math/commutative-algebra-notes.tex
commutative-algebra-notes.texRaw
1\documentclass{article}
2\usepackage[utf8]{inputenc}
3\usepackage{amsfonts}
4% \usepackage{yfonts} % WIP
5\usepackage{amsthm}
6\usepackage{amsmath}
7\usepackage{enumerate}
8\usepackage{hyperref}
9\usepackage{amssymb}
10\usepackage{tikz} % diagram
11
12\begin{filecontents}[overwrite]{commutative-algebra-notes.bib}
13@misc{am,
14 author = {M. F. Atiyah and I. G. MacDonald},
15 title = {{Introduction to Commutative Algebra}},
16 year = {1969}
17}
18@misc{reid,
19 author = {Miles Reid},
20 title = {{Undergraduate Commutative Algebra}},
21 year = {1995}
22}
23@misc{mit-course,
24 author = {Steven Kleiman},
25 title = {{Commutative Algebra - MIT OpenCourseWare}},
26 year = {2008},
27 note = {\url{https://ocw.mit.edu/courses/18-705-commutative-algebra-fall-2008/}},
28 url = {https://ocw.mit.edu/courses/18-705-commutative-algebra-fall-2008/}
29}
30\end{filecontents}
31\nocite{*}
32
33
34\theoremstyle{definition}
35
36\newtheorem{innerdefn}{Definition}
37\newenvironment{defn}[1]
38{\renewcommand\theinnerdefn{#1}\innerdefn}
39{\endinnerdefn}
40
41\newtheorem{innerthm}{Theorem}
42\newenvironment{thm}[1]
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44{\endinnerthm}
45
46\newtheorem{innerlemma}{Lemma}
47\newenvironment{lemma}[1]
48{\renewcommand\theinnerlemma{#1}\innerlemma}
49{\endinnerlemma}
50
51\newtheorem{innerprop}{Proposition}
52\newenvironment{prop}[1]
53{\renewcommand\theinnerprop{#1}\innerprop}
54{\endinnerprop}
55
56\newtheorem{innercor}{Corollary}
57\newenvironment{cor}[1]
58{\renewcommand\theinnercor{#1}\innercor}
59{\endinnercor}
60
61\newtheorem{innereg}{Example}
62\newenvironment{eg}[1]
63{\renewcommand\theinnereg{#1}\innereg}
64{\endinnereg}
65
66\newtheorem{innerex}{Exercise}
67\newenvironment{ex}[1]
68{\renewcommand\theinnerex{#1}\innerex}
69{\endinnerex}
70
71\newcommand{\aA}{\mathfrak{a}} % TODO: use goth font
72\newcommand{\mM}{\mathfrak{m}}
73
74\title{Commutative Algebra notes}
75\author{arnaucube}
76\date{}
77
78\begin{document}
79
80\maketitle
81
82\begin{abstract}
83 Notes taken while studying Commutative Algebra, mostly from Atiyah \& MacDonald book \cite{am} and Reid's book \cite{reid}.
84
85 Usually while reading books and papers I take handwritten notes in a notebook, this document contains some of them re-written to $LaTeX$.
86
87 The proofs may slightly differ from the ones from the books, since I try to extend them for a deeper understanding.
88\end{abstract}
89
90\tableofcontents
91
92\section{Ideals}
93
94\subsection{Definitions}
95
96\begin{defn}{ideal}
97 $I \subset R$ ($R$ ring) such that $0 \in I$ and $\forall x \in I,~ r \in R,~ xr, rx \in I$.\\
98 \hspace*{2em} ie. $I$ absorbs products in $R$.
99\end{defn}
100
101\begin{defn}{prime ideal}
102 if $a, b \in R$ with $ab \in P$ and $P \neq R$ ($P$ a prime ideal), implies $a in P$ or $b \in P$.
103\end{defn}
104
105\begin{defn}{principal ideal}
106 generated by a single element, $(a)$.
107
108 $(a)$: principal ideal, the set of all multiples $xa$ with $x \in R$.
109\end{defn}
110
111\begin{defn}{maximal ideal}
112 $\mM \subset A$ ($A$ ring) with $m \neq A$ and there is no ideal $I$ strictly between $\mM$ and $A$. ie. if $\mM$ maximal and $\mM \subseteq I \subseteq A$, either $\mM=I$ or $I=A$.
113\end{defn}
114
115
116\begin{defn}{unit}
117 $x \in A$ such that $xy=1$ for some $y \in A$. ie. element \emph{which divides 1}.
118\end{defn}
119
120\begin{defn}{zerodivisor}
121 $x \in A$ such that $\exists 0 \neq y \in A$ such that $xy=0 \in A$. ie. $x$ \emph{divides 0}..
122
123
124 If a ring does not have zerodivisors is an integral domain.
125\end{defn}
126
127\begin{defn}{prime spectrum - $Spec(A)$}
128 set of prime ideals of $A$. ie.
129
130 $$Spec(A) = \{ P ~|~ P \subset A~ \text{is a prime ideal} \}$$
131\end{defn}
132
133\begin{defn}{integral domain}
134 Ring in which the product of any two nonzero elements is nonzero.
135
136 ie. no zerodivisors.
137
138 ie. $\forall~ 0 \neq a,~ 0 \neq b \in A,~ ab \neq 0 \in A$.
139
140 Every field is an integral domain, not the converse.
141\end{defn}
142
143\begin{defn}{principal ideal domain - PID}
144 integral domain in which every ideal is principal. ie.
145 ie. $\forall I \subset R,~ \exists~ a \in I$ such that $I = (a) = \{ ra ~|~ r \in R \}$.
146\end{defn}
147
148\begin{defn}{nilpotent}
149 $a \in A$ such that $a^n=0$ for some $n>0$.
150\end{defn}
151\begin{defn}{nilrad A}
152 set of all nilpotent elements of $A$; is an ideal of $A$.
153
154 if $nilrad A = 0 ~\Longrightarrow$ $A$ has no nonzero nilpotents.
155
156
157 $$nilrad A = \bigcap_{P \in Spec(A)} P$$
158\end{defn}
159
160\begin{defn}{idempotent}
161 $e \in A$ such that $e^2=e$.
162\end{defn}
163
164\begin{defn}{radical of an ideal}
165 $$rad I = \{ f \in A | f^n \in I~ \text{for some} n \}$$
166
167 $rad I$ is an ideal.
