)$, then ${\rm Zar}(K|D)$ is
homeomorphic to ${\rm Spec}(H)$. The preorder induced by a topology.
Classification of the topologies inducing a fixed partial order.
Examples and exercises. Any order on a finite set $X$ is induced by
a unique spectral topology on $X$.
December 12 (2 hours, afternoon). There is at most one Noetherian
spectral topology inducing a given order. If $X,Y$ are Noetherian
spectral spaces and $f:X\longrightarrow Y$ is an order isomorphism
(with respect to the orders induced by the spectral topologies of
$X,Y$), then $f$ is a homeomorphism. Some techniques to construct a
ring whose prime spectrum is order isomorphic to some finite (and
easy) partially ordered set. The inverse topology on a spectral
space $X$ and its properties. The space $X^{\rm inv}$ (i.e., $X$,
endowed with the inverse topology) is T$_0$, has a basis of open and
compact subspaces and the order induced by the inverse topology is
opposite to the order induced by $X$. Hochster duality: given a
spectral space $X$, then $X^{\rm inv}$ is spectral and $(X^{\rm
inv})^{\rm inv}$ is $X$.
January 9, (2 hours, morning). The specialization of a subset of a
topological space. If $X$ is a spectral space and $Y$ is a subset of
$X$, the the closure of $Y$ (in the given spectral topology) is the
specialization of the closure of $Y$, with respect to the patch
topology of $X$. Topological approach, due to B. Olberding, to
irredundant intersections. If $D$ is a set and $A\subsetneq
C\subseteq D$ are fixed, a collection $X$ of subsets of $D$ is
called a $C$-representation of $A$ if $A=C\cap \bigcap_{B\in X}B$.
Spectral $C$-representations, existence of minimal closed
$C$-representations.
January 9 (2 hours, afternoon). Let $X$ be a spectral
$C$-representation of $A$ and let $Z\subseteq X$ be a
$C$-representation of $A$. Then, a set $B\in Z$ is irredundant in
$Z$ if and only if $B$ is irredundant in the closure of $Z$, with
respect to the patch topology of $X$. Moreover, if $B$ is
irredundant in $Z$, then $B$ is isolated in $Z$, with respect to
both the spectral and the patch subspace topology (of $Z$). Any
point of a spectral space is $\geq$ than a minimal point (with
respect to the order induced by the topology). Main properties of
minimal $C$-representations of a set.
January 16 (2 hours, morning). If $X$ is a spectral space, then the
spectral topology and the patch topology of $X$ induce the same
topology on the subspace of minimal points of $X$. If $X$ is a
spectral space and $Y$ is a subspace of $X^{\rm patch}$, then $Y$ is
a spectral subspace of $X$ and the patch topology of $Y$ is equal to
the subspace topology induced on $Y$ by the patch topology of $X$.
Strongly irredundant representations. Topological conditions on
existence and uniqueness of irredundant and strongly irredundant
representations.
January 16 (2 hours, afternoon). Scattered spaces.
Mazurkiewicz-Sierpinski's Theorem: a compact Hausdorff and countable
space is scattered. Applications. Critical sets and their
properties.
January 23 (2 hours, morning). Further properties of critical sets.
Closure of a subset of a spectral space with respect to the inverse
topology. Applications to Riemann-Zariski spaces.
January 23 (2 hours, afternoon). Irredundance of valuation domains.
Topological conditions on representations of a Prufer domain.
Introduction to Kronecker function rings.
January 30 (2 hours, morning). An order reversing bijection between
Kronecker function rings and closed representations of an integral
domain. Vacant domains. Essential domains.
January 30 (2 hours, afternoon). A vacant domain that is not Prufer.
Star operations. Prufer $v$-multiplication domains. Examples. A
topological criterion for P$v$MDs.
Problem set 1.
Problem set 2.
Problem set 3.
Problem set 4.
Problem set 5.
Problem set 6.
Problem set 7.
Problem set 8.
Problem set 9.
Problem set 10.
Problem set 11.
Deadline for the submission of the solutions of Problem set
11: January 25, 2017.
