Commutative Algebra (Lecture)

I followed this course during my 4th Semester at the LMU Munich for my B.Sc. Mathematics (LMU).

Following the lecture series on Algebra (Lecture), I proceed taking notes for this course too. The lecture notes begin with a general 1. Introduction, I summarise here the core of the content that is relevant for the final exam.

2. Rings and Ideals

• 2.1 def. ring, 2.2 def. ring.hom, 2.3 def. subring
  • $f:A\to B$ ring.hom, $im(f)$ subr. of $B$.
• $A,B$ rings, $B$ is a $A$-algebra (with str. map $f$) if there is an hom. $f$ s.t. $f:A\to B$. 2.4
  • if $B^{\prime }$ also $A$ but with str.map $f^{\prime }:A\to B^{\prime }$, a hom. of $A$-alg. is a func. $g:B\to B^{\prime }$ s.t. $g\circ f=f^{\prime }$
• $A$ ring, $f:A\to B$ ring.hom., for $b_1,...,b_n\in B$, def. $\tilde{f}:A[X_1,...,X_n]\to B$, s.t. $\tilde{f}{|}_A=f$, ${\forall }_{i\in [0,n]}\tilde{f}:X_i\mapsto b_i$, i.e. $\tilde{f}\circ i=f$, 2.5
  • $A$ ring, $i:A\to A[X]$ incl., if $f:A\to B$ a ring hom. and $b\in B$, then $X\mapsto b$ defines a unique ring hom. $\tilde{f}:A[X]\to B,{\sum }_{i=0}^n{a}_i{X}^i\mapsto {\sum }_{i=0}^{n}f(a_i)b^i$, if $a$ const.pol. $\tilde{f}(a)=f(a)b^0=f(a)$, hence $\tilde{f}{|}_A=f$, i.e. $\tilde{f}\circ i=f$, hence $\tilde{f}$ is an hom. of $A$-alg. (Universal Mapping Property)
  • Example: $A=k$ a field, $B=k[X,1/X]$, $X_1\mapsto X,X_2\mapsto 1/X$ def. a uniq.surj. hom of $k$-alg. $k[X_1,X_2]\to k[X,1/X]$, 2.5.a
• ideals, 2.7
  • $f$ ring.hom., $ker(f)$ is id.
  • $A/\mathfrak{a}=\{x+\mathfrak{a}:x\in A\}$ is a ring, by $(x+\mathfrak{a})(y+\mathfrak{a})=xy+\mathfrak{a}$
• for $A$ ring, $\mathfrak{a}\subseteq A$ id., $\pi :A\to A/\mathfrak{a}$ proj., $T\subseteq A$ s.t. $\overline{T}=\pi (T)=T+\mathfrak{a}$, then $\pi$ induces $\{\mathfrak{b}:\mathfrak{b}\subseteq A$ id. $\mathfrak{a}\subseteq \mathfrak{b}\}{\to }^{\cong }\{\overline{\mathfrak{b}}:\overline{\mathfrak{b}}\subseteq A/\mathfrak{a}$ id. $\}$, 2.8
• tfae: (i) $A$ fi., (ii) only id. are $(0)$ and $A$, (iii) every ring.hom $f:A\to B$ is inj (if $B\ne 0$). 2.10
  • in $\mathbb{Z}$: $(n)\cap (m)=(lcm(m,n))$, $(n)+(m)=(gcd(m,n))$, $(n)(m)=(nm)$
• $\mathfrak{a},\mathfrak{b}\subseteq A$ are coprime if $\mathfrak{a}+\mathfrak{b}=A$, 2.12
  • then ${\exists }_{x\in \mathfrak{a}}{\exists }_{y\in \mathfrak{b}}x+y=1$
• $A$ ring, ${\mathfrak{a}}_1,...,{\mathfrak{a}}_n$ ideals, $\Phi :A\to {\prod }_{i=1}^{n}A(A/{\mathfrak{a}}_i),x\mapsto (x+{\mathfrak{a}}_1,...,x+{\mathfrak{a}}_n)$ then A: i. ${\mathfrak{a}}_i,{\mathfrak{a}}_j$ are copr. for $i\ne j\Rightarrow {\prod }_{i=1}^n{\mathfrak{a}}_{\mathfrak{i}}={\bigcap }_{i=1}^n{\mathfrak{a}}_i$ \ ii. $\Phi$ is surj. $\iff {\mathfrak{a}}_i,{\mathfrak{a}}_j$ are copr. for $i\ne j$ iii. $\Phi$ is inj. $\iff {\bigcap }_{i=1}^n{\mathfrak{a}}_i=\{0\}$ (Chinese Reminder Theorem), 2.14
• for $f:A\to B$ ring.hom. $\mathfrak{a}$ id. in $A$, $\mathfrak{b}$ id. in $B$: 2.16
  • ${\mathfrak{b}}^c=f^{-1}(\mathfrak{b})\subseteq A$ is an ideal called contraction or pullback of $\mathfrak{b}$
  • ${\mathfrak{a}}^e:=\mathfrak{a}B:=\{{\sum }_{i=1}^n{y}_{i}f(x_i):x_i\in \mathfrak{a},y_i\in B\}\subseteq B$ is an ideal (generated by $f(\mathfrak{a})$), called extension or pushforward of $\mathfrak{a}$.
  • Furhtermore, it holds: 2.17
    • $\mathfrak{a}\subseteq {\mathfrak{a}}^{ec}$, ${\mathfrak{b}}^{ce}\subseteq \mathfrak{b}$
    • ${\mathfrak{b}}^c={\mathfrak{b}}^{cec}$, ${\mathfrak{a}}^e={\mathfrak{a}}^{ece}$
    • let $C=\{\mathfrak{a}:\mathfrak{a}\subseteq A$ ideal, ${\exists }_{\mathfrak{b}\subseteq B}\mathfrak{a}={\mathfrak{b}}^c\}=\{\mathfrak{a}:{\mathfrak{a}}^{ec}=\mathfrak{a}\}$, $E=\{\mathfrak{b}:\mathfrak{b}\subseteq B$ ideal, ${\exists }_{\mathfrak{a}\subseteq A}\mathfrak{b}={\mathfrak{a}}^e\}=\{\mathfrak{b}:{\mathfrak{b}}^{ce}=b\}$, set of ideals in $A$ that are contractions and extensions respectively, then $\mathfrak{a}\mapsto {\mathfrak{a}}^e$ induces a bij. C \to^\tilde E with inverse $\mathfrak{b}\mapsto {\mathfrak{b}}^c$
  • $f:A\to B$ ring.hom.: 2.18
    • $({\mathfrak{a}}_1+{\mathfrak{a}}_2{)}^e={\mathfrak{a}}_1^e+{\mathfrak{a}}_1^e$, $({\mathfrak{b}}_1+{\mathfrak{b}}_2{)}^c\subseteq {\mathfrak{b}}_1^c+{\mathfrak{b}}_2^c$
    • $({\mathfrak{a}}_1\cap {\mathfrak{a}}_2{)}^e\subseteq {\mathfrak{a}}_1^e\cap {\mathfrak{a}}_1^e$, $({\mathfrak{b}}_1\cap {\mathfrak{b}}_2{)}^c={\mathfrak{b}}_1^c\cap {\mathfrak{b}}_2^c$
    • $({\mathfrak{a}}_1{\mathfrak{a}}_2{)}^e={\mathfrak{a}}_1^e{\mathfrak{a}}_1^e$, $({\mathfrak{b}}_1{\mathfrak{b}}_2{)}^c\subseteq {\mathfrak{b}}_1^c{\mathfrak{b}}_2^c$

3. Prime Ideals

• 3.1 pr.id., max.id.
  • (i) $A/\mathfrak{p}$ ID $\iff$ $\mathfrak{p}$ pr.id., (ii) $A/\mathfrak{m}$ $\iff \mathfrak{m}$ is max.id., (iii) max.id. $\Rightarrow$ pr.id.
  • $\mathfrak{a}\subseteq \mathfrak{p}\subseteq A$, then $A/\mathfrak{p}\cong (A/\mathfrak{a})/(\mathfrak{p}/\mathfrak{a})$ and ($\mathfrak{p}$ is pr.id. $\iff \mathfrak{p}/\mathfrak{a}\subseteq A/\mathfrak{a}$ is pr.id.)
  • $\mathfrak{p}$ pr.id. $\iff$ (for any id. $\mathfrak{a},\mathfrak{b}$ then $\mathfrak{ab}\subseteq \mathfrak{p}\Rightarrow \mathfrak{a}\in \mathfrak{p}\vee \mathfrak{b}\in \mathfrak{p}$), exe.2.1.a
  • $\{{\mathfrak{p}}_{\mathfrak{i}}{\}}_{i\in [1,n]}$ all but max. 2 pr.id., $\mathfrak{a}$ id.: $\mathfrak{a}\subseteq {\bigcup }_{i=1}^n{\mathfrak{p}}_i\Rightarrow {\exists }_{i\in [1,n]}\mathfrak{a}\subseteq {\mathfrak{p}}_i$, exe.2.1.b (Prime Avoidance)
  • every ideal lies in a max.id. 3.3
  • $\mathfrak{p}⊊A$ pr.id. $\mathfrak{ab}\subseteq \mathfrak{p}\to (\mathfrak{a}\subseteq \mathfrak{p}\vee \mathfrak{b}\subseteq \mathfrak{p})$ e.2.1.a
• 3.4 def. PID
  • ($k[X]$ PID $\iff k$ fi.) but $n\ge 2\Rightarrow k[X_1,...,X_n]$ not PID
• $A$ loc.ring if $\exists {!}_{\mathfrak{m}⊊A}max.id(\mathfrak{m})$, 3.6
  • $\kappa :=A/\mathfrak{m}$ (residue field)
    • $A$ fin.gen.$k$-alg, $\mathfrak{p}\in spec(A)$, then res.fi. of $\mathfrak{p}$ is $k(\mathfrak{p}):=quot(A/\mathfrak{p})$, and tfae: exe.5.2
      1. $\mathfrak{p}$ cl.point in $spec(A)$
      2. $k\subseteq k(\mathfrak{p})$ is a fin.fi.ext.
      3. $k\subseteq k(\mathfrak{p})$ is an alg.fi.ext.
  • tfae: (i) $A$ loc.ring, (ii) ${\exists }_{\mathfrak{a}⊊A}(id.(\mathfrak{a})\wedge {\forall }_{x\in A\setminus A^{\times }}x\in \mathfrak{a})$, (iii) $A\setminus A^{\times }$ is id.
    • example: ${\mathbb{F}}_{p^k}$ is loc.ring with ${\mathbb{F}}_{p^{k-1}}$ max.id., ${\mathbb{F}}_p$ res.fi.
• $A$ ring, $X=spec(A):=\{\mathfrak{p}\subseteq A:\mathfrak{p}$ prime ideal$\}$, called the spectrum of $A$, 3.9
• $T\subseteq A$ subset, then $V(T):=\{\mathfrak{p}\in spec(A):T\subseteq \mathfrak{p}\}$
  • if $\mathfrak{a}=(T)$ id. then $V(T)=V(\mathfrak{a})$
  • $V(0)=X$, $V(1)=V(A)=\varnothing$
  • $S\subseteq T\subseteq A\Rightarrow V(T)\subseteq V(S)$
  • $V({\bigcup }_i{a}_i)=V({\sum }_i{a}_i)={\bigcap }_{i}V(a_i)$
  • $V(a\cap a^{\prime })=V(a{a}^{\prime })=V(a)\cup V(a^{\prime })$
  • $X=spec(A),\mathfrak{p}\in X\Rightarrow [\mathfrak{p}]=V(p)$
• $\mathfrak{p}\in X$ is not a closed point ( $[\mathfrak{p}]⊊\overline{[\mathfrak{p}]}$) $\iff [p]⊊V(p)\iff p⊊A\wedge \mathfrak{p}$ is not max.
• $A$ ring, $X=spec(A)$, zar.top.$X$ is s.t. cl.sets are $\{V(\mathfrak{a}):\mathfrak{a}$ id.$\}$, 3.12
  • $\{$irr. and cl. subs. in $X\}$ correspond to $\{$max ideals in $A\}$ correspond to $\{$closed points in $X\}$
  • $x\in X$ is cl. if $\{x\}\subseteq X$ is cl., i.e. if $\overline{\{x\}}=\{x\}$, 3.12
    • $\mathfrak{p}\in spec(A)$, $V(\mathfrak{p})=\overline{\{\mathfrak{p}\}}=\bigcap \{V(\mathfrak{a}):\mathfrak{a}\subseteq \mathfrak{p}\}$
    • $\mathfrak{p}$ pr.max.id. $\iff \mathfrak{p}$ cl.point in $X$ top.sp.
• $X$ top.sp. $\varnothing \ne T\subseteq X$ subs., $T$ is red. if $T=T_1\cap T_2$ and $T_i⊊T$ closed.
  • $T$ irr. $\iff$ for any $\varnothing \ne U\subseteq T$ then $T=\overline{U}\cup (T\setminus U)$
    • $\varnothing$ is not irr.
    • $\mathfrak{p}\in spec(A)$, then $\{\mathfrak{p}\}$ is irr.
      • $V(\mathfrak{p})$ is irr.cl.subs.
  • $X=spec(A)$, since $[\mathfrak{p}]\subseteq x$ is irred. (bc it’s a singleton); $[\mathfrak{p}]=V(\mathfrak{p})$ is irr.
  • $T\subseteq X$ irr. $\Rightarrow \overline{T}\subseteq X$ is irr.
  • $\mathfrak{p}\subseteq A$ pr.id. $\iff V(\mathfrak{p})\subseteq X$ is irr. and cl.
• $A$ ring: $A$ has one pr.id. $\iff$ any $a\in A$ is nilp. or $a\in A^{\times }$, exe.4.1
• $A$ ring: $spec(A)$ irr. $\iff \sqrt{(0)}$ pr.id. $\iff {\mathcal{N}}_A$ pr.id., link
• examples, 3.13
  1. $k$ fi., then $spec(k)=\{(0)\}$
  2. $p\in \mathbb{Z}$ pr.num., $\mathfrak{p}=(p)$, $spec({\mathbb{Z}}_{\mathfrak{p}})=\{(0),{\mathfrak{p}}^e\}$
     • $V({\mathfrak{p}}^e)=\{{\mathfrak{p}}^e\}$, hence it’s cl.point
     • ${\mathbb{Z}}_{\mathfrak{p}}:=\{p/q:p,q\in \mathbb{Z}\wedge q\notin \mathfrak{p}\}$, 3.8.e
       • ${\mathbb{Z}}_{\mathfrak{p}}$ is loc.ring., ${\mathfrak{p}}^e:=\{a/s:a\in \mathfrak{p}\wedge s\notin \mathfrak{p}\}$ uniq.max.id.
  3. $spec(\mathbb{Z})=\{(p):p$ pr.num.$\}\cup \{(0)\}$, generally:
     • $A$ ID $\Rightarrow spec(A)=\{$pr.id.$\}\cup \{(0)\}$ and $\{$pr.id.$\}$ are the cl.points.
       • cl.subsets are finite
• Moving from $A$ ring. to $spec(A)$ top.sp. with $V(-)$ yields the following correspondences:
  • prime ideals in $A$ $\leftrightarrow$ irreducible closed subsets of $spec(A)$
  • maximal ideals in $A$ $\leftrightarrow$ closed points of $spec(A)$