168
169 $nilrad A = rad 0$
170
171 $rad I = \bigcap_{\substack{P \in \operatorname{Spec}(A)\\ P \supset I}} P$
172\end{defn}
173
174\begin{defn}{local ring}
175 A \emph{local ring} has a unique maximal ideal.
176
177 Notation: local ring $A$, its maximal ideal $\mM$, residue field $K=A/\mM$:
178 $$A \supset \mM ~\text{or}~ (A, \mM) ~\text{or}~ (A, \mM, K)$$
179\end{defn}
180
181\subsection{$\mathbb{Z}$ and $K[X]$, two Principal Ideal Domains}
182
183
184\begin{lemma}{}
185 $\mathbb{Z}$ is a PID.
186\end{lemma}
187\begin{proof}
188 Let $I$ a nonzero ideal of $\mathbb{Z}$.
189
190 Since $I \neq \{0\}$, there is at least one nonzero integer in $I$. Choose the smallest element of $I$, namely $d$.
191
192 Observe that $(d) \subseteq I$, since $d \in I$. Then, every multiple $nd \in I$, since $I$ is an ideal.
193
194 Take $a \in I$. By the Euclidean division algorithm in $\mathbb{Z}$, $a=qd+r$, with $q,r \in \mathbb{Z}$ and $0 \leq r \leq d$.
195
196 Then $r = a - qd \in I$, but $d$ was chosen to be the smallest positive element of $I$, so the only possibility is $r=0$.
197
198 Hence, $a=qd$, so $a \in (d)$, giving $I \subseteq (d)$.
199
200 Since we had $(d) \subseteq I$ and now we got $I \subseteq (d)$, we have $I = (d)$, so every ideal of $\mathbb{Z}$ is principal. Thus $\mathbb{Z}$ is a Principal Ideal Domain(PID).
201\end{proof}
202
203\begin{lemma}{}
204 $K[X]$ is a PID.
205\end{lemma}
206\begin{proof}
207 This proof follows very similarly to the previous proof.\\
208
209 Let $K$ be a field, $K[X]$ a polynomial ring.
210
211 Take $\{0\} \neq I \subseteq K[X]$.
212
213 Since $I \neq \{0\}$, there is at least one non-zero polynomial in $I$.
214
215 Let $p(X) \in I$ be of minimal degree among nonzero elements of $I$.
216
217 Observe that $(p(X)) \subseteq I$, because $p(X) \in I$ and $I$ is an ideal.
218
219 Let $f(X) \in I$. By Euclidean division algorithm in $K[X]$, $\exists q, r \in K[X]$ such that $f(X) = q(X) \cdot p(X) + r(X)$ with eithr $r(X)=0$ or $deg(r) < deg(p)$.
220
221 Since $f,p \in I$, then $r(X) = f(X) - q(X)\cdot p(X) \in I$
222
223 If $r(X) \neq 0$, then $deg(r) < deg(p)$, which contradicts the minimality of $deg(p)$ in $I$.
224
225 Therefore, $r(X)=0$, thus $f(X)=q(X)\cdot p(X)$, hence $f(X) \in (p(X))$.
226 Henceforth, $I \subseteq (p(X))$.
227
228 Then, since $(p(X)) \subseteq I$ and $I \subseteq (p(X))$, we have that $I = (p(X))$.
229
230 So every ideal of $K[X]$ is principal; thus $K[X]$ is a PID.
231
232\end{proof}
233
234
235
236\subsection{Lemmas, propositions and corollaries}
237
238Let $\Sigma$ be a partially orddered set. Given subset $S \subset \Sigma$, an \emph{upper bound} of $S$ is an element $u \in \Sigma$ such that $s<u \forall s \in S$.
239
240A \emph{maximal element} of $\Sigma$, is $m \in \Sigma$ such that $m<s$ does not hold for any $s \in \Sigma$.
241
242A subset $S \subset \Sigma$ is \emph{totally ordered} if for every pair $s_1,s_2 \in S$, either $s_1 \leq s_2$ or $s_2 \leq s_1$.
243
244\begin{lemma}{R.1.7}{Zorn's lemma} \label{zorn}
245 suppose $\Sigma$ a nonempty partially ordered set (ie. we are given a relation $x \leq y$ on $\Sigma$), and that any totally ordered subset $S \subset \Sigma$ has an upper bound in $\Sigma$.
246
247 Then $\Sigma$ has a maximal element.
248\end{lemma}
249
250\begin{thm}{AM.1.3} \label{1.3}
251 Every ring $A \neq 0$ has at least one maximal ideal.
252\end{thm}
253\begin{proof}
254 By Zorn's lemma \ref{zorn}.
255\end{proof}
256
257\begin{cor}{AM.1.4} \label{1.4}
258 if $I \neq (1)$ an ideal of $A$, $\exists$ a maximal ideal of $A$ containing $I$.
259\end{cor}
260
261\begin{cor}{AM.1.5} \label{1.5}
262 Every non-unit of $A$ is contained in a maximal ideal.
263\end{cor}
264
265\begin{defn}{Jacobson radical}
266 The \emph{Jacobson radical} of a ring $A$ is the intersection of all the maximal ideals of $A$.
267
268 Denoted $Jac(A)$.
269
270 $Jac(A)$ is an ideal of $A$.
271\end{defn}
272
273\begin{prop}{AM.1.9} \label{1.9}
274 $x \in Jac(A)$ iff $(1 - xy)$ is a unit in $A$, $\forall y \in A$.
275\end{prop}
276\begin{proof}
277 Suppose $1-xy$ not a unit.
278
279 By \ref{1.5}, $1-xy \in \mM$ for $\mM$ some maximal ideal.
280
281 But $x \in Jac(A) \subseteq \mM$, since $Jac(A)$ is the intersection of all maximal ideals of $A$.
282
283 Hence $xy \in \mM$, and therefore $1 \in \mM$, which is absurd, thus $1-xy$ is a unit.
284
285 Conversely:\\
286 Suppose $x \not\in \mM$ for some maximal ideal $\mM$.
287
288 Then $\mM$ and $x$ generate the unit ideal $(1)$, so that we have $u + xy = 1$ for some $u \in \mM$ and some $y \in A$.
289
290 Hence $1 -xy \in \mM$, and is therefore not a unit.
291\end{proof}
292
293\section{Modules}
294
295\subsection{Modules concepts}
296
297Let $A$ be a ring. An $A$-module is an Abelian group $M$ with a multiplication
298map
299\begin{align*}
300 A \times M &\longrightarrow M\\
301 (f, m) &\longmapsto fm
302\end{align*}
303satisfying $\forall~ f,g \in A,~~ m, n \in M$.
304\begin{enumerate}[i.]
305 \item $f(m \pm n)=fm \pm fn$
306 \item $(f \pm g) m = fm \pm gm$
307 \item $(fg) m = f(gm)$
308 \item $1_A m = m$
309\end{enumerate}
310
311Let $\psi: M \longrightarrow M$ an $A$-linear endomorphism of $M$.\\
312$A[\psi] \subset End M$ is the subring geneerated by $A$ and the action of $\psi$.
313\begin{itemize}
314\item since $\psi$ is $A$-linear, $A[\psi]$ is a commutative ring.
315\item $M$ is a module over $A[\psi]$, so $\psi$ beomes multiplication by a ring element.
316\end{itemize}
317
318\subsection{Cayley-Hamilton theorem, Nakayama lemma, and corollaries}
319
320\begin{prop}{AM.2.4}(Cayley-Hamilton Theorem) \label{2.4}
321 Let $M$ a fingen $A$-module. Let $\aA$ an ideal of $A$, let $\psi$ an
322 $A$-module endomorphism of $M$ such that $\psi(M) \subseteq \aA M$.
323
324 Then $\psi$ satisfies
325
326 $$\psi^n + a_1 \psi^{n-1} + \ldots + a_{n-1} \psi + a_n = 0$$
327
328 with $a_i \in \aA$.
329\end{prop}
330\begin{proof}
331 Since $M$ fingen, let $\{ x_1, \ldots, x_n \}$ be generators of $M$.\\
332 By hypothesis, $\psi(M) \subseteq \aA M$; so for any generator $x_i$, it's image $\psi(x_i) \in \aA M$.
333
334 Any element in $\aA M$ is a linear combination of the generators with coefficients in the ideal $\aA$, thus
335 $$\psi(x_i)= \sum_{j=1}^n a_{ij} x_j$$
336 with $a_{ij} \in \aA$.
337
338 Thus, for a module with $n$ generators, we have $n$ different $\psi(x_i)$ equations:
339
340 $$
341 \left.
342 \begin{aligned}
343 \psi(x_1) &= a_{1,1} x_1 + a_{1,2} x_2 + \ldots + a_{1,n} x_n\\
344 \psi(x_2) &= a_{2,1} x_1 + a_{2,2} x_2 + \ldots + a_{2,n} x_n\\
345 \ldots\\
346 \psi(x_n) &= a_{n,1} x_1 + a_{n,2} x_2 + \ldots + a_{n,n} x_n
347 \end{aligned}
348 \right\}
349 \begin{aligned}
350 &\text{n elements $\psi(x_i) \in \aA M$ which}\\
351 &\text{are linear combinations of the}\\
352 &\text{$n$ generators of $M$}
353 \end{aligned}
354 $$
355
356 Next step: rearrange in order to use matrix algebra.
357
358 Observe that each row equals $0$, and rearranging the elements at each row we get
359
360 $$
361 \left.
362 \begin{aligned}
363 &\psi(x_1) - (a_{1,1} x_1 + a_{1,2} x_2 + \ldots + a_{1,n} x_n) = 0\\
364 &\psi(x_2) - (a_{2,1} x_1 + a_{2,2} x_2 + \ldots + a_{2,n} x_n) = 0\\
365 &\ldots\\
366 &\psi(x_n) - (a_{n,1} x_1 + a_{n,2} x_2 + \ldots + a_{n,n} x_n) = 0
367 \end{aligned}
368 \right\}
369 $$
370
371 Then, group the $x_i$ terms together; as example, take the row $i=1$:
372 $$(\psi - a_{1,1})x_1 - a_{1,2} x_2 - \ldots - a_{1,n} x_n = 0$$
373
374 $$
375 \left.
376 \begin{aligned}
377 &~~~~(\psi - a_{1,1})x_1 - a_{1,2} x_2 - \ldots - a_{1,n} x_n = 0\\
378 &-a_{2,1} x_1 + (\psi - a_{2,2}) x_2 - \ldots - a_{2,n} x_n = 0\\
379 &\ldots\\
380 &-a_{1,1} x_1 - a_{1,2} x_2 - \ldots + (\psi - a_{1,n}) x_n = 0\\
381 \end{aligned}
382 \right\}
383 $$
384
385 So, $\forall i \in [n]$, as a matrix:
386
387 $$
388 \begin{pmatrix}
389 \psi - a_{1,1} & -a_{1,2} & \ldots & -a_{1,n}\\
390 -a_{2,1} & \psi-a_{2,2} & \ldots & -a_{2,n}\\
391 \vdots\\
392 -a_{n,1} & -a_{n,2} & \ldots & \psi-a_{n,n}\\
393 \end{pmatrix}
394 \begin{pmatrix}
395 x_1\\ x_2\\ \vdots\\ x_n
396 \end{pmatrix}
397 =
398 \begin{pmatrix}
399 0\\ 0\\ \vdots\\ 0
400 \end{pmatrix}
401 $$
402
403
404
405 Denote the previous matrix by $\Phi$. Let $m$ denote the vector $(x_1, x_2, \ldots, x_n)^T$ (ie. the vector of generators of the $A$-module $M$).\\
406 \hspace*{4em}Then we can write the previous equality as
407 \begin{equation}
408 \Phi \cdot m = 0
409 \label{eq:2.4.1}
410 \end{equation}
411
412 We know that
413 \begin{equation}
414 adj(\Phi) \Phi = det(\Phi) I
415 \label{eq:2.4.2}
416 \end{equation}
417 (aka. \emph{fundamental identity for the adjugate matrix}).