4. Noetherian Rings

• $A$ ring, is a Noetherian ring if every ideal $\mathfrak{a}\subseteq A$ is finally generated, 4.1
  • (a) $A$ is $PID\Rightarrow A$ is noeth.
    • fields, $\mathbb{Z}$, $\mathbb{Z}[i]$ and $k[X]$ are noeth. rings
  • (b) $f:A\to B$ ring.hom., $A$ noeth. then $im(f)\le B$ is noeth.
    • hence quotients of noeth. are noeth.
    • if $\overline{\mathfrak{a}}\subseteq f(A)$ ideal, there is fin.gen.id. $\mathfrak{a}=(x_1,...,x_r)$ s.t. $f(\mathfrak{a})=\overline{\mathfrak{a}}$, then $\overline{\mathfrak{a}}=(f(x_1),...,f(x_r))$
  • (c) $A=k[X_1,X_2,...]$ pol.ring. with infinite many variables, then $\mathfrak{a}:=(X_1,X_2,...)\subseteq A$ is not fin.gen.
    • if $B=k[X_1,X_2,...]$ is quot.field. for $A\to B,f\mapsto \frac{f}{1}$, so subrings with noeth.rings must not be noeth.
• for $A$ a ring: ($\mathfrak{p}$ pr.id. $\Rightarrow$ $\mathfrak{p}$ fin.gen) $\Rightarrow A$ noeth., exe.2.4.b
• $A$ noeth. then (i) $A$ has fin. many min.pr.id., (ii) every ideal in $A$ contains a min.pr.id., (iii) ${\mathcal{N}}_A={\bigcap }_{i=1}^r{\mathfrak{p}}_i$ for $\{{\mathfrak{p}}_1,...,{\mathfrak{p}}_r\}$ min.pr.id. of $A$, 8.5
• For $A$ ring, tfae: 4.3
  1. $A$ noeth
  2. every ascending chain of ideals ${\mathfrak{a}}_1\subseteq {\mathfrak{a}}_2\subseteq ...$ in $A$ becomes stationary,
     • i.e. ${\exists }_{n\in \mathbb{N}}{\forall }_{k\ge 0}{\mathfrak{a}}_n={\mathfrak{a}}_{n+k}$ (Ascending Chain Condition)
  3. any $S\ne \varnothing$ set of ideals in $A$ has a max. elem.
     • (then you can use the [[Linear Algebra II (Lecture)#Recap#12.2 Zorn’s Lemma]])
• $A$ noeth., $f:A\to A$ surj.ring.hom. $\Rightarrow f$ is aut.
• $A$ noeth. $\Rightarrow A[X_1,...,X_n]$ noeth. (Hilbert’s Basis Theorem), 4.5
• let $X$ top.sp., $X$ is noeth. if $X$ has $DCC$ on cl.subsets, 4.6
  • for $Y_i$ closed subset of $X$, $X\supset Y_1\supset Y_2\supset ...\Rightarrow {\exists }_{n\in \mathbb{N}}{\forall }_{k\in \mathbb{N}}{Y}_n=Y_{n+k}$ (DCC)
  • $X$ top.sp. tfae: 4.7
    1. $X$ is noeth.top.sp.
    2. every non empty fam. of cl.subsets has a minimal element
    3. $X$ satisfies $ACC$ for open subsets
    4. every non empty fam. of open subsets has a max.elem.
  • example 4.8:
    • $\mathbb{R}$ and $\mathbb{C}$ in usual top. are not noeth.top.sp.
    • $A$ noeth.ring. $\Rightarrow X=spec(A)$ noeth.top.sp.
    • you can have $A$ not noeth. but $spec(A)$ noeth.top.sp.
• $T\subseteq X$ irr.subs. iff $T$ cannot be written as $T=Y_1\cup Y_2$ for $Y_1,Y_2⊊T$ top.sp.
  • let $X$ top.sp. $Z\subseteq X$ is irr.comp. if $Z$ maximally irr.subset.
    • $\{$min.pr.ideals$\}$ correspond to $\{$irr.comp (i.e. max.ir.cl.subset$\}$
    • every irr.comp. is cl. (i.e. $T\subseteq X$ cl. $\Rightarrow \overline{T}\subseteq X$ cl.) exe.3.3.b
    • every irr.subset is contained in an irr.comp.
    • $A$ ID $\Rightarrow spec(A)$ irr., tut.12.4.i
    • $A$ red.ring, $spec(A)$ irr. $\Rightarrow A$ ID tut.,12.4.i
    • $spec(A)$ irr., $\Rightarrow spec(A)$ has a uniq.min.irr.comp $\Rightarrow A$ has a uniq.min.pr.id. tut.12.4.i
• $X$ noeth.top.sp. $\varnothing \ne Y\subseteq X$ cl., then there are irr.cl.subsets $Y_1,...,Y_r\subseteq Y$ s.t. (i) $Y={\bigcup }_{i=1}^r{Y}_i$, (ii) for $i\ne j$ then $Y_i⊈Y_j$, then $(Y_i{)}_{i\in [1,.r]}$ are the irr.comp. of $Y$ 4.9
  • $X$ noeth.top.sp. $\Rightarrow$ finitely many irr.comp.
  • $A$ noeth.ring., fin.min.pr.id, 4.10
    • $A$ noeth.ring., $X=spec(A)$ noeth.top.sp. $\Rightarrow X$ has fin. many irr.comp $\Rightarrow A$ has fin. many min. pr.ideals.
    • Compute irr.comp. for $A$ noeth.ring, $X=spec(A)$, find all min.pr.ideals
• for $f:A\to B$, $B$ is of finite type over $A$ if ex. surj.map $A[X_1,...,X_n]\to B$, tut.3.1.a
  • $A$ neoth, $B$ fin.typ.ov.$A$ $\Rightarrow B$ noeth.

5. Radicals

• let $A$ ring, $x\in A$ is nilp. if ${\exists }_{n\in \mathbb{N}}{x}^n=0$
  • ${\mathcal{N}}_A=\{x\in A:{\exists }_{n\in \mathbb{N}}{x}^n=0\}=\cap \{\mathfrak{p}:\mathfrak{p}\subseteq A$ is pr.id$\}=\sqrt{(0)}$ ideal in $A$
  • $A$ is red. if ${\mathcal{N}}_A=\{0\}$
    • $A$ ID $\Rightarrow A$ red.
    • $A$ red.ring, $spec(A)$ irr. $\Rightarrow A$ ID tut.,12.4.i
  • $0\ne x\in A$ nilp. $\Rightarrow x$ zerodiv.
  • $A$ has one! pr.id $\iff$ any $a\in A$ is nilp. or $a\in A^{\times }\iff A/{\mathcal{N}}_A$ fi., exe.4.1
  • $ID\iff (0)$ pr.id. wiki
• let $A$ a ring. $A_{red}:=A/\mathcal{N}$ contains no non-triv. nilp.elem., 5.3
• for $A$ ring, $\mathfrak{a}$ id.: $\mathfrak{r}(\mathfrak{a}):=\sqrt{\mathfrak{a}}:=\{x\in A:{\exists }_{n\in \mathbb{N}}{x}^n\in \mathfrak{a}\}={\bigcap }_{\mathfrak{p}\in V(\mathfrak{a})}\mathfrak{p}$ id. in $A$, 5.4
  • $\pi :A\to A/\mathfrak{a}$ quot.map., $\sqrt{\mathfrak{a}}={\pi }^{-1}({\mathcal{N}}_{A/\mathfrak{a}})$
    • $\mathfrak{a}=\sqrt{\mathfrak{a}}\iff \mathfrak{a}={\pi }^{-1}(\mathcal{N}{\ast }_{\mathcal{A}/\mathfrak{a}})\iff {\mathcal{N}}_{A/\mathfrak{a}}=\{0\}$, i.e. $\mathfrak{a}$ is rad.id. iff $A/\mathfrak{a}$ red, 5.4
  • $\mathfrak{a}$ pr.id. $\Rightarrow \mathfrak{a}$ rad.id.
  • $\mathfrak{a},{\mathfrak{a}}^{\prime }$ rad. $\Rightarrow \mathfrak{a}\cap {\mathfrak{a}}^{\prime }$ rad., tut.
• For $\mathfrak{a}$, ${\mathfrak{a}}^{\prime }\subseteq A$ ideals:, 5.5
  • $\mathfrak{a}\subseteq \mathfrak{r}(\mathfrak{a})$
  • $\mathfrak{r}(\mathfrak{r}(\mathfrak{a}))=\mathfrak{r}(\mathfrak{a})$
  • $\mathfrak{r}(\mathfrak{a}{\mathfrak{a}}^{\prime })=\mathfrak{r}(\mathfrak{a}\cap {\mathfrak{a}}^{\prime })=\mathfrak{r}(\mathfrak{a})\cap \mathfrak{r}({\mathfrak{a}}^{\prime })$
  • $\mathfrak{r}(\mathfrak{a})=(1)\iff \mathfrak{a}=A$
  • $\mathfrak{r}(\mathfrak{a}+{\mathfrak{a}}^{\prime })=\mathfrak{r}(\mathfrak{r}(\mathfrak{a})+\mathfrak{r}({\mathfrak{a}}^{\prime }))$
  • $\mathfrak{r}(\mathfrak{a}),\mathfrak{r}({\mathfrak{a}}^{\prime })$ coprime $\Rightarrow \mathfrak{a},{\mathfrak{a}}^{\prime }$ coprime
  • $\mathfrak{a}\subseteq {\mathfrak{a}}^{\prime }\Rightarrow \mathfrak{r}(\mathfrak{a})\subseteq \mathfrak{r}({\mathfrak{a}}^{\prime })$
  • for $f:A\to B$ is a ring.hom., $\mathfrak{a}\subseteq A,\mathfrak{b}\subseteq B$ id., then
    • ${\mathfrak{r}(\mathfrak{a})}^e\subseteq {\mathfrak{r}(\mathfrak{a}}^e)$
    • ${\mathfrak{r}(\mathfrak{b})}^c={\mathfrak{r}(\mathfrak{b}}^c)$
• $\mathfrak{p}$ fin.gen.pr.id.: $\sqrt{\mathfrak{a}}=\mathfrak{p}\iff {\exists }_{n\in \mathbb{N}}{\mathfrak{p}}^n\subseteq \mathfrak{a}\subseteq \mathfrak{p}$, exe.4.3.b
• $A$ PID, $\sqrt{(p_1^{n_1}\cdot ...\cdot p_r^{n_r})}=(p_1\cdot ...\cdot p_r)$, 5.6
• $A$ ring, the Jacobson rad.id. of $A$, $\mathcal{J}:={\mathcal{J}}_A:=\cap \{\mathfrak{m}:\mathfrak{m}\subseteq A$ max.id.$\}$ id. in $A$, 5.7
  • ${\mathcal{N}}_A\subseteq {\mathcal{J}}_A$
  • $A$ ring, $x\in \mathcal{J}\iff {\forall }_{y\in A}1-xy\in A^{\times }$, 5.8
• $A$ jacob.ring. if ${\forall }_{\mathfrak{p}\in spec(A)}\mathfrak{p}=\bigcap \{\mathfrak{m}:\mathfrak{m}\subseteq A$ max.id.$\wedge \mathfrak{p}\subseteq \mathfrak{m}\}$, 5.10
  • fields are jacob.rings, $\mathbb{Z}$ too.
  • $A$ jacob.ring. $\Rightarrow {\mathcal{N}}_A={\mathcal{J}}_A$
  • $A$ jacob.ring. $\Rightarrow \{$max.id.$\}$ are dense in $spec(A)$, tut.4.3
  • $A$ jacob.ring., $\mathfrak{a}$ id., $A/\mathfrak{a}$ is jacob. ring, hence:
    • ${\mathcal{N}}_{A/\mathfrak{a}}={\mathcal{J}}_{A/\mathfrak{a}}$, for each id. $\sqrt{a}=\cap \{m:\mathfrak{m}\subseteq A$ max.id.$\wedge \mathfrak{a}\subseteq \mathfrak{m}\}$, 5.10
  • $A$ jacob.ring. $\iff {\forall }_{\mathfrak{a}\subseteq A}id.(\mathfrak{a})\to {\mathcal{N}}_{A/\mathfrak{a}}={\mathcal{J}}_{A/\mathfrak{a}}$, 5.11
  • for $A$ ring s.t. (i) $A$ is ID, (ii) $A$ is noeth.ring., (iii) $\mathfrak{p}$ pr.id. $\Rightarrow \mathfrak{p}$ max.id. (iv) $|spe{c}_{max}(A)|<{\aleph }_0$, $\Rightarrow A$ is Jacob, 5.12
    • for $A$ ring, $spe{c}_{max}(A):=\{\mathfrak{m}:\mathfrak{m}\subseteq A$ max.id.},
    • $spe{c}_{rab}(A):=\{\mathfrak{m}\cap A:\mathfrak{m}\subseteq A[X]$ max.id.$\}$ 5.14
      • $spe{c}_{max}(A)\subseteq spe{x}_{rab}(A)\subseteq spec(A)$
• for $A$ ring, $\mathfrak{a}⊊A$ id., then $\sqrt{\mathfrak{a}}=\bigcap \{\mathfrak{p}:\mathfrak{p}\in spe{c}_{rab}(A)\wedge \mathfrak{a}\subseteq \mathfrak{p}\}$, 5.15