418
419 So if at \eqref{eq:2.4.1} we multiply both sides by $adj(\Phi)$,
420 \begin{align*}
421 adj(\Phi) \cdot \Phi \cdot &m = 0\\
422 (\text{recall from \eqref{eq:2.4.2}:}~ &adj(\Phi)\Phi=det(\Phi)\cdot I ~)\\
423 =det(\Phi) \cdot I \cdot &m = 0
424 \end{align*}
425
426 Thus,
427 \begin{align*}
428 det(\Phi) \cdot I \cdot &m = 0:\\
429 \begin{pmatrix}
430 det(\Phi) & 0 & \ldots & 0\\
431 0 & det(\Phi) & \ldots & 0\\
432 \vdots\\
433 0 & 0 & \ldots & det(\Phi)
434 \end{pmatrix}
435 \cdot
436 &\begin{pmatrix}
437 x_1\\ x_2\\ \vdots\\ x_n
438 \end{pmatrix}
439 =
440 \begin{pmatrix}
441 0\\ 0\\ \vdots\\ 0
442 \end{pmatrix}
443 \end{align*}
444
445 $\Longrightarrow$
446 \begin{equation}
447 det(\Phi) \cdot x_i = 0 ~~\forall i \in [n]
448 \label{eq:2.4.3}
449 \end{equation}
450
451 ie. $det(\Phi)$ is an \emph{annihilator} of the generators $x_i$ of $M$, thus
452 is an annihilator of the entire module $M$.
453
454
455 So, we're interested into calculating the $det(\Phi)$.
456
457 By the Leibniz formula,
458 $$\det(A) = \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i=1}^n a_{i, \sigma(i)}$$
459
460 thus,
461 $$det(\Phi) = \underbrace{(\psi - a_{11}) (\psi - a_{22}) \ldots (\psi - a_{nn})}_{\text{diagonal of $\Phi$, leading term of the determinant}} - \ldots$$
462
463 The \emph{determinant trick} is that the terms that go after the "leading term of the determinant", will belong to $\aA$ and their combinations with $\psi$ will not be bigger than $\psi^n$. Furthermore, when expanding it
464 \begin{itemize}
465 \item highest power is $1 \cdot \psi^n$
466 \item coefficient of $\psi^{n-1}$ is $-( \underbrace{ a_{11} + a_{22} + \ldots + a_{nn} }_{a_1})$,\\
467 where, since each $a_{ii} \in \aA,~~ a_1 \in \aA$
468 \item the rest of coefficients of $\psi^k$ are also elements in $\aA$
469 \end{itemize}
470
471 Therefore we have
472 $$det(\Phi) = \psi^n + a_1 \psi^{n-1} + a_2 \psi^{n-2} + \ldots + a_{n-1} \psi + a_n$$
473 with $a_i \in \aA$.
474
475 \vspace{0.5cm}
476
477 Now, notice that we had $det(\Phi) \cdot x_i = 0 ~\forall~ i\in [n]$.
478
479 The matrix $\Phi$ is the \emph{characteristic matrix}, $xI-A$, viewed as an operator. Then,
480 $$det(\Phi) = det(xI-A) = p(x)$$
481 where $p(x)$ is the \emph{characteristic polynomial}.
482
483 If a linear transformation turns every basis vector ($x_i$) into zero, then that transformation is the zero transformation. So in our case, $det(\Phi)$ is the zero transformation, thus $det(\Phi)=0$.
484 Therefore,
485 $$\psi^n + a_1 \psi^{n-1} + a_2 \psi^{n-2} + \ldots + a_{n-1} \psi + a_n = 0$$
486
487
488 %%%%%% OLD START
489
490 % \vspace{3cm}
491 %
492 % Kronecker delta:
493 % $\delta_{ij} =
494 % \begin{cases}
495 % 1 & \text{if } i = j,\\
496 % 0 & \text{otherwise}
497 % \end{cases}$
498 %
499 % With the Kronecker delta, $\psi(x_i)$ can be expressed as
500 % $$\psi(x_i) = \sum_{j=1}^n \delta_{ij} \psi(x_j)$$
501 % so the previous matrix can be characterized as
502 % $$\sum_{j=1}^n (\delta_{ij} \psi - a_{ij}) x_j = 0$$
503 %
504 % The entries of the matrix are \emph{endomorphisms} (elements of the ring $A[\psi]$)
505 % \begin{itemize}
506 % \item the term $(\psi - a_{11})$ is an operator that acts on $x_1$; as $(\psi(x_1)-a_{11}\cdot x_1)$
507 % \item the term $(-a_{12})$ is an operator that acts on $x_2$; as multiplication by it, ie. $(-a_{12} \cdot x_2)$
508 % \end{itemize}
509 %
510 % We need a single element $x \in A$ that \emph{annihilates} every $m \in M$ simultaneously, ie. $xM=0$. We're going to use the determinant for getting $x$.
511 %
512 % Since $A$ is a commutative ring, and $\psi$ commutes with any $a \in A$,
513 % the ring of operators $A[\psi]$ is a commutative ring.
514 %
515 % $\Longrightarrow~$ so we can treat the matrix as a matrix of real numbers and calculate its determinant.
516 %
517 %
518 % \vspace{0.75cm}
519 % This is called \emph{"the determinant trick"}.\\
520 % We're interested in the determinant because it is the only way to turn a system of multiple equations in a single scalar-like equation that describes the endomorphism $\psi$.\\
521 % $\rightarrow$ Because in module theory, we lack of "division", so can not "solve for $\psi$" the system of equations.\\
522 % $\rightarrow$ The determinant provides a way to find a polynomial that \emph{annihilates} the module; the \emph{characteristic polynomial}, which related $\psi$ to the ideal $\aA$
523 %
524 % $$det(M) \cdot x_i = 0~~ \forall i$$
525 % where $x_i$ are the generators of $M$.
526 %
527 % Use $\Phi$ to denote the previous matrix. The determinant is the only function that can take that matrix $\Phi$ and produce a single scalar $x=det(\Phi)$ such that the following identity holds: $adj(\Phi)\cdot \Phi=det(\Phi) \cdot \aA$.
528 % \vspace{0.5cm}
529 %
530 % Since $det(M)$ kills every generator, it must kill every element in $M$\\
531 % $\Longrightarrow~~ det(M)$ is the zero map.
532 %
533 % Leibniz formula of the determinant of an $n \times n$ matrix:
534 % $$
535 % det(M) = \sum_{\sigma \in S_n} sign(\sigma) \prod_{i=1}^n M_{i, \sigma(i)}
536 % $$
537 %
538 % so,
539 % $$(\psi - a_{11}) (\psi - a_{22}) \ldots (\psi - a_{nn})$$
540 % expanding it,
541 % \begin{itemize}
542 % \item highest power is $1 \cdot \psi^n$
543 % \item coefficient of $\psi^{n-1}$ is $-( \underbrace{ a_{11} + a_{22} + \ldots + a_{nn} }_{a_1})$,\\
544 % where, since each $a_{ii} \in \aA,~~ a_1 \in \aA$
545 % \item the rest of coefficients of $\psi^k$ are also elements in $\aA$
546 % \end{itemize}
547 %
548 % So we have
549 % $$p(\psi) = \psi^n + a_1 \psi^{n-1} + a_2 \psi^{n-2} + \ldots + a_{n-1} \psi + a_n$$
550 % with $a_i \in \aA$.
551 %
552 % Since this determinant annihilates the generators (ie. $det(M)x_i=0$), the resulting enddomorphism $p(\psi)$ is the zero map on the entire module $M$, so:
553 % $$\psi^n + a_1 \psi^{n-1} + a_2 \psi^{n-2} + \ldots + a_{n-1} \psi + a_n = 0$$
554 % with $a_i \in \aA$, as stated in the Cayley-Hamilton theorem.
555
556 %%%%% OLD END
557\end{proof}
558
559
560\vspace{0.5cm}
561
562\begin{cor}{AM.2.5} \label{2.5}
563 Let $M$ a fingen $A$-module, let $\aA$ an ideal of $A$ such that $\aA M = M$.
564
565 Then, $\exists~ x \equiv 1 \pmod \aA$ such that $xM = 0$.
566\end{cor}
567\begin{proof}
568 take $\psi = \text{identity}$. Then in Cayley-Hamilton (\ref{2.4}):
569 \begin{align*}
570 &\psi^n + a_1 \psi^{n-1} + a_2 \psi^{n-2} + \ldots + a_{n-1} \psi + a_n = 0\\
571 \Longrightarrow~ &id_M + a_1 id_M + a_2 id_M + \ldots + a_{n-1} id_M + a_n = 0\\
572 \Longrightarrow~ &(1 + a_1 + \ldots + a_n) id_M = 0
573 \end{align*}
574 apply it to $m \in M$, where since $id_M(m)=m$ (by definition of the identity), we then have
575 $$(1 + a_1 + \ldots + a_n) \cdot m = 0$$
576 with $a_i \in \aA$.
577
578 \begin{enumerate}[\text{part} i.]
579 \item $xM=0$:\\
580 Thus the scalar $x = (1 + a_1 + \ldots + a_n)$ annihilates every $m \in M$, ie. the entire module $M$.
581
582 \item $x \equiv 1 \pmod \aA$:\\
583 $x \equiv 1 \pmod \aA ~~ \Longleftrightarrow (x-1) \in \aA$\\
584 then from $x = (1 + \underbrace{a_1 + \ldots + a_n}_b) \in \aA$, set $b=a_1 + \ldots + a_n$,\\
585 so that $x=(1+b) \in \aA$.\\
586
587 Then $x-1 = (1+b)-1 = b \in \aA$\\
588 so $x-1 \in \aA$, thus $x \equiv 1 \pmod \aA$ as stated.
589 \end{enumerate}
590\end{proof}
591
592
593
594\begin{prop}{AM.2.6}{Nakayama's lemma} \label{2.6}
595 Let $M$ a fingen $A$-module, let $\aA$ an ideal of $A$ such that $\aA \subseteq Jac(A)$.
596
597 Then $\aA M = M$ implies $M=0$.
598\end{prop}
599\begin{proof}
600 By \ref{2.5}: since $\aA M = M$, we have $x M =0$ for some $x \equiv 1 \pmod {Jac(A)}$. (notice that at \ref{2.5} is $\pmod \aA$ but here we use $\pmod {Jac(A)}$, since we have $\aA \subseteq Jac(A)$).
601
602 (recall \ref{1.9}: $x \in Jac(A)$ iff $(1 - xy)$ is a unit in $A$, $\forall y \in A$).\\
603 By \ref{1.9}, $x$ is a unit in $A$ (thus $x^{-1}\cdot x=1$).
604
605 Hence $M = x^{-1} \cdot \underbrace{x~ \cdot M}_{=0~ \text{(by \ref{2.5})}} = 0$.
606
607 Thus, if $\aA M = M$ then $M=0$.
608\end{proof}
609
610
611
612\begin{cor}{AM.2.7} \label{2.7}
613 Let $M$ a fingen $A$-module, let $N \subseteq M$ a submodule of $M$, let $\aA \subseteq Jac(A)$ an ideal.
614
615 Then $M = \aA M + N \stackrel{\text{implies}}{\Longrightarrow} M=N$.
616\end{cor}
617\begin{proof}
618 The idea is to apply Nakayama (\ref{2.6}) to $M/N$.
619
620 Since $M$ fingen $\Longrightarrow~~ M/N$ is fingen and an $A$-module.
621
622 Since $\aA \subseteq Jac(A) ~\Longrightarrow~$ Nakayama applies to $M/N$ too.
623
624 By definition,
625 $$\aA M = \left\{ \sum a_i \cdot m_i ~~|~~ a_i \in \aA, m_i \in M \right\}$$
626 where $m_i$ are the generators of $M$.
627
628 Then, for $M/N$,
629 $$\aA (\frac{M}{N}) = \left\{ \sum a_i \cdot (m_i + N) ~~|~~ a_i \in \aA, m_i \in M \right\}$$
630
631 observe that $a_i(m_i+N)= a_i m_i +N$, thus
632 $$\sum_i a_i \cdot (m_i + N) = \underbrace{(\sum_i a_i \cdot m_i)}_{\in \aA M} + N \in \aA M + N$$
633
634 Hence,
635
636 \begin{equation}
637 \aA (\frac{M}{N}) = \left\{ x + N ~~|~~ x \in \aA M \right\} = \aA M + N
638 \label{eq:2.7.1}
639 \end{equation}
640
641 By definition, if we take $\frac{\aA M + N}{N}$, then
642 $$\frac{\aA M + N}{N} = \left\{ y + N ~~|~~ y \in \aA M +N \right\} = \aA M + N$$
643
644 thus every $y \in \aA M +N$ can be written as
645 $$y=x+n,~~ \text{with}~ x \in \aA M,~ n\in N$$
646 which comes from \eqref{eq:2.7.1}.
647
648 Thus, $y + N = (x+n)+N = x+N$, since $n \in N$ is zero in the quotient.
649
650 Hence, every element of $\frac{\aA M +N}{N}$ has the form
651 $$\frac{\aA M + N}{N} = \left\{ x + N ~~|~~ x \in \aA M \right\}$$
652 as in \eqref{eq:2.7.1}.