6. Affine Algebraic Geometry

• for $\iota :A\to B$ $A$-alg., then $B^{\prime }$ is $A$-subalg. if $B^{\prime }\subseteq B$ subr. s.t. $\iota (A)\subseteq V^{\prime }$, 6.1
  • for $T\subseteq B$ subset, $A[T]$ $A$-subalg. gen. by $T$, $A[T]:=\bigcap \{B^{\prime }:B^{\prime }$ $A$-subgalg. of $B$ $\wedge T\subseteq B^{\prime }\}$
    • $A[T]$ is the smallest $A$-subalg. of $B$ containing $T$
    • $A[t_1,...,t_n]:=A[T]=\{{\sum }_{i_1,...,i_n\in \mathbb{N}}{a}_{i_1,...,i_n}{t}_1^{i_1}\cdot ...\cdot t_n^{i_n}:a_{i_1,...,i_n}\in A\wedge t_i\in T$ alm. all$a_{i_1,...,i_n}=0\}$ (see Answers A. 1)
• $B$ is a fin.gen. $A$-alg. $\iff {\exists }_{n\in \mathbb{N}}A[X_1,...,X_n]/a\cong B$, 6.3
  • every ring $A$ is fin.gen. to $i{d}_A$, since $A[X]/(X)\cong A$
• for $k$ fi., $A\ne 0$ a $k$-alg, then $a\in A$ alg. is alg.over.$k$ if ex. $0\ne f\in k[X]$ s.t. $f(a)=0$, 6.4
  • if each $a\in A$ is alg.over.$k$, then $A$ is alg.over.$k$.
  • if $L/k$ is a fi.ext., then $L$ alg.over.$k$ iff ring $L$ alg.over.$k$.
• exists $f:A\to B$ ring.hom. $\iff B$ is a $A$-alg.
• $k$ field, $k\to A$ is $k$-alg., then: 6.5
  1. if $A$ ID and alg. over $k$ $\Rightarrow A$ is a field
  2. $A$ is a field and ex. $A^{\prime }$ s.t. $A^{\prime }$ fin.gen. $k$-alg and $A\subseteq A^{\prime }$ and $A^{\prime }$ ID, then $A$ is alg. over $k$.
  3. $f:A\to B$ hom. of $k$-alg., $B$ fin.gen. $k$-alg, if $\mathfrak{m}\subseteq B$ max.id., then $f(m)\subseteq A$ is max.id.
• 6.5 generalised: $A$ jac.ring., $B$ fin.gen.$A$-alg. $\Rightarrow B$ jac.ring., exe.5.4
  • for any $\mathfrak{m}\subseteq B$, ${\mathfrak{m}}^c\subseteq A$ max.id., and $A/{\mathfrak{m}}^c\to B/\mathfrak{m}$ is fin.fi.ext.
• $k$ fi., ${\mathbb{A}}_k^n:=\{(a_1,...,a_n):a_i\in k\}$ with structure as exposed below, 6.6
  • as sets ${\mathbb{A}}_k^n=k^n$, though we give them a different structure
  • for $P=(a_1,...,a_n)\in {\mathbb{A}}_k^n\Rightarrow {\mathfrak{m}}_p:=(X_1-a_1,...,X_n-a_n)$ max.id., (3.5.d)
  • if $k=\overline{k}$ and $\mathfrak{m}\subseteq k[X_1,...,X_n]$ is max.id., then ${\exists }_{P\in {\mathbb{A}}_k^n}\mathfrak{m}={\mathfrak{m}}_P$, 6.7
• $k$ fi., $T:={\mathcal{V}}_{{\mathbb{A}}_k^n}(S):=\{P\in {\mathbb{A}}_k^n:{\forall }_{f\in S}f(P)=0\}$, 6.8
  • if $\mathfrak{a}=(S)$, then ${\mathcal{V}}_{{\mathbb{A}}_k^n}(\mathfrak{a})={\mathcal{V}}_{{\mathbb{A}}_k^n}(S)$
  • if $\mathfrak{a}\subseteq k[X_1,...,X_n]$ id., $\mathfrak{a}=(f_1,...,f_n)$
    • $k[X_1,...,X_n]$ is noeth.ring.
  • $\mathcal{V}(f)$ for $f\in k[X_1,...,X_n]$
  • $\mathcal{V}((f_1,...,f_r))\times \mathcal{V}((g_1,...,g_s))=\mathcal{V}((f_1,...,f_r,g_1,...,g_s))$ for $f_i\in k[X_1,...,X_m]$ and $g_i\in k[Y_1,...,Y_n]$
• $k$ fi., $S\subseteq k[X_1,...,X_n]$, ${\mathcal{M}}_S:=\{\mathfrak{m}:\mathfrak{m}\subseteq k[X_1,...,X_n]$ max.id. $S\subseteq \mathfrak{m}\}$, $\Phi :{\mathcal{V}}_{{\mathbb{A}}_k^n}(S)\to {\mathcal{M}}_S,P\mapsto {\mathfrak{m}}_P$ is inj., 6.10
  • in particular: ${\mathcal{V}}_{{\mathbb{A}}_k^n}(S)\subseteq spe{c}_{max}(k[X_1,...,X_n])\subseteq spec(k[X_1,...,X_n])$
• $k$ alg.cl.fi. then: (a) $\Phi$ is bij., (b) $\mathfrak{a}⊊k[X_1,...,X_n]$ id., then ${\mathcal{V}}_{{\mathbb{A}}_k^n}\ne \varnothing$, 6.11
  • $\mathfrak{a}=(x^2+1)⊊\mathbb{R}[X]$ has ${\mathcal{V}}_{{\mathbb{A}}_{\mathbb{R}}^2}=\varnothing$
    • (b) says that there always are common solutions, and it is necessary that $k$ is alg.cl.
  • $\mathcal{I}(T)$ is rad.id. (by 5.6.b)
  • for $T=\{p\}$, $p\in {\mathbb{A}}_k^n\Rightarrow \mathcal{I}(T)=I(p)={\mathfrak{m}}_p$
    • (if we took ${\mathcal{I}}^{\ast }(T):=\{f\in k[X_1,...,X_n]:{\exists }_{p\in T}f(p)=0\}={\mathcal{V}}^{-1}(T)$, I think)
  • $k$ fi., $V\subseteq {\mathbb{A}}_k^n$: $V$ irr.top.sp. $\iff \mathcal{I}(V)$ pr.id., exe.5.3
• $k$ alg.cl.fi., $\mathcal{I}(\mathcal{V}(\mathfrak{a}))=\sqrt{a}$ , (Hilb. Nullstellensatz, II), 6.13
• $k$ fi., $A$ fin.gen. $k$-alg., $A$ is jac.ring., 6.14
  • if $\mathfrak{a}\subseteq A$ id., then $\sqrt{\mathfrak{a}}:=\bigcap \{\mathfrak{m}:\mathfrak{m}\subseteq A$ max.id. $\mathfrak{a}\subseteq \mathfrak{m}\}$
• $k$ alg.cl.fi., $n\ge 1$, then $\mathfrak{a}\mapsto \mathcal{V}(\mathfrak{a})$ and $V\mapsto \mathcal{I}(V)$ induce a bij.: $\{\mathfrak{a}\subseteq k[X_1,...,X_n]:\mathfrak{a}$ rad.id.$\}\leftrightarrow \{$affine $k$-vanches $V\subseteq {\mathbb{A}}_k^n\}$
• $k$ fi., $V\subseteq {\mathbb{A}}_k^n$, aff.$k$-var., if $\mathcal{I}(V)\subseteq k[X_1,...,X_n]$ is van.id of $V$, then the coord.ring. of $V$ is the $k$-alg. $A(V)=k[X_1,...,X_n]/\mathcal{I}(V)$, 6.16
  • coord.ri. is fin.gen. and red. $k$-alg.
  • $V=\mathcal{V}(\mathfrak{a})$ for $\mathfrak{a}\subseteq k[X_1,...,X_n]$, then $\mathfrak{a}\subseteq \mathcal{I}(\mathcal{V}(\mathfrak{a}))=\mathcal{I}(V)$
    • if $k$ alg.cl.fi. identity holds: $k[X_1,...,X_n]/\mathfrak{a}=k[X_1,...,X_n]/\mathcal{I}(V)=A(V)$
• $k$ alg.cl.fi., $A$ fin.gen.red.$k$-alg., then ex. $V$ aff.$k$-var. s.t. $A\cong A(V)$, 6.17
  • fin.gen.red.$k$-alg. are called aff.$k$-alg.
  • $A(V)\cong A(W)\iff V\cong W$ (p. 34)
• $k$ fi., $V\subseteq {\mathbb{A}}_k^n$ aff.$k$-var., a subvar. $W\subseteq V$ s.t. $W$ $k$-var., i.e. $W=\mathcal{V}(\mathfrak{a})$ for $\mathfrak{a}$ id. in $k[X_1,...,X_n]$, 6.18
  • $W\subseteq V$ subv., then $\mathcal{I}(V)\subseteq \mathcal{I}(W)\subseteq k[X_1,...,X_n]$
    • and: $\mathcal{I}(W)/\mathcal{I}(V)$ id. in $A(V)$
    • $A(V)/(\mathcal{I}(W)/\mathcal{I}(V))\cong A(W)$ is red., hence $\mathcal{I}(W)/\mathcal{I}(V)$ is rad.id.
  • for $\bar{f}=f+\mathcal{I}(V)\in A(V)$, ${\mathcal{V}}_V(\mathcal{J})=\{P\in V|{\forall }_{\bar{f}\in \mathcal{J}}\bar{f}(P)=0\}\subseteq V$ subvar.
• for $k$ alg.cl.fi. $V\subseteq {\mathbb{A}}_k^n$ aff.$k$-var., then $W\mapsto \mathcal{I}(W)/\mathcal{I}(V)$ and $\mathcal{J}\mapsto {\mathcal{V}}_V(\mathcal{J})$ are bijections, 6.19
  • $\{W:W\subseteq V$ subvar. $\}\leftrightarrows \{\mathcal{J}:\mathcal{J}\subseteq A\subseteq A(V)$ rad.id. $\}$
  • if $\mathcal{J}(W)\subseteq A(V)$ id. in subvar. $W\subseteq V$, then $A(V)/\mathcal{J}(W){\to }^{\cong }A(W),f+\mathcal{J}(W)\mapsto f{|}_W$ isom.
• $\varnothing \ne V\subseteq {\mathbb{A}}_k^n$ aff.$k$-var., then $V$ irr. $\iff \mathcal{I}(V)\subseteq k[X_1,...,X_n]$ is pr.id., (cf. exe.5.3) 6.20
• Finally, the maps $\mathcal{I}(-)$ (left to right), ${\mathcal{V}}_{{\mathbb{A}}_n^k}(-)$ (right to left) induce the following correspondences:
  1. closed subsets of $V$ $\leftrightarrow$ radical ideals in $A(V)$
  2. irreducible closed subsets of $V\leftrightarrow$ prime ideal in $A(V)$
  3. irreducible components in of $V\leftrightarrow$ minimal prime ideals in $A(V)$
  4. points of $V\leftrightarrow$ minimal ideals in $A(V)$