653
654 Thus
655 \begin{equation}
656 \aA (\frac{M}{N}) = \aA M + N = \frac{\aA M +N}{N}
657 \label{eq:2.7.2}
658 \end{equation}
659
660 By the Collorary assumption, $M = \aA M + N$; quotient it by $N$:
661 \begin{equation}
662 \frac{M}{N} = \frac{\aA M +N}{N}
663 \label{eq:2.7.3}
664 \end{equation}
665
666 So, from \eqref{eq:2.7.2} and \eqref{eq:2.7.3}:
667 $$\aA (\frac{M}{N}) = \aA M +N = \frac{\aA M +N}{N} = \frac{M}{N}$$
668 thus, $\aA (\frac{M}{N}) = \frac{M}{N}$.
669
670 By Nakayama's lemma \ref{2.6}, if $\aA (\frac{M}{N}) = \frac{M}{N} ~\stackrel{implies}{\Longrightarrow}~ \frac{M}{N}=0$
671
672 Note that
673 $$\frac{M}{N} = \{ m + N ~|~ m \in M \}$$
674 (the zero element in $\frac{M}{N}$ is the coset $N=0+N$)
675
676 Then, $\frac{M}{N}=0$ means that the quotient has exactly one element, the zero coset $N$.
677
678 Thus, every coset $m + N$ equals the zero coset $N$, so $m-0 \in N ~\Longrightarrow~ m \in N$.
679
680 Hence every $m \in M$ lies in $N$, ie. $\forall m \in M,~ m \in N$.
681
682 So $M \subseteq N$. But notice that by the Corollary, we had $N \subseteq M$, therefore $M = N$.
683
684 Thus, if $M = \aA M + N \stackrel{implies}{\Longrightarrow} M = N$.
685\end{proof}
686
687
688
689\begin{prop}{AM.2.8} \label{2.8}
690 Let $x_i \forall i \in [n]$ be elements of $M$ whose images $\frac{M}{m M}$ from a basis of this vector space. Then the $x_i$ generate $M$.
691\end{prop}
692\begin{proof}
693 Let $N$ submodule $M$, generated by the $x_i$.
694
695 Then the composite map $N \longrightarrow M \longrightarrow \frac{M}{m M}$ maps $N$ onto $\frac{M}{m M}$, hence $N + \aA M = M$, which by \ref{2.7} implies $N = M$.
696\end{proof}
697
698
699\begin{prop}{AM.2.10} \label{2.10}
700 Split exact sequence. TODO
701\end{prop}
702
703\section{Noetherian rings}
704
705\begin{defn}{}{Ascending Chain Condition}
706 A partially orddered set $\Sigma$ has the \emph{ascending chain condition} (a.c.c.) if every chain
707 $$s_1 \leq s_2 \leq \ldots \leq s_k \leq \ldots$$
708 eventually breaks off, that is, $s_k = s_{k+1} = \ldots$ for some $k$.
709\end{defn}
710
711$\Longrightarrow~ \Sigma$ has the a.c.c. iff every non-empty subset $S \subset \Sigma$ has a maximal element.\\
712\hspace*{2em} if $\empty \neq S \subset \Sigma$ does not have a maximal element, choose $s_1 \in S$, and for each $s_k$, an element $s_{k+1}$ with $s_k < s_{k+1}$, thus contradicting the a.c.c.
713
714\begin{defn}{R.3.2}{Noetherian ring}
715 Let $A$ a ring; 3 equivalent conditions:
716 \begin{enumerate}[i.]
717 \item the set $\Sigma$ of ideals of $A$ has the a.c.c.; in other words, every increasing chain of ideals
718 $$I_1 \subset I_2 \subset \ldots \subset I_k \subset \ldots$$
719 eventually stops, that is $I_k = I_{k+1}=\ldots$ for some $k$.
720 \item every nonempty set $S$ of iddeals has a maximal element
721 \item every iddeal $I \subset A$ is finitely generated
722 \end{enumerate}
723 If these conditions hold, then $A$ is \emph{Noetherian}.
724\end{defn}
725\begin{proof}
726 TODO
727\end{proof}
728
729\begin{defn}{R.3.4.D}{Noetherian modules}
730 An $A$-module $M$ is Noetherian if the submoles of $M$ have the a.c.c.,\\
731 that is, ay increasing chain
732 $$M_1 \subset M_2 \subset \ldots \subset M_k \subset \ldots$$
733 of submodules eventually stops.
734\end{defn}
735
736As in with rings, it is equivalent to say that
737\begin{enumerate}[i.]
738 \item any nonempty set of modulesof $M$ has a maximal element
739 \item every submodule of $M$ is finite
740\end{enumerate}
741
742
743
744\begin{prop}{R.3.4.P}
745 Let $0 \longrightarrow L \xrightarrow{\ \alpha \ } M \xrightarrow{\ \beta \ } N \longrightarrow 0$ be a s.e.s. (split exact sequence, \ref{2.10}).
746
747 Then, $M$ is Noetherian $\Longleftrightarrow~ L$ and $N$ are Noetherian.
748\end{prop}
749\begin{proof}
750 $\Longrightarrow$: trivial, since ascending chains of submodules in $L$ and $N$ correspond one-to-one to certain chains in $M$.
751
752 $\Longleftarrow$: suppose $M_1 \subset M_2 \subset \ldots \subset M_k \subset \ldots$ is an increasing chain of submodules of $M$.
753
754 Then identifying $\alpha(L)$ with $L$ and taking intersection gives a chain
755 $$L \cap M_1 \subset L \cap M_2 \subset \ldots \subset L \cap M_k \subset \ldots$$
756 of submodules of $L$, and applying $\beta$ gives a chain
757 $$\beta(M_1) \subset \beta(M_2) \subset \ldots \beta(M_k) \subset \ldots$$
758 of submodules of $N$.
759
760 Each of these two chains eventually stop, by the assumption on $L$ and $N$, so that we only need to prove the following lemma which completes the proof.
761\end{proof}
762
763\begin{lemma}{R.3.4.L}
764 for submodules $M_1 \subset M_2 \subset M$,
765 $$L \cap M_1 = L \cap M_2 ~\text{and}~ \beta(M_1) = \beta(M_2) ~\Longrightarrow~ M_1 = M_2$$
766\end{lemma}
767\begin{proof}
768 if $m\in M_2$, then $\beta(m) \in \beta(M_1) = \beta(M_2)$, so that there is an $n \in M_1$ such that $\beta(m)=\beta(m)$.
769
770 Then $\beta(m-n)=0$, so that
771 $$m - n \in M_2 \cap ker(\beta)=M_1 \cap ker(\beta)$$
772
773 Hence $m \in M_1$, thus $M_1 = M_2$.
774\end{proof}
775
776\newpage
777
778\section{Exercises}
779
780For the exercises, I follow the assignments listed at \cite{mit-course}.
781
782The exercises that start with \textbf{R} are the ones from the book \cite{reid}, and the ones starting with \textbf{AM} are the ones from the book \cite{am}.
783
784\subsection{Exercises Chapter 1}
785
786\begin{ex}{R.1.1}
787 Ring $A$ and ideals $I, J$ such that $I \cup J$ is not an ideal. What's the smallest ideal containing $I$ and $J$?
788\end{ex}
789\begin{proof}
790 Take ring $A= \mathbb{Z}$. Set $I = 2 \mathbb{Z},~ J=3 \mathbb{Z}$.
791
792 $I,~J$ are ideals of $A$ ($=\mathbb{Z}$). And $I \cup J = 2 \mathbb{Z} \cup 3 \mathbb{Z}$.\\
793 Observe that for $2 \in I,~ 3 \in J ~\Longrightarrow~ 2,3 \in I \cup J$, but $2+3 = 5 \not\in I \cup J$.
794
795 Thus $I \cup J$ is not closed under addition; thus is not an ideal.
796
797
798 Smallest ideal of $\mathbb{Z}$ ($=A$) containing $I$ and $J$ is their sum:
799
800 $$I+J = \{ a+b | a \in I, b \in J \}$$
801
802 $gcd(2,3)=1$, so $I+J = \mathbb{Z}$.
803
804 Therefore, smallest ideal containing $I$ and $J$ is the whole ring $\mathbb{Z}$.
805\end{proof}
806
807\begin{ex}{R.1.5}
808 let $\psi: A \longrightarrow B$ a ring homomorphism. Prove that $\psi^{-1}$ takes prime ideals of $B$ to prime ideals of $A$.\\
809 In particular if $A \subset B$ and $P$ a prime ideal of $B$, then $A \cap P$ is a prime ideal of $A$.
810\end{ex}
811\begin{proof}
812 (Recall: prime ideal is if $a,b \in R$ and $a \cdot b \in P$ (with $R \neq P$), implies $a \in P$ or $b \in P$).
813
814 Let
815 $$\psi^{-1}(P) = \{ a \in A | \psi(a) \in P \} = A \cap P$$
816 The claim is that $\psi^{-1}(P)$ is prime iddeal of $A$.
817
818 \begin{enumerate}[i.]
819 \item show that $\psi^{-1}(P)$ is an ideal of $A$:\\
820 $0_A \in \psi^{-1}(P)$, since $\psi(0_A)=0_B \in P$ (since every ideal contains $0$).
821
822 If $a,b \in \psi^{-1}(P)$, then $\psi(a), \psi(b) \in P$, so
823 $$\psi(a-b)= \psi(a) - \psi(b) \in P$$
824 hence $a-b \in \psi^{-1}(P)$.
825
826 If $a \in \psi^{-1}(P)$ and $r \in A$, then $\psi(ra) = \psi(r) \psi(a) \in P$, since $P$ is an ideal.\\
827 Thus $ra \in \psi^{-1}(P)$.
828
829 $\Longrightarrow$ so $\psi^{-1}$ is an ideal of $A$.
830
831 \item show that $\psi^{-1}(P)$ is prime:\\
832 $\psi^{-1}(P) \neq A$, since if $\psi^{-1}(P)=A$, then $1_A \in \psi^{-1}(P)$, so $\psi(1_A)=1_B \in P$, which would mean that $P=B$, a contradiction since $P$ is prime ideal of $B$.
833
834 Take $a,b \in A$ with $ab \in \psi^{-1}(P)$; then $\psi(ab) \in P$, and since $\psi$ is a ring homomorphism, $\psi(ab) = \psi(a)\psi(b)$.
835
836 Since $P$ prime ideal, then $\psi(a)\psi(b) \in P$ implies either $\psi(a) \in P$ or $\psi(b) \in P$.\\
837 Thus $a \in \psi^{-1}(P)$ or $b \in \psi^{-1}(P)$.
838
839 Hence $\psi^{-1}(P)$ ($=A \cap P$) is a prime ideal of $A$.
840 \end{enumerate}
841\end{proof}
842
843
844\begin{ex}{R.1.6}
845 prove or give a counter example:
846 \begin{enumerate}[a.]
847 \item the intersection of two prime ideals is prime
848 \item the ideal $P_1+P_2$ generated by $2$ prime ideals $P_1,P_2$ is prime
849 \item if $\psi: A \longrightarrow B$ ring homomorphism, then $\psi^{-1}$ takes maximal ideals of $B$ to maximal ideals of $A$
850 \item the map $\psi^{-1}$ of Proposition 1.2 takes maximal ideals of $A/I$
851 to maximal ideals of $A$
852 \end{enumerate}
853\end{ex}
854\begin{proof}
855 \begin{enumerate}[a.]
856 \item let $I = 2 \mathbb{Z} = (2)$, $J = 3 \mathbb{Z} = (3)$ be ideals of $\mathbb{Z}$, both prime.
857
858 Then $I \cap J = (2) \cap (3) = (6)$.
859
860 The ideal $(6)$ is not prime in $\mathbb{Z}$, since $2 \cdot 3 \in (6)$, but $2 \neq (6)$ and $3 \neq (6)$.
861
862 Thus the intersection of two primes can not be prime.
863
864 \item $P_1=(2),~ P_2=(3)$, both prime.
865
866 Then,
867 $$P_1 + P_2 = (2)+(3)=\{ a+b | a \in P_1, b \in P_2 \}$$
868
869 $\longrightarrow~$ in a principal ideal domain (like $\mathbb{Z}$), the sum of two principal ideals is again principal, and given by $(m)+(n)=(gcd(m,n))$.