7. Modules

• $A$ ring., a $A$-mod. is a ab.gr. on + and a scal.mult. $A\times M\to M,(a,m)\mapsto am$ s.t., 7.1
  1. $a(m+m^{\prime })=am+a{m}^{\prime }$
  2. $(a+a^{\prime })m=am+a^{\prime }m$
  3. $(a{a}^{\prime })m=a(a^{\prime }m)$
  4. $1m=m$
  • for $M$ $A$-mod., $N$ $A$-submod. of $M$ if $N\subseteq M$ subgr. s.t. ${\forall }_{a\in A}{\forall }_{n\in N}an\in N$, 7.1
  • examples, 7.2:
    • (i) $(\mathfrak{a},{\cdot }_A)$ with scalar mult., (ii) $A$ fi., mod. are exactly vec.sp., (iii) $\mathbb{Z}$-mod. are exactly ab.gr. (iv) $(A^n,{+}_A)$ with comp.wise mult.
• for $M,N$ $A$-mod, $A$ morph./lin.map. is $f:M\to N$ s.t., 7.3
  1. ${\forall }_{m,m^{\prime }\in M}f(m+m^{\prime })=f(m)+f(m^{\prime })$
  2. ${\forall }_{a\in A}{\forall }_{m\in M}f(am)=af(m)$
  • $(Hom(M,N),{+}_f)$is $A$-mod. for $(f+g)(m)=f(m)+g(m)$, $(af)(m)=af(m)$
    • $Ho{m}_A(M,N):=\{f:M\to N:f$ is $A$-lin.$\}$
      • $M$ an $A$-mod., $M\cong Ho{m}_A(A,M)$, exe.6.2.a
      • $\mathfrak{a}\subseteq A$ id., $Ho{m}_A(A/\mathfrak{a})\cong M[\mathfrak{a}]:=\{m\in M:{\forall }_{a\in \mathfrak{a}}am=0\}$, exe.6.2.b
      • $m,n\in \mathbb{N}$, then $Ho{m}_{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/n)\cong \mathbb{Z}/gcd(m,n)$ as $A$-mod., exe.6.2.c
    • $(f{+}_{f}g)(m):=f(m){+}_{N}g(m)$ and $(a{\cdot }_{f}f)(m)=a{\cdot }_{N}f(m)$
  • for $f\in Ho{m}_A(M,N)$, then $ker(f)\subseteq M$ and $im(f)\subseteq N$ are $A$-submod.
  • if $M$ is $A$-mod, $N\subseteq M$ $A$-submod., then $M/N$ is $A$ mod. with scal.mult. $a(m+N)=am+N$.
    • $\pi :M\to M/N,m\mapsto m+N$ is surj.$A$-lin.map
    • Homomorphism Theorem holds too
  • (i) ${\bigcap }_{i\in I}{M}_i$, (ii) ${⨁}_{i\in I}{M}_i$, (iii) ${\prod }_{i\in I}{M}_i$, are $A$-mod., (v) $M_1\cup M_2$ is not.
    • ${⨁}_{i\in I}{M}_i:=\{\{m_i{\}}_I\in {\prod }_{i\in I}{M}_i:|\{i\in I:m_i\ne 0\}|<{\aleph }_0\}$,
      • hence $|I|<{\aleph }_0\Rightarrow {⨁}_{i\in I}{M}_i={\prod }_{i\in I}{M}_i$, check Answers, A.3
    • $\mathfrak{a}M:=\{{\sum }_{i=1}^n{a}_i{m}_i:a_i\in \mathfrak{a}\wedge m_i\in M\}\subseteq M$ sub.$A$-mod.
• $\alpha :{⨁}_{i\in I}Ho{m}_A(M,N_i)\to Ho{m}_A(M,{⨁}_{i\in I}{N}_i$ inj., tut.6.2
  • $M$ fin.gen. $\Rightarrow \alpha$ isom.
• for $M,N$ $A$-mod., $(N:M):=\{a\in A:aM\subseteq N\}\subseteq A$ id., 7.4
  • $An{n}_A(M):=(0:M)=\{a\in A:aM=0\}\subseteq A$ id.
  • call $M$ $A$-mod. faithful if $An{n}_A(M)=\{0\}$
  • if $M$ is $A$-mod. $\mathfrak{a}\subseteq A$ is id. s.t. $\mathfrak{a}\subseteq An{n}_A(M)$, then $M$ is a faithful $A/\mathfrak{a}$-mod.
    • if $\mathfrak{a}=An{n}_A(M)$ then $An{n}_{A/\mathfrak{a}}(M)=\{a+\mathfrak{a}:(a+\mathfrak{a})M=0\}=\{a+\mathfrak{a}:aM=0\}=\{0\}$
• for $M$ a $A$-mod., then:, 7.5
  1. for $m\in M$, $Am:=\{am:a\in A\}\subseteq M$ is a cyclic $A$-submod.gen.by.$m$
  2. $\{m_i{\}}_{i\in I}\subseteq M$ generates $M$ if ${\forall }_{m\in M}{\exists }_{\{a_i{\}}_I\subseteq A}m={\sum }_{i\in I}{a}_i{m}_i$.
  3. $\{m_i{\}}_{i\in I}\subseteq M$ is a basis if ${\forall }_{m\in M}\exists {!}_{\{a_i{\}}_I\subseteq A}|\{i\in I:a_i\ne 0\}|<{\aleph }_0\wedge m={\sum }_{i\in I}{a}_i{m}_i$, $rank(M):=|I|$, check Answers A.4
     • if $M$ has a basis, call it free
       • example ${⨁}_{i=1}^n{A}_i$ is free.$A$-mod., $rank({⨁}_{i=1}^n{A}_i)=n$
       • $M$ free, then each $f\in Hom(M,N)$ is uniq.det. by $\{f(m_i):i\in I\}$.
         • like vec.sp., then you can use matrixes, see Linear Algebra II (Lecture).
       • $M$ free $\Rightarrow {\exists }_{n\in \mathbb{N}}{⨁}_{i=1}^n{A}_i\cong M$
         • like vec.sp., $di{m}_k(V)=n\iff V\cong k^n$, Linear Algebra II (Lecture)
       • examples, 7.6:
         • $M$ fin.abl.gr. is $\mathbb{Z}$-mod. $\Rightarrow M$ fin.gen. and not free
         • $A$-submod of free $A$-mod. must not be free
• $A$-mod. $P$ is proj. if ex. $Q$ $A$-mod. s.t. ${\exists }_{n\in \mathbb{N}}{⨁}_{i=1}^n{A}_i\cong P\oplus Q$, 7.7
  • free $\Rightarrow$ proj.
  • $A$ PID, $P$ proj.$A$-mod. $\Rightarrow P$ is free, 7.8.b
  • $P$ proj.$k[X_1,...,X_n]$-mod $\Rightarrow P$ is free 7.8.d
  • $P$ proj. $A$-mod., $S$ mult.cl.subs., then $S^{-1}P$ is proj.$S^{-1}A$-mod., tut.12.4.xi
  • $M,N$ $A$-mod., $M{\otimes }_{A}N=A\Rightarrow M$ proj., tut.12.4.xii
• for $A$ ring, $P$ an $A$-mod. tfae:. 7.9
  1. $P$ is proj.
  2. if $M,N$ are $A$-mod., $g:M\to N$ is surj.$A$-lin.map., $h:P\to N$ is $A$-lin.map., then ex. $A$-lin.map $f:P\to M$ s.t. $g\circ f=h$.
  3. if $M$ is $A$-mod., $f:M\to P$ is surj.map. $A$-lin.map., then ex. $s:P\to M$ s.t. $s$ $A$-lin.map. s.t. $f\circ s=i{d}_P$.
  4. if $M,N$ are $A$-mod., $g:M\to N$ is surj.$A$-lin., $g_{\ast }:Ho{m}_A(P,M)\to Ho{m}_A(P,N)$ surj., exe.7.3
• $M/N$ proj., $L$ $A$-mod., the incl. $i:N\to M$ induces a surj.map. $Ho{m}_A(M,L)\to Ho{m}_A(N,L)$, exe.7.4
• $A$ ring., $M$ a fg.$A$-mod., $\mathfrak{a}\subseteq A$ id. s.t. $\mathfrak{a}M=M$, then ${\exists }_{a\in \mathfrak{a}}(1-a)\cdot M=0$, (Nakayama) 7.10
  • for $M$ fg.$A$-mod., $\mathfrak{a}\subseteq A$ id. s.t. $\mathfrak{a}\subseteq {\mathcal{J}}_A$, if $\mathfrak{a}M=M$, then $M=0$, 7.11
  • for $M$ fg.$A$-mod., $\mathfrak{a}\subseteq A$ id. s.t. $\mathfrak{a}\subseteq {\mathcal{J}}_A$, if $N\subseteq M$ is $A$-submod. s.t. $M=\mathfrak{a}M+N$, then $N=M$
  • $A$ loc.ring. with $\mathfrak{m}$ max.id. $\kappa =A/\mathfrak{m}$, $M$ fin.gen.$A$-mod., $\{m_1,...,m_n\}\subseteq M$ s.t. $\{\overline{m_1},...,\overline{m_n}\}\subseteq M/\mathfrak{m}$ form a $\kappa$-bas. of $M/\mathfrak{m}M$, then $\{m_1,...,m_n\}$ gen. $M$ as an $A$-mod., 7.13
  • $A$ ring., $M$ fin.gen.$A$-mod., $f:M\to M$ $A$-lin. and surj. then $f$ is isom., 7.14
• for $M,N,P$ $A$-mod $f:M\times N\to P$ is $A$-bil. if (i) $f(m+m^{\prime },n)=f(m,n)+f(m+n)$, (ii) $f(m,n+n^{\prime })=f(m,n)+f(m,n^{\prime })$, (iii) $f(am,n)=af(m,n)=f(m,an)$, (see Linear Algebra II (Lecture)) 7.16
• for $M,N$ $A$-mod., then, 7.17
  1. ex. $A$-mod. $T=M{\otimes }_{A}N$ and a $A$-bil. $g:M\times N\to T$ s.t. for $P$ $A$-mod., $f:M\times N\to P$ $A$-bil., then ex! $f^{\prime }:T\to P$ st. $f=f^{\prime }\circ g:M\times N\to P$. (see diagram on notes)
  2. pair $(T,g)$ is unique up to isom.
  3. if $f:M\to M^{\prime }$ is $A$-lin. then $f$ induces a $A$-lin.map $f_{\ast }:M{\otimes }_{A}N\to M^{\prime }{\otimes }_{A}N$, if $g:M^{\prime }\to M^{\prime \prime }$ is $A$-lin. then $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$
     • $M,N$ fin.gem$A$-mod., then $M{\otimes }_{A}N$ fin.gen.
       • since $M$ gen by $\{e_i:i\in I\}$, $N$ by $\{f_j:j\in J\}$ then $M{\otimes }_{A}N$ is gen by $\{e_i\otimes f_j:i\in I\wedge j\in J\}$.
     • for $A$ ring., $M,M^{\prime },N$ $A$-mod., $f\in Ho{m}_A(M,M^{\prime })$, then: exe.6.1
       1. $f^{\ast }:Ho{m}_A(M^{\prime },N)\to Ho{m}_A(M,N),g\mapsto g\circ f$
          • $f$ surj. $\Rightarrow f^{\ast }$ inj., exe.6.1
       2. $f_{\ast }:Ho{m}_A(N,M)\to Ho{m}_A(N,M^{\prime }),g\mapsto f\circ g$
• example, 7.19:
  • $m\otimes 0=0\otimes n=0$ by linearity $g$
  • $M{\otimes }_{A}N=0\Rightarrow {\forall }_{m\in M}{\forall }_{n\in N}m\otimes n=0$
  • $A$ ab.tor.gr., then $\mathbb{Q}{\otimes }_{\mathbb{Z}}A=0$, since ${\forall }_{a\in A}{\exists }_{n\in \mathbb{N}}na=0$.
    • $A$ tor.gr. if ${\forall }_{a\in A}{\exists }_{n\in \mathbb{N}}n\cdot a=0$
• $A/\mathfrak{a}{\otimes }_{A}M\cong M/\mathfrak{a}M$, exe.8.1
• $M,N,P$ $A$-mod., then:, 7.20
  • (i) $M{\otimes }_{A}N\cong N{\otimes }_{A}M$, $m\otimes n\mapsto n\otimes m$
  • (ii) $(M{\otimes }_{A}N){\otimes }_{A}P\cong M{\otimes }_A(N{\otimes }_{A}P)$, $(m\otimes n)\otimes p\mapsto m\otimes (n\otimes p)$
  • (iii) $(M\oplus N){\otimes }_{A}P\cong (M{\otimes }_{A}P)\oplus (N{\otimes }_{A}P)$, $(m,n)\otimes p\mapsto (m\otimes p,n\otimes p)$
  • (iv) $(A{\otimes }_{A}M)\cong M$, $a\otimes m\mapsto am$
• $M,N$ fin.gen.$A$-mod. s.t. $M{\otimes }_{A}N=0$, then $M=0\vee N=0$, exe.9.2
• for $F$, $F^{\prime }$, with bases $\{e_i{\}}_I$, $\{e_j{\}}_J$, then $F{\otimes }_A{F}^{\prime }$ has basis $\{e_i\otimes e_j:i\in I,j\in J\}$, 7.20.a
  • $A^m{\otimes }_A{A}^n\cong A^{mn}$ (see fin.gen.free.$A$-mod., 7.5.3)
  • $M$ $A$-mod. and $F$ free.$A$-mod. with basis $\{e_i{\}}_I$, then ${\forall }_{m\otimes f\in M{\otimes }_{A}F}\exists {!}_{\{m_i{\}}_I\subseteq M}m\otimes f={\sum }_{i=1}^n{m}_i\otimes e_i$
• for $k$ alg.cl.fi. $V\subseteq {\mathbb{A}}_k^m$, $W\subseteq {\mathbb{A}}_k^n$ aff.$k$-var., $V\times W\subseteq {\mathbb{A}}_k^{m+n}$, then $A(V\times W)\cong A(V){\otimes }_{k}A(W)$, 7.22
• $A,B$ rings, $(A,B)$-bimod. $M$ is an ab.gr., is a $A$-mod. and $B$-mod. s.t. $a(mb)=(am)b$, 7.23
  • example, 7.24: (i) $M$ $A$-mod, then $M$ $(A,A)$-bimod., (ii) $\mathbb{Q}$ is $(\mathbb{Z},\mathbb{Q})$-bimod.
• $A,B$, $M$ an $A$-mod., $N$ an $(A,B)$-bimod., then $(M{\otimes }_{A}N){\otimes }_{B}P\cong M{\otimes }_A(N{\otimes }_{B}P)$ ab.subgr., 7.25
• $\Phi :ho{m}_A(M{\otimes }_{A}N,P)\to ho{m}_A(M,ho{m}_A(N,P)),\Phi (f)(m)(n)=f(m\otimes n)$ is $A$-lin. isom., 7.26
• let $...\to M_{i-1}{\to }^{f_{i-1}}{M}_i{\to }^{f_i}{M}_{i+1}{\to }^{f_{i+1}}{M}_{i+1}\to ...$ be a seq. $A$-mod. and $A$-lin.fun.:, 7.27
  • is exact at $M_i$, if $im(f_{i-1})=ker(f_i)$.
  • is exact if it is in every module.
  • $0\to M^{\prime }{\to }^{f}M$ is exact $\iff f$ inj
  • $M{\to }^g{M}^{\prime \prime }\to 0$ is exact $\iff g$ surj.
  • $0\to M^{\prime }{\to }^{f}M{\to }^g{M}^{\prime \prime }\to 0$ is exact $\iff f$ inj., $g$ surj., $im(f)=ker(g)$
  • $...\to M_{i-1}{\to }^{f_{i-1}}{M}_i{\to }^{f_i}{M}_{i+1}\to ...$ is exact $\iff {\forall }_{i}exac.(0\to im(f_{i-1})\to M_i\to im(f_i)\to 0)$
  • $...\to M_{i-1}\to M\to M_{i+1}\to ...$ exact if there is an pointw.comm.isom. to an exc.seq.
  • if $0\to M^{\prime }{\to }^u\to M{\to }^v{M}^{\prime \prime }\to 0$ is exact, for $N$ $A$-mod., then: 7.28
    • $0\to Ho{m}_A(N,M^{\prime }){\to }^{u_{\ast }}Ho{m}_A(N,M){\to }^{v_{\ast }}Ho{m}_A(N,M^{\prime \prime })$ is exact
      • i.e. $Ho{m}_A(N,-)$ is left exact
    • $M^{\prime }{\otimes }_{A}N{\to }^{u_{\ast }}M{\otimes }_{A}N{\to }^{v_{\ast }}{M}^{\prime \prime }{\otimes }_{A}N\to 0$ is exact
      • i.e. $-{\otimes }_{A}N$ is right exact
• $f:M\to N$, hom., $coker(f):=N/im(f)$, 7.29
  • $coker(f)=0\iff f$ surj.
  • for each $f$, exists $0\to ker(f)\to M{\to }^{f}N\to coker(f)\to 0$ that is exact
• for a snake comm.diagram (7.30) with exact rows, then the following is exact (snake lemma) $0\to ker(f^{\prime })\to ker(f)\to ker(f^{\prime \prime }){\to }^{\delta }coker(f^{\prime })\to coker(f)\to coker(f^{\prime \prime })\to 0$, 7.30
• for a 5-lemma.comm.diagr. with exact rows: 7.31
  • $f_1$ surj., $f_2,f_4$ inj., $\Rightarrow f_3$ inj.
  • $f_5$ inj., $f_2,f_4$ surj., $\Rightarrow f_3$ surj.
• for ($\star$): $0\to M^{\prime }{\to }^{i}M{\to }^p{M}^{\prime \prime }\to 0$ sh.exa.seq. of $A$-mod., tfae: 7.32
  1. $M\cong M^{\prime }\oplus M^{\prime \prime }$, $i$ is incl, $p$ proj. on the sec. summand.
  2. exists $t:M\to M^{\prime }$ s.t. $t\circ i=i{d}_{M^{\prime }}$
  3. exists $s:M^{\prime \prime }\to M$ s.t. $t\circ s=i{d}_{M^{\prime \prime }}$
  4. ($\star$) is split exact
  • $M^{\prime \prime }$ proj., $\Rightarrow$ ($\star$) split exact
• if ($\star$) sh.spl.exa.seq., $N$ $A$-mod., $0\to M^{\prime }{\otimes }_{A}N{\to }^{i\otimes 1}M{\otimes }_{A}N{\to }^{p\otimes 1}{M}^{\prime \prime }{\otimes }_{A}N\to 0$ exact
• $N$ is $M$-flat if every inj.$A$-lin.map. $M^{\prime }\to M$, then $M^{\prime }{\otimes }_{A}N\to M{\otimes }_{A}N$ inj., 7.34
  • $N$ is $M$-fl. $\iff$ for any fin.gen.subm. $M^{\prime }\subseteq M$, the nat.map. $M^{\prime }{\otimes }_{A}N\to M{\otimes }_{A}N$ inj., exe.9.4.b
  • for all $i\in I$, $N$ is $M_i$-fl. $\Rightarrow$ $N$ is $({⨁}_{i\in I}{M}_i)$-fl., exe.9.4.c, 7.37
  • $N$ is flat if it is $M$-flat for every $A$-mod.
  • free $\Rightarrow$ proj. $\Rightarrow$ flat $\Rightarrow$ tor.fr, 7.35-39
  • $A$ PID: free $\iff$ proj $\Rightarrow$ flat $\iff$ tor.fr. 7.39.f
  • $A$ loc.ring: proj. $\Rightarrow$ free (Kaplansky), exe.9.3
  • quot. of $M$-fl is $M$-fl., 7.36
    • $N$ is $M$-fl. $\Rightarrow N$ is $M^{\prime }$-fl. for $M^{\prime }$ quot. of $M$, exe.9.4
  • $N$ $A$-mod., tfae: (i) $N$ is $A$-fl., (ii) $N$ is fl., (iii) $-{\otimes }_{A}N$ preserves exactness on sh.exa.seq., 7.38
    • (iii): $M^{\prime }{\to }^{u}M{\to }^v{M}^{\prime \prime }$ exact $\Rightarrow M^{\prime }{\otimes }_{A}N{\to }^{u\otimes 1}M{\otimes }_{A}N{\to }^{v\otimes 1}{M}^{\prime \prime }\otimes N$ exact.