870
871 (recall: principal= generated by a single element)
872
873 So, $P_1+P_2= (2)+(3) = (gcd(2,3))=(1)=\mathbb{Z}$.
874
875 The whole ring is not a prime ideal (by the definition of the prime ideal), so $P_1+P_2$ is not a prime ideal.
876
877 Henceforth, the sum of two prime ideals is not necessarily prime.
878
879 \item let $A=\mathbb{Z},~ B=\mathbb{Q},~ \psi: A \longrightarrow B$.
880
881 Since $\mathbb{Q}$ is a field, its only maximal ideal is $(0)$.
882
883 Then
884 \begin{align*}
885 \psi^{-1}( (0) ) &= (0) \subset \mathbb{Z}\\
886 \text{ie.}~ \psi^{-1}( m_B ) &= (m_B) \subset A
887 \end{align*}
888
889 But $(0)$ is not maximal in $\mathbb{Z}$, because $\mathbb{Z}/(0) \cong \mathbb{Z}$ is not a field.
890
891 Thus the preimages of maximal ideals under arbitrary ring homomorphisms need not be maximal.
892
893 \item $\psi: A \longrightarrow A/I$ quotient homomorphism, $I \subseteq A$ an ideal.
894
895 Let $M$ a maximal ideal of $A/I$, then $\frac{(A/I)}{M}$ is a field (Proposition 1.3).
896
897 By the isomorphism theorems,
898 $$\frac{(A/I)}{M} \cong \frac{A}{\psi^{-1}(M)}$$
899
900
901 Since $\frac{(A/I)}{M}$ is a field, the quotient $\frac{A}{\psi^{-1}(M)}$ is a field, so $\psi^{-1}(M)$ is a maximal ideal of $A$.
902
903 $\Longrightarrow~$ under $\psi$, preimages of maximal ideals are maximal.
904 \end{enumerate}
905\end{proof}
906
907\begin{ex}{R.1.12.a}
908 if $I,J$ ideals and $P$ prime ideal, prove that
909 $$IJ \subset P ~\Longleftrightarrow~ I \cap J \subset P ~\Longleftrightarrow~ I ~\text{or}~ J \subset P$$
910\end{ex}
911\begin{proof}
912 assume $I \subseteq P$ (for $J \subseteq P$ will be the same, symmetric), take $x \in IJ$,
913
914 then
915 $$x = \sum_{k=1}^n a_k b_k$$
916 with $a_k \in I,~ b_k \in J$.
917
918 Each $a_k \in I \subseteq P$. Since $P$ an ideal,
919 $$\sum_{k=1}^n a_k b_k \in P$$
920
921 thus $x \in P$, hence $IJ \subseteq P$.
922
923 So $I \subseteq P$ or $J \subseteq P$ $~\Longrightarrow IJ \subseteq P$.
924
925 \vspace{0.5cm}
926 Conversely,\\
927 assume $P$ prime and $IJ \subseteq P$.
928
929 Suppose by contradiction that $I \not\subseteq P$ and $J \not\subseteq P$.
930 \begin{itemize}
931 \item[-] since $I \not\subseteq P,~ \exists a \in I$ with $a \not\in P$
932 \item[-] since $J \not\subseteq P,~ \exists b \in J$ with $b \not\in P$
933 \end{itemize}
934 Since $a \in I,~ b \in J,~~ ab \in IJ \subseteq P$, but $P$ is prime, so $ab \in P$ implies that $a \in P$ or $b \in P$. This contradicts $a,b$ being taken outside of $P$.
935
936 Thus $I \not\subseteq P$ and $J \not\subseteq P$ are false.
937
938 \vspace{0.3cm}
939 So both directions are proven, hence
940 $$IJ \subseteq P ~\Longrightarrow~ I \subseteq P ~\text{or}~ J \subseteq P$$
941
942\end{proof}
943
944\begin{ex}{R.1.18}
945 Use Zorn's lemma to prove that any prime ideal $P$ contains a minimal prime ideal.
946\end{ex}
947\begin{proof}
948 Let $P$ prime ideal of $R$.
949
950 $$S = \{ Q \subseteq R ~|~ Q ~\text{a prime ideal AND}~ Q \subseteq P \}$$
951
952 Goal: show that $S$ has a minimal element, the minimal ideal contained in $P$.
953
954 $P \subset S$, so $S$ is nonempty.
955
956 Let $C \subseteq S$ be a chain (= totally ordered subset) with respect to inclusion.
957
958 Define
959 $$Q_C = \bigcap_{Q \in C} Q$$
960
961 Clearly $Q_C \subseteq P$, since each $Q \in C$ is $Q \subseteq P$.
962
963 Since $C$ is ordered by inclusion, it is a decreasing chain of prime ideals.
964
965 Intersection of a decreasing chain of prime ideals is again a prime ideal:
966 \begin{itemize}
967 \item[-] if $ab \in Q_C$, then $ab \in Q ~\forall Q \in C$
968 \item[-] since $Q$ prime, $\forall Q \in C$ either $a \in Q$ or $b \in Q$
969 \end{itemize}
970
971 If there were some $Q_1,~ Q_2 \in C$ with $a \in Q_1$ and $b \not\in Q_2$, then by total ordering, either $Q_1 \subseteq Q_2$ or $Q_2 \subseteq Q_1$.
972
973 In either case: contradiction, since the smaller one would have to contain the element that was assumed to be excluded.
974
975 Thus $\forall Q \in C$ the same element $a, b$ must lie in all $Q$. $\Longrightarrow~$ lies in the intersection of them, $Q_C$.
976
977 Henceforth, $Q_C$ is a prime ideal and lies in $S$, and its a lower bound of $C$ in $S$.
978
979 Now, $S$ is nonempty, and every chain in $S$ has a lower bound in $S$ (its intersection).\\
980 Therefore, $S$ has a minimal element $P_{min}$.
981
982 By construction, $P_{min}$ is a prime ideal $P_{min} \subseteq P$, and by minimality there are no strictly smaller prime ideals inside $P$.
983
984 So $P_{min}$ is a minimal prime ideal, contained in $P$.
985\end{proof}
986
987\begin{ex}{R.1.10}
988\end{ex}
989\begin{proof}
990\end{proof}
991
992\begin{ex}{R.1.11}
993\end{ex}
994\begin{proof}
995\end{proof}
996
997\begin{ex}{R.1.4}
998\end{ex}
999\begin{proof}
1000\end{proof}
1001
1002\subsection{Exercises Chapter 2}
1003
1004\bibliographystyle{unsrt}
1005\bibliography{commutative-algebra-notes.bib}
1006
1007\end{document}