8. Dimensions

• for $\varnothing \ne X$ top.sp., then $dim(X)=\sup \{n:X_0⊊X_1⊊...⊊X_n\subseteq X$ irr.cl.$\}$, 8.1
  • for $A$ ring, $dim(A):=dim(spec(A))$
    • equivalently: $dim(A)=\sup \{n:{\mathfrak{p}}_0⊊...⊊{\mathfrak{p}}_n\wedge {\forall }_{i\in [1,n]}{\mathfrak{p}}_i\subseteq A$ pr.id$\}$
  • for $T\subseteq {\mathbb{A}}_k^n$, $T={\bigcup }_{i=1}^r{T}_i$, then $dim(T)=\max \{dim(T_i):i\in [1,r]\}$
    • both $dim(T)$ and $dim(T_i)$ are defined as subspaces with irr.cl.subsets.
  • for $V\subseteq {\mathbb{A}}_k^n$ aff.var.ov.$k$ for $k$ alg.cl.fi., then $dim(V)=dim(A(V))$
  • examples: 8.2
    • $A$ PID and not a field, $dim(A)=1$
      • $dim(k)=0$
      • $A$ noeth.ring $\Rightarrow dim(A[X_1,...,X_n])=dim(A)+n$, 8.5
    • $dim(k[X_1,...,X_{\omega }])=\infty$
    • $A$ noeth.loc.ring $\Rightarrow dim(A)<\infty$, 9.15
    • $A$ noeth., $dim(A)\ge 2\Rightarrow |spec(A)|=\infty$, exe.14.2
• $\{a_1,...,a_n\}\subseteq A$ alg.ind. if ${\forall }_{0\ne f\in k[X_1,...,X_n]}f(a_1,...,a_n)=0$
  • equivalently: ${\forall }_{0\ne f\in k[X_1,...,X_n]}k[X_1,...,X_n]\to A,f\mapsto f{a}_1,...,a_n$ is inj.ring.hom.
• $k$ fi. $A$ $k$-alg., then $tr(A):=\sup \{|T|:T\subseteq A$ finite alg.ind.subset$\}$ (transcendence degree), 8.3
  • if $A=0$, then $tr(A)=-1$
  • if $A=k[X_1,...,X_n]$, then $tr(A)=n$
    • if $A=k[X]$, then $\{X\}\subseteq A$ is max. fin. alg.ind.subset and $tr(A)=1$.
• $k$ fi., $A$ a $k$-alg., $S\subseteq A$ subset s.t. it gen. $A$ as a $k$-alg., then $dim(A)\le \sup \{|T|:T\subseteq S$ f.alg.ind.subset$\}$
  • hence $dim(K[S])\le tr(S)$, 8.4
    • $S=A\Rightarrow dim(A)\le tr(A)$
  • example: 8.5
    • $A$ noeth. then (i) $A$ has fin. many min.pr.id., (ii) every ideal in $A$ contains a min.pr.id., (iii) ${\mathcal{N}}_A={\bigcap }_{i=1}^r{\mathfrak{p}}_i$ for $\{{\mathfrak{p}}_1,...,{\mathfrak{p}}_r\}$ min.pr.if. of $A$, 8.5
• $k$ fi., $A$ fin.gen.$k$-alg., then $dim(A)=tr(A)$, 8.6
• $M$ is an art.$A$-mod. if for $A$-submod. any chain $M\subseteq ...⫅M_n\subseteq ...$ is stat., 8.7
  1. every non-empty set of $A$-submod. of $M$ has a min.el., 8.7
  2. $A$ is art.ring if $A$ is an art.$A$-mod., 8.7
  • fin.$A$-mod. $\Rightarrow$ art.$A$-mod., 8.8.a
  • $V$ $k$-mod., $di{m}_k(V)<{\aleph }_0\Rightarrow V$ is art.$k$-mod., 8.8.b
  • $\mathbb{Z}$ is not art.ring. (since $\{(n^k):k\ge 1\}$ has no min.el.), 8.8.c
• $A\ne 0$ ring, s.t. $0={\mathfrak{m}}_1\cdot ...\cdot {\mathfrak{m}}_k$ for ${\mathfrak{m}}_i\subseteq A$ max.id., then $A$ is art.ring $\iff A$ is noeth., 8.9
• $A$ art.ring, then: 8.10
  1. pr.id. $\iff$ max.id.
     • $dim(A)=0$
  2. fin.many max.id. in $A$
• art.ring $\iff$ noeth. and $dim(A)=0$, 8.10
  • art.ID $\iff$ fi.
• $M$ art.$A$-mod., $f\in En{d}_A(M)$, $f$ inj. $\Rightarrow f$ surj., tut.10.2
• $k$ fi., $A$ fin.gen.$k$-alg., tfae: 8.11
  1. $dim(A)=.0$
  2. $A$ alg.ov.$k$
  3. $di{m}_k(A)<\infty$ (as a vec.sp.)
  4. $A$ is art.ring
  5. $|spe{c}_{max}(A)|<{\aleph }_0$
• $A$ ring, $\mathfrak{p}\in spec(A)$, then $ht(\mathfrak{p})=\sup \{m:{\mathfrak{p}}_0⊊...⊊{\mathfrak{p}}_m=\mathfrak{p}\wedge {\forall }_{i\in [0,m]}{\mathfrak{p}}_i\in spec(A)\}$, 8.12
  • if $\mathfrak{a}\subseteq A$ id., then $ht(\mathfrak{a})=\inf \{ht(\mathfrak{p}):\mathfrak{p}\in spec(A)\wedge \mathfrak{a}\subseteq \mathfrak{p}\}$
  • $ht(A)=dim(A)+1$
  • $ht(\mathfrak{p})+dim(A/\mathfrak{p})\le dim(A)$ (often equal)
  • $A$ fact.ring, $\mathfrak{p}\in spac(A)$ s.t. $ht(\mathfrak{p})\Rightarrow \mathfrak{p}$ princ.id., 8.13
• $k$ fi., $\mathfrak{a}\subseteq k[X_1,...,X_n]$ id., then tfae:, 8.14
  1. $k[X_1,...,X_n]/\mathfrak{a}$ has pure dim $n-1$
     • every irr.comp. $T$ of $X$ has $dim(T)=n-1$
  2. $ht(\mathfrak{a})=1$ (in particular $\mathfrak{a}⊊k[X_1,...,X_n]$)
  3. $\sqrt{\mathfrak{a}}$ princ.id.
• $X\subseteq {\mathbb{A}}_k^m$, $Y\subseteq {\mathbb{A}}_k^n$ aff.var. for $k$ alg.cl.fi., then $dim(X\times Y)=dim(X)+dim(Y)$, 8.16

9. Localization

• $A$ ring, $S\subseteq A$ mult.cl. if $S^{-1}A$ in which the element of $S$ have a multiplicative inverse.
  • $A$ ID, $S=A\setminus \{0\}\Rightarrow S^{-1}A=Quot(A)$
  • $A$ ring, $S=A\setminus \mathfrak{p}\Rightarrow S^{-1}=:A_{\mathfrak{p}}$ loc.ring.
  • $A$ ring, $S=\{f^k:k\ge 0\}\Rightarrow S^{-1}A=A_f$ for some $f\in A$.
  • There is a ring.hom. $f:A\to S^{-1}A,a\mapsto a/1$ (always)
    • if $A$ ID, then $f$ inj., $f:A\to Quot(A)$
    • $S\subseteq A$, $(a,s)\sim (a^{\prime },s^{\prime }):={\exists }_{t\in S}(t(a{s}^{\prime }-a^{\prime }s)=0)$
      • note consequences of $t=0\in S$, every tuple is similar.
• $S^{-1}A$ fin.gen.$A$-alg. $\iff {\exists }_{f\in A}{S}^{-1}A=A_f$, tut.11.1.a
• $A$ ring., $\mathfrak{a}\subseteq A$ id., $S^{-1}\sqrt{\mathfrak{a}}=\sqrt{S^{-1}\mathfrak{a}}$, tut.11.2.a
• $\overline{S}$ is im. of $S$ in $A/\mathfrak{a}$, then
• $A$ ring., $S\subseteq A$, mult.cl. $f:A\to S^{-1}A$, $g:A\to B$ ring.hom. s.t. ${\forall }_{s\in S}g(s)\in B^{\times }$ then: ex! ring.hom $h:S^{-1}A\to B$ s.t. $g=h\circ f$ (Universal Property), 9.4
  • $P$ proj. $A$-mod., $S$ mult.cl.subs., then $S^{-1}P$ is proj.$S^{-1}A$-mod., tut.12.4.xi
• $S^{-1}A$ int.ov.$A$ $\Rightarrow S^{-1}A=A$, exe.14.3
• for $S^{-1}A$ and $f:A\to S^{-1}A$, 9.4
  • $s\in S\Rightarrow f(s)\in (S^{-1}A{)}^{\times }$
  • ${\exists }_{s\in S}(f(a)=0\to as=0)$
  • ${\forall }_{x\in S^{-1}A}{\exists }_{a\in A}{\exists }_{s\in S}x=f(a)f(s{)}^{-1}$
• For $f:A\to S^{-1}A,a\mapsto \frac{a}{1}$, 9.5
  • $\mathfrak{a}\subseteq A$ id. $\Rightarrow {\mathfrak{a}}^e=\{a/s:a\in \mathfrak{a}\}\subseteq S^{-1}A$
    • $\mathfrak{a}\cap S\ne \varnothing \Rightarrow {\mathfrak{a}}^e=S^{-1}A$
  • if $\mathfrak{b}\subseteq S^{-1}A$ is id., then ${\mathfrak{b}}^{ce}=\mathfrak{b}$
  • ext. and contr. give bijections: $\{$pr.id. of $S^{-1}A\}{\leftrightarrow }^{\cong }\{$pr.id.$\mathfrak{p}\subseteq A$ s.t. $\mathfrak{p}\cap S=\varnothing \}$
• example (Localisation as a Functor): 9.6
  • for $A$ ring, $\mathfrak{p}\in spec(A)$, define $A_{\mathfrak{p}}$ loc. with $\mathfrak{p}$ as uniq.max.id.:
  • $M$ $A$-mod., ${\exists }_{t\in S}(m,s)\sim (m^{\prime },s^{\prime })\iff t(s{m}^{\prime }-s^{\prime }m)=0$
  • $m/s+m^{\prime }/s^{\prime }=(s^{\prime }m+s{m}^{\prime })/s{s}^{\prime }$ and $a/s\cdot m/s^{\prime }=am/s{s}^{\prime }$
  • $f:M\to N$ $A$-lin.map., $S^{-1}f:S^{-1}M\to S^{-1}N,m/s\mapsto f(m)/s$ is $S^{-1}A$-lin.
    • $S^{-1}(f)\circ S^{-1}(g)=S^{-1}(f\circ g)$
  • Hence localisation defines a functor $S^{-1}:{\mathcal{M}od}_A\to {\mathcal{M}od}_{S^{-1}A}$
    1. $M\mapsto S^{-1}M,$
    2. $(f:M\to N)\mapsto (S^{-1}f:S^{-1}M\to S^{-1}N)$
• the functor $S^{-1}-$ is exact (Exact Localisation), 9.7
  1. if $M^{\prime }{\to }^{f}M{\to }^g{M}^{\prime \prime }$ exa, then $S^{-1}{M}^{\prime }{\to }^{S^{-1}f}{S}^{-1}M{\to }^{S^{-1}g}{S}^{-1}{M}^{\prime \prime }$ is exact.
• $N,P\subseteq M$ $A$-mod., then, 9.8
  1. $S^{-1}(N+P)=S^{-1}N+S^{-1}P$
  2. $S^{-1}(N\cap P)=S^{-1}N\cap S^{-1}P$
  3. $S^{-1}(M/N)\cong S^{-1}M/S^{-1}N$
• $f:S^{-1}A{\otimes }_{A}M\to S^{-1}M,a/s\otimes m\mapsto am/s$ is the unique $S^{-1}A$-lin.isom., 9.9
  • consider the two functors $S^{-1}(-)$ and $S^{-1}A{\otimes }_A(-):M\mapsto S^{-1}A{\otimes }_{A}M,f\mapsto 1\otimes f_{\ast }$
    • 9.9 implies $S^{-1}M\cong S^{-1}A{\otimes }_{A}M$, also the mapping of functions is equivalent.
    • hence the two distinct functors are isomorphic.
• $f:S^{-1}M{\otimes }_{S^{-1}A}{S}^{-1}N\to S^{-1}(M{\otimes }_{A}N),(m/s)\otimes (n/s^{\prime })\mapsto (m\otimes n)/s{s}^{\prime }$ is $S^{-1}A$-lin.isom. 9.10
  • for $S=A\setminus \mathfrak{p}$, $M_{\mathfrak{p}}{\otimes }_{A_{\mathfrak{p}}}{N}_{\mathfrak{p}}\cong (M{\otimes }_{A}N{)}_{\mathfrak{p}}$
• a property $\mathcal{P}$ of rings (or mod.) is loc. if $\mathcal{P}(A)\iff {\forall }_{\mathfrak{p}\in spec(A)}\mathcal{P}(A_{\mathfrak{p}})$ (Local Property), 9.11
• "$M=0$" is a local property, 9.12
  • $A$ ring, $M$ $A$-mod., tfae: (i) $M=0$, (ii) ${\forall }_{\mathfrak{p}\in spec(A)}{M}_{\mathfrak{p}}=0$, (iii) ${\forall }_{\mathfrak{m}\text{ max.id.}}{M}_{\mathfrak{m}}=0$, 9.12
• ”$f:M\to N$ is ink. (surj.)” is a local property, 9.12
• ”$M^{\prime }{\to }^{f}M{\to }^g{M}^{\prime \prime }$ is an exact sequence” is a local property, 9.12
• ”$M$ is flat” is a local property, 9.13
• “being normal” is a local property, 11.11
• ”$A$ reduced” is a local property, tut.11.2.b
  • $A$ ring, $M$ $A$-mod. tfae: (i) $M$ flat, (ii) ${\forall }_{\mathfrak{p}\in spec(A)}flat(M_{\mathfrak{p}})$, (iii) ${\forall }_{\mathfrak{m}\text{ max.id.}}flat(M_{\mathfrak{m}})$, 9.13
• $A$ noeth.ring., $\mathfrak{a}⊊A$ princ.id., if $\mathfrak{p}$ min.pr.id. s.t. $\mathfrak{a}\subseteq \mathfrak{p}$, then $ht(\mathfrak{p})\le 1$ (Krull’s princ.id. Th.), 9.14
  • in particular: $ht(\mathfrak{a})\le 1$
• $A$ noeth.ring., $\mathfrak{a}=(a_1,...,a_r)\subseteq A$ id., if $\mathfrak{p}$ is min.pr.id. s.t. $\mathfrak{a}\subseteq \mathfrak{p}$, then $ht(\mathfrak{p})\le r$, (K. Height Th.), 9.15
  • in particular: $ht(\mathfrak{a})\le r$
  • $A$ noeth.loc.ring, $\Rightarrow dim(A)<\infty$
• $A$ noeth.ring., $\mathfrak{p}\in spec(A)$ s.t. $ht(\mathfrak{p})=r$, then ${\exists }_{a_1,...,a_r\in \mathfrak{p}}\mathfrak{p}.min.ov.(a_1,...,a_r)$, 9.16
  • $A$ noeth.loc.ring with $\mathfrak{m}$ max.id., for $(a_1,...,a_r)$ s.t. $\mathfrak{m}=\sqrt{(a_1,...,a_r)}$, then:
    • $dim(A)=r$ (for the minimal set of generators)
    • $a_1,...,a_r$ is a local coordinate system or system of parameters of $A$
    • $\sqrt{(a_1,...,a_r)}=\bigcap \{\mathfrak{p}\in spec(A):(a_1,...,a_r)\subseteq \mathfrak{p}\}$
    • note: $\mathfrak{m}=\sqrt{(a_1,...,a_r)}\iff \mathfrak{m}$ is min.ov.$(a_1,...,a_r)$

10. Primary Decomposition

• $A$ ring., $\mathfrak{q}⊊A$ primary.id. iff ${\forall }_{x,y\in A}{\exists }_{n\in \mathbb{N}}xy\in \mathfrak{q}\to x\in \mathfrak{q}\vee y^n\in \mathfrak{q}$, 10.1
  • equivalently, $A/\mathfrak{q}\ne 0$, and every zero.div. in $A/\mathfrak{q}$ is nilpot.
    • i.e. $\bar{y}\in A/\mathfrak{q}$ zero.div. $\bar{x}\cdot \bar{y}=0$ for $\bar{x}\ne 0\Rightarrow {\bar{y}}^n=0$ for $n\ge 1$
  • for $\mathfrak{a}\subseteq \mathfrak{q}⊊A$, then: $\mathfrak{q}$ primary.id. $\iff \mathfrak{q}/\mathfrak{a}\subseteq A/\mathfrak{q}$ primary.id.
  • prime.id. $\Rightarrow$ primary.id.
  • $\{(p^n):p\in \mathbb{P}\wedge n\in \mathbb{N}\}$ are primary.id. in $\mathbb{Z}$
• $\mathfrak{q}⊊A$ primary id. $\Rightarrow \mathfrak{p}=\sqrt{\mathfrak{q}}⊊A$ prime.id. (say $\mathfrak{q}$ is $\mathfrak{p}$-primary) 10.3-4
  • $\sqrt{\mathfrak{q}}$ is smallest s.t. $\mathfrak{q}\subseteq \sqrt{\mathfrak{q}}$
• $\mathfrak{m}\subseteq A$ max.id., $\mathfrak{q}⊊A$ id., if $\sqrt{\mathfrak{q}}=\mathfrak{m}$ or if ${\exists }_{n\in \mathbb{N}}{\mathfrak{m}}^n\subseteq \mathfrak{q}\subseteq \mathfrak{m}$, then $q$ is $\mathfrak{m}$-primary, 10.5
  • ${\mathfrak{m}}^n$ is $\mathfrak{m}$-primary, but not all primary ideals are a power of max.id.
  • powers of prime ideals are not necessarily primary ideals 10.6
• $A$ ring, $\mathfrak{a}\subseteq A$ id., a PD of $\mathfrak{a}$ is a fin.set of primary.id. ${\mathfrak{q}}_1,...,{\mathfrak{q}}_r$ s.t. $\mathfrak{a}={\mathfrak{q}}_1\cap ...\cap {\mathfrak{q}}_r$, 10.7
• $A$ noeth.ring., $\mathfrak{a}⊊A$ id., $\Rightarrow \mathfrak{a}$ has a (min.)PD. 10.8(.12)
  • $A$ PID, $\mathfrak{a}=(p_1^{n_1},...,p_r^{n_r})$, then $\mathfrak{a}=(p_1^{n_1})\cap ...\cap (p_r^{n_r})$
• ${\mathfrak{q}}_1,...,{\mathfrak{q}}_r$ are $\mathfrak{p}$-primary $\Rightarrow {\bigcap }_{i=1}^r{\mathfrak{q}}_i$ is $\mathfrak{p}$-primary, 10.10
• A PD is a min.PD if ${\forall }_{i\in [1,r]}({\bigcap }_{j\ne i}{\mathfrak{q}}_j)⊈{\mathfrak{q}}_i$ and ${\forall }_{i\ne j}\sqrt{{\mathfrak{q}}_i}\ne \sqrt{{\mathfrak{q}}_j}$, 10.11
  • $A$ ring, $\mathfrak{a}\subseteq A$ id., $a\in A$, then $(\mathfrak{a}:a):=\{x\in A:xa\in \mathfrak{a}\}\subseteq A$ id.
• $A$ ring, $\mathfrak{p}\in spec(A)$, $\mathfrak{q}$ a $\mathfrak{p}$-prim.id., then: 10.14
  1. $\sqrt{(\mathfrak{q}:a)}=A$ if $a\in \mathfrak{q}$
  2. $\sqrt{(\mathfrak{q}:a)}=\mathfrak{p}$ if $a\notin \mathfrak{q}$
• $A$ ring, $\mathfrak{a}⊊A$ id, then $\mathfrak{p}\in spec(A)$ is ass.to.$\mathfrak{a}$ if ${\exists }_{a\in A}\sqrt{(\mathfrak{a}:a)}=\mathfrak{p}$, 10.15
  • define $AP(\mathfrak{a})=\{\mathfrak{p}\in spec(A):\mathfrak{p}$ ass.to.$\mathfrak{a}\}$
  • $x\in AP(\mathfrak{a})$ is an isolated.prime. if it is min${}_{\subseteq }$. in $AP(\mathfrak{a})$
    • not isolated elements are embedded
• $A$ ring, $\mathfrak{a}⊊A$ id., $\mathfrak{a}={\mathfrak{q}}_1\cap ...\cap {\mathfrak{q}}_r$ min.PD, then $AP(\mathfrak{a})=\{\sqrt{{\mathfrak{q}}_1},...,\sqrt{{\mathfrak{q}}_r}\}$ (Uniqueness I), 10.16
  • $r$ and ${\mathfrak{p}}_1,...,{\mathfrak{p}}_r$ depend only on $\mathfrak{a}$.
  • isol.pr.id. of $\mathfrak{a}$ are exactly the min.pr.id.ov.$\mathfrak{a}$, 10.17
• $A$ ring, $\mathfrak{a}⊊A$ id., $\mathfrak{a}={\mathfrak{q}}_1\cap ...\cap {\mathfrak{q}}_r$ min.PD, $\mathfrak{p}=\sqrt{{\mathfrak{q}}_i}$, if ${\mathfrak{p}}_i$ isol.pr.(/min.pr.id.ov.$\mathfrak{a}$) and $A\to A_{{\mathfrak{p}}_i}$ loc.map., then ${\mathfrak{q}}_i=({\mathfrak{a}}^e{)}^c$ (Uniq. II), 10.19
  • also, pr.comp. corresponding isol.pr.id. in a min.PD do not depend on the PD.

11. Integral Ring Extensions

• Motivation
  • recall from Algebra (Lecture): $L/K$ fi.ext.: fin.gen & alg. $\leftrightarrow$ finite
  • here: $R/A$ ring.ext.: fin.gen & int. $\leftrightarrow$ finite
• $A\subseteq B$ subring., $b\in B$ is int.ov.$A$ if ${\exists }_{f\in A[X]}f\text{ mon. }\wedge f(b)=0$, 11.1
  • i.e. ${\exists }_{a_1,...,a_n\in A}{b}^n+a_1{b}^{n-1}+...+a_n=0$
  • $B$ (or $B/A$) is int. if any $b\in B$ is int.ov.$A$
• $S^{-1}A$ int.ov.$A$ $\Rightarrow S^{-1}A=A$, exe.14.3
• $A$ ring, $\mathfrak{a}\subseteq A$, $M$ fin.gen.$A$-mod., if $\Phi :M\to M$ $A$-lin. s.t. $\Phi (M)\subseteq \mathfrak{a}M$, then ex. $f\in \mathfrak{a}[X]$ mon. s.t. $f(\Phi )=0\in En{d}_A(M)$ (Caley Hamilton, see 17.4 Cayley-Hamilton), 11.3
  • for $\mathfrak{a}=A=k$ fi., $M$ a fin.dim.vex.sp., then pick a basis $\{b_1,..,b_n\}$ and let $\Phi$ have a matrix, then $f$ is the usual characteristic polynomial.
• $A$ ring.ext. $A\subseteq B$ is fin., if $B$ is a fin.gen $A$-mod., 11.4
• $A\subseteq B$ ring.ext., then $A\subseteq B$ fin. $\leftrightarrow A\subseteq B$ int. $\wedge$ fin.gen, 11.5
  • i.e. ex. $b_1,...,b_n\in B$ int.ov.$A$ s.t. $B=A[b_1,...,b_n]$
• $A\subseteq B$ ring.ext., then $C:=\{b\in B:b$ int.ov.$A\}\subseteq B$ subring. (of $A$ too), (Integral Closure) 11.6-7
  • if $C=A$, then A is Integrally Closed, if $C=B$, $B$ is integral over $A$
• $A\subseteq B$, $B\subseteq D$ ring.ext., then: 11.9
  1. $A\subseteq B$, $B\subseteq D$ int. $\Rightarrow A\subseteq D$ int.
  2. $C$ int.cl. of $A$ in $B\Rightarrow C$ is max.int.cl. of $A$ in $B$
  3. $A\subseteq B$, $\mathfrak{b}\subseteq b$, $\mathfrak{a}=f^{-1}(\mathfrak{b})\subseteq A$ id. $\Rightarrow A/\mathfrak{a}\subseteq B/\mathfrak{b}$ int.
  4. $A\subseteq B$ int., $S\subseteq A$ mult.cl. $\Rightarrow S^{-1}A\subseteq S^{-1}B$ int.
     • if also $C$ int.cl. of $A$ in $B$ $\Rightarrow S^{-1}C$ int.cl. of $S^{-1}A$ in $S^{-1}B$
• $A$ ID, then the norm.cl. $A^{\nu }$ of $A$ in $Quot(A)$ is the int.cl of $A$ in $Quot(A)$, 11.10
  • hence: $A\subseteq A^{\nu }\subseteq Quot(A)$
  • if $A=A^{\nu }$, then $A$ is norm.
• “being normal” is a local property, 11.11
  • tfae: (i) $A$ norm. (ii) ${\forall }_{\mathfrak{p}\in spec(A)}{A}_{\mathfrak{p}}$ norm., (iii) ${\forall }_{\mathfrak{m}\in spe{c}_{max}(A)}{A}_{\mathfrak{m}}$ norm.
  • fact.ring $\Rightarrow$ norm.
• $A\subseteq B$ int.ring.ext., $B$ ID, then: $A$ fi. $\iff B$ fi., 11.13
  • $A\subseteq B$ int.ring.ext., then: 11.14
    1. $\mathfrak{q}\in spec(B)$, $\mathfrak{p}=\mathfrak{q}\cap A\in spec(A)$, then: $\mathfrak{p}$ max.id. $\iff \mathfrak{q}$ max.id.
    2. ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in spec(B)$ s.t. ${\mathfrak{q}}_1\subseteq {\mathfrak{q}}_2$, $\mathfrak{p}={\mathfrak{q}}_1\cap A={\mathfrak{q}}_2\cap A\in spec(A)$, then ${\mathfrak{q}}_1={\mathfrak{q}}_2$
       • ${\mathfrak{q}}_1⊊{\mathfrak{q}}_2\Rightarrow {\mathfrak{q}}_1\cap A⊊{\mathfrak{q}}_2\cap A$
• $A\subseteq B$ int.ring.ext.: $\mathfrak{p}\in spec(A)\Rightarrow {\exists }_{\mathfrak{q}\in spec(B)}\mathfrak{p}=\mathfrak{q}\cap A$ (Lying-Over), 11.15
  • over every prime ideal of $A$ lies a prime ideal of B
• $A\subseteq B$ int.ring.ext., ${\mathfrak{p}}_1\subseteq ...\subseteq {\mathfrak{p}}_m\subseteq ...\subseteq {\mathfrak{p}}_n$ pr.id. in $A$, ${\mathfrak{q}}_1\subseteq ...\subseteq {\mathfrak{q}}_m$ pr.id. in $B$ s.t. ${\forall }_{i\in [1,m]}{\mathfrak{p}}_i={\mathfrak{q}}_i\cap A$, then: $\{{\mathfrak{q}}_i{\}}_{[1,m]}$ can be extended to $\{{\mathfrak{q}}_i{\}}_{[1,n]}$ by ${\forall }_{i\in [1,n]}{\mathfrak{p}}_i={\mathfrak{q}}_i\cap A$ (Going-Up), 11.16
• $A\subseteq B$ int.ring.ext, $\Rightarrow$ $dim(A)=dim(B)$, 11.17
• $A\subseteq B$ int.ring.ext. $A$ norm., $B$ ID, ${\mathfrak{p}}_n\subseteq ...\subseteq {\mathfrak{p}}_m\subseteq ...\subseteq {\mathfrak{p}}_1$ pr.id. in $A$, ${\mathfrak{q}}_m\subseteq ...\subseteq {\mathfrak{q}}_1$ pr.id. in $B$ s.t. ${\forall }_{i\in [1,m]}{\mathfrak{p}}_i={\mathfrak{q}}_i\cap A$, then: $\{{\mathfrak{q}}_i{\}}_{[1,m]}$ can be ext. to $\{{\mathfrak{q}}_i{\}}_{[1,n]}$ by ${\forall }_{i\in [1,n]}{\mathfrak{p}}_i={\mathfrak{q}}_i\cap A$ (Going-Down), 11.16
  • $f:A\to B$ ring.hom has going.up. prop. if $f(A)\subseteq B$ satisfies the conditions above
  • $C(\mathfrak{a}):=\{n\in B:b$ int.ov.$\mathfrak{a}\}$
• $A\subseteq B$ ring.ext., $\mathfrak{a}$ id. in $A$, $C$ int.cl. of $A$ in $B$, then $C(\mathfrak{a})=\sqrt{C\cdot \mathfrak{a}}$
  • $C\cdot \mathfrak{a}=a^e$ for $A\to C$
  • hence $C(\mathfrak{a})$ id. in $A$
  • $A=C\Rightarrow C(\mathfrak{a})=\sqrt{\mathfrak{a}}$
• $A\subseteq B$ ring.ext., $B$ ID, $A$ norm., $\mathfrak{a}\subseteq A$ id., $b\in B$ int.ov.$\mathfrak{a}$. then: $b$ alg.ov.$Quot(A)$, 11.20
  • the min.pol. of $f$ has the form: $m+b=X^n+a_1{X}^{n-1}+...+a_{n-1}X+a_n$ s.t. ${\forall }_{i\in [1,n]}{a}_i\in \sqrt{\mathfrak{a}}$
• $k$ fi., $A$ fg.$k$-alg.: ex. $\{c_1,..,c_n\}\in A$ alg.ind. s.t. $k[X_1,...,X_n]\cong k[c_1,...,c_n]$, $C\subseteq A$ int.ext., 11.21
  • $|k|\ge {\aleph }_0$, $\{a_1,...,a_n\}\subseteq k$, s.t. $A=k[a_1,...,a_n]$, then for ${\gamma }_{ij}\in k$, $c_i=a_i+{\sum }_{j=n+1}^m{\gamma }_{ij}\cdot a_j$
• for $A$ a ring, ${\mathfrak{p}}_0⊊...⊊{\mathfrak{p}}_n⊊A$ is max. if it can’t be extended.
• $k$ fi., $A$ fin.gen $k$-alg., ${\mathfrak{p}}_0⊊...⊊{\mathfrak{p}}_n⊊A$ is max.ch.pr.id., then $dim(A/{\mathfrak{p}}_0)=n$, 11.23
  • also, $A$ has pure.dim. $n$
    • i.e. every irr.comp. of $spec(A)$ has dim. $n$
  • every max.ch.pr.id. in $A$ has len. $n$.
  • if $\mathfrak{p}\subseteq \mathfrak{q}$ pr.id. in $A$, then all max.ch.pr.id. between $\mathfrak{p}$ and $\mathfrak{q}$ have same lenght
    • i.e. a finitely generated $k$-algebra $A$ is a catenary ring
• $A$ fin.gen.$k$-alg., of pure dim., if $\mathfrak{p}\subseteq A$ pr.id. then $dim(A)=ht(\mathfrak{p})+dim(A/\mathfrak{p})$, 12.24
  • hence for $\mathfrak{a}\subseteq A$ id., $dim(A)=ht(\mathfrak{a})+dim(\mathfrak{a}/A)$

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Appendix

Introduction

• Commutative Algebra and Algebraic Geometry
  • Example:
    • $V:=\{(a,a^2):a\in \mathbb{C}\}$, complex parabola $Y=X^2$
    • $A(V):=\mathbb{C}[X,Y]/(Y-X^2)$, i.e. functions $V\to \mathbb{C}$
    • $I(V):=\{h\in \mathbb{C}[X,Y]:{\forall }_{P\in V}h(P)=0\}$, i.e. the eq.class of polynomial of $V$, in the ex. $(Y-X^2)$.
  • a ring, like $A(V)$ is:
    • finitely generated, if it is a quotient of $\mathbb{C}[X,Y]$, call it $\mathbb{C}$-algebra.
    • reduced, if it contains nilpotent elements, call it affine $\mathbb{C}$-algebra.
  • for varieties $V$, $W$ define:
    • $\psi :V\to W,(a,a^2)\mapsto f(a)$ for $f\in \mathbb{C}[X]$.
    • $\varphi :W\to V$ s.t. $\varphi (a)=(f(a),g(a))\in V$ (i.e. $g(a)=f(a{)}^2$ in the ex.) for $f,g\in \mathbb{C}[X]$.
  • Back to the example with $W=\mathbb{C}$, $\varphi :W\to V,a\mapsto (a,a^2)$, $\psi :V\to W,(a,a^2)\mapsto a$
    • $\varphi \circ \psi =i{d}_V$, $\psi \circ \psi =i{d}_W$ i.e. $V\cong W$ (as affine varieties).
  • each such $\psi$ induces the def. of ${\psi }^{\ast }:A(W)\to A(V)$.
    • in the ex. we get $A(W)=\mathbb{C}[X]\cong \mathbb{C}[X,Y]/(Y-X^2)=A(V)$
  • on p.3 there is an example of non-isom. varieties.
• Commutative Algebra and Algebraic Number Theory
  • Question: Which ID are factorial? Look at Ring extensions
  • $A\le B$ a ring.ext., say $b\in B$ int. over $A$ if there is a monic pol. $f\in A[X]$ s.t. $f(b)=0$.
  • $C:=\{b\in B:b$ int. over $A\}$ is a ring s.t. $A\le C\le B$, call it the integral closure. There are further remarks in the introduction.

Questions

Here I collect the questions I ask to the tutor concerning the lecture material, I expressively make reference to the script file.

1. In the definition of ${\mathfrak{b}}^c$, 2.16, I cannot understand the notion of $\mathfrak{b}\cap A$ since the function $f$ is an arbitrary ring homomorphism and can therefore have a domain completely different from its codomain. Intuitively $f^{-1}(\mathfrak{b})$ is a notion that must differ from $\mathfrak{b}\cap A$, since the latter is empty in many more cases, than the former (take $f:A\to B$ and $f^{\prime }:A^{\prime }\to B$ for $\Phi :A\to A^{\prime }$ an isomorphism, $\mathfrak{b}\cap A\ne \varnothing$ but $\mathfrak{b}\cap A^{\prime }=\varnothing$, hence $f^{\prime -1}(\mathfrak{b})=\varnothing$ but $\varnothing \ne \Phi (f^{-1}(\mathfrak{b}))=f^{\prime -1}(\mathfrak{b})$). Searching online I often found the preimage of the ideal to be the correct definition of ${\mathfrak{b}}^c$ and couldn’t clarify what exactly $\mathfrak{b}\cap A$ means (other than the proper set theoretic meaning that does not convince me due to the reason exposed above).
2. In the definition 6.1, I cannot understand what “almost all $a_{i_1,...,i_n}=0$” exactly means. Considering that that sequence is clearly meant to be finite (i.e. for $n\in \mathbb{N}$), I don’t see how the notion of “almost all” can be defined. On this, I found from wikipedia that: “The meaning of “negligible” depends on the mathematical context; for instance, it can mean finite, countable, or null.” Given this finte context, I suppose the only plausible meaning is that it applies to all elements (i.e. negligible is null), but then why not just write "${\forall }_{j\in [1,n]}{a}_{i_j}=0$"?
3. A similar issue presents itself in 7.3, where the index can be infinite, in that case I suppose that “almost all” means “all but finitely many”, hence: the definition of direct sum results as the following: ${⨁}_{i\in I}{M}_i:=\{\{m_i{\}}_I\in {\prod }_{i\in I}{M}_i:|\{i\in I:m_i\ne 0\}|<{\aleph }_0\}$; is this definition correct?
4. On the definition 7.5.c of an $A$-basis, I wanted to ask whether the understanding I have of it is correct: for $A$ a ring, $M$ an $A$-module, $\{m_i{\}}_{i\in I}\subseteq M$ is a basis if ${\forall }_{m\in M}\exists {!}_{\{a_i{\}}_I\subseteq A}m={\sum }_{i=1}^n{a}_i{m}_i$, here I use the symbol "$\exists !$" defined as follows: $\exists {!}_x\phi (x):={\exists }_x\phi (x)\wedge {\forall }_y\phi (y)\to x=y$.
5. On the lemma 7.9.b it is assumed that ”$h:P\to N$ is an arbitrary $A$-linear map”, I wanted therefore to ask whether the notion of “being arbitrary” could be clarified and whether it has been previously defined or more clearly stated.
6. In the solution of Exercise Sheet 14, Exercise 1 c, it is stated that there is a map $Spec(S^{-1}A)\to Spec(A)$ that is injective and inclusion preserving. In the lecture we have seen the map $f:A\to S^{-1}A,a\to \frac{a}{1}$ though I cannot see how that induces the map claimed in the solution or where that could otherwise come from. Could you either show me how the mapping of the function $Spec(S^{-1}A)\to Spec(A)$ works or where it compares in the script? Also, I suppose that the map $f$ would not induce directly any mapping between the two spectra, since for $\mathfrak{p}\in spec(A)$, then $\{f(p):p\in \mathfrak{p}\}\subseteq S^{-1}A$ would not be a prime ideal.

Answers

1. Here, $\mathfrak{b}\cap A$ is just a notation to emphasize to which ring you contract the ideal $\mathfrak{b}$. Hence, in 2.16, you have to interpret the equality $\mathfrak{b}\cap A=f^{-1}(\mathfrak{b})$ as a definition of the left hand side. As you write, this equality does usually not make sense set-theoretically.
2. Here, “almost all” refers to the set $I:=\{(i_t{)}_{t\in T^{\prime }}\mid T^{\prime }\subseteq T\text{ finite},i_t\in \mathbb{N}\}.$ The notation in 6.1 means that the sum is taken over the index set $I$. In particular, for each $i=(i_t{)}_{t\in T^{\prime }}\in I$, you have an element $a_i\in A$ (written as $a_i=a_{i_1,\dots ,i_n}$ if $T^{\prime }=\{t_1,\dots ,t_n\}$) and we demand that $a_i=0$ for all but finitely many $i$.
3. This is correct.
4. Your definition is almost correct. However, you have to additionally demand that $a_i=0$ for all but finitely many $i$. Also, you should write $m={\sum }_{i\in I}{a}_i{m}_i$ (and not ${\sum }_{i=1}^n$).
5. Here, “arbitrary” is just there to emphasize that while $g$ is surjective, $h$ need not be surjective. You can replace “an arbitrary” by “any”.
6. in general, if $f:A\to B$ is a ring homomorphism, then $f$ induces a continuous map $Spec(f):Spec(B)\to Spec(A),\mathfrak{p}\mapsto f^{-1}(\mathfrak{p})$ (see e.g. exercise sheet 2, exercise 3). Here, we just have the special case $B=S^{-1}A$.

Continuations

I may continue to study related fields such as Algebraic Geometry (yt) and (Algebraic) Number Theory. In particular this source by Columbia University may be interesting: Stack Project and builds up on the already familiar material.

Exam Review

Here I list the remarks and questions I have regarding the correction of the exam by following the order of the exercises:

1a) Where I state $\bigcap A=$ max.elem. of $A\Rightarrow |A|=1$, I used a confusing notation, I mean tin fact that this holds in general (as I state before) that when the minimal and maximal element of a partial order coincide, then the cardinality of the underlying set must be one. So $A$ as stated in that proposition is a general set and not $A$ that is used in the exercise. Since what I wrote about $A$ in that proposition holds for $spec(A)$ (here referring to the ring $A$), then I can conclude $|spec(A)|=1$. If I got 5/6 in that exercise because of this misleading notation, one could reconsider it by reading this explanation and underlying that “in general”.

Secondly, it is remarked that I did not prove taht $A\ne 0$, this though immediately follows from the fact that…?