Throughout, rings are commutative with identity.
Zorn’s lemma
Chain in a poset (Tag D-P6XOT)
Let \((X,\leq)\) be a partially ordered set. A chain in \(X\) is a subset \(C\subseteq X\) that is totally ordered by \(\leq\): for all \(c,c'\in C\), \(c\leq c'\) or \(c'\leq c\).
Let \((X,\leq)\) be a partially ordered set and \(S\subseteq X\). An upper bound of \(S\) is an element \(x\in X\) such that \(s\leq x\) for all \(s\in S\).
Zorn's lemma (Tag T-V7JRG)
If \(P\) is a poset in which every chain has an upper bound in \(P\), then \(P\) has a maximal element.
Zorn's lemma (Tag FT-PUVIQ)
Let \((P,\le)\) be a partially ordered set in which every chain has an upper bound in \(P\). Then \(P\) has a maximal element: an element \(m\in P\) such that \(m\le x\) with \(x\in P\) implies \(x=m\).
The empty chain is a chain, so the hypothesis forces \(P\) to be nonempty.
Krull's theorem on maximal ideals (Tag T-QSTRJ)
Let \(R\) be a nonzero ring. Then every proper ideal of \(R\) is contained in a proper maximal ideal; in particular, \(R\) has a proper maximal ideal.
Krull's theorem on maximal ideals (Tag FF-QELG7)
Let \(R\) be a commutative ring. Every proper ideal \(I\subsetneq R\) is contained in a maximal ideal of \(R\). In particular, applying this to \(I=0\), every nonzero commutative ring has a maximal ideal.
Let \(P\) be the set of proper ideals of \(R\) containing \(I\), ordered by inclusion; it contains \(I\). The union of a nonempty chain in \(P\) is an ideal containing \(I\), and it is proper because it does not contain \(1\), so it is an upper bound in \(P\); the empty chain has the upper bound \(I\). By Zorn's lemma, \(P\) has a maximal element, which is a maximal ideal of \(R\) containing \(I\).
Existence of maximal ideals (Tag PR-UTFSY)
Let \(R\) be a ring and \(I\subsetneq R\) a proper ideal. Then \(I\) is contained in a maximal ideal of \(R\).
Let \(\mathcal S\) be the set of proper ideals of \(R\) containing \(I\), ordered by inclusion; it is nonempty since \(I\in\mathcal S\). If \(\mathcal C\subseteq\mathcal S\) is a nonempty chain, then \(J\coloneqq\bigcup_{J'\in\mathcal C}J'\) is an ideal containing \(I\), and \(1\notin J\) because \(1\) lies in no proper ideal, so \(J\in\mathcal S\) is an upper bound for \(\mathcal C\). By Zorn’s lemma \(\mathcal S\) has a maximal element \(\mathfrak m\). Any ideal strictly containing \(\mathfrak m\) contains \(I\) and is not in \(\mathcal S\), so it equals \(R\); thus \(\mathfrak m\) is a maximal ideal.
Every non-unit of $R$ is contained in a maximal ideal (Tag E-NXHG6)
Show that every non-unit of \(R\) is contained in a maximal ideal.
Solution
This follows because if \(x\in R\sm \unitsof{R}\), then \(Rx \normal R\) and \(Rx\neq R\) implies \(R/Rx \neq 0\). Then there exists some \(\bar \mfm \in \mspec R/Rx\), and by the correspondence theorem this lifts to some \(\mfm \in \mspec R\) containing \(Rx\).
(Existence statements proved by Zorn's lemma)
Every nonzero ring has a maximal ideal, every proper ideal is contained in a maximal ideal, and every vector space has a basis. Each proof applies Zorn’s lemma to a poset of partial objects ordered by inclusion – proper ideals containing a given ideal, or linearly independent subsets – in which the union of a chain is again an element of the poset and is an upper bound for the chain.
Nakayama’s lemma
Let \((R,\mfm)\) be a local ring and \(M\) a finitely generated \(R\)-module. If \(\mfm M=M\), then \(M=0\). Equivalently, elements \(m_1,\ldots,m_n\in M\) generate \(M\) if and only if their images span the \(R/\mfm\)-vector space \(M/\mfm M\).
Nakayama's lemma (Tag FF-NREXC)
Let \(R\) be a commutative ring, let \(I\subseteq R\) be an ideal, and let \(M\) be a finitely generated \(R\)-module with \(IM=M\). Then there is \(i\in I\) with \(im=m\) for every \(m\in M\). In particular, if \(I\) is contained in the Jacobson radical \(J(R)\), then \(M=0\).
Let \((R,\mfm)\) be a local ring, let \(M\) and \(N\) be \(R\)-modules with \(N\) finitely generated, and let \(\varphi\colon M\to N\) be \(R\)-linear. Then \(\varphi\) is surjective if and only if the induced map \(M/\mfm M\to N/\mfm N\) of \(R/\mfm\)-vector spaces is surjective.
Let \(m_1,\ldots,m_n\) generate \(M\). Since \(M=IM\), there are \(a_{jk}\in I\) with \(m_j=\sum_ka_{jk}m_k\), so the matrix \(\mathrm{I}_n-A\), with \(A=(a_{jk})\), kills the column vector \((m_1,\ldots,m_n)^t\). Multiplying by the adjugate of \(\mathrm{I}_n-A\) gives \(\det(\mathrm{I}_n-A)\,m_j=0\) for every \(j\). Expanding the determinant, \(\det(\mathrm{I}_n-A)=1-i\) with \(i\in I\), so \(m=im\) for every \(m\in M\). If \(I\subseteq J(R)\), then \(1-i\) is a unit by FF-QWCKR, so \(M=0\).
For the local statement, only the converse needs proof. If \(M/\mfm M\to N/\mfm N\) is surjective, then \(N=\varphi(M)+\mfm N\), so the finitely generated module \(C=N/\varphi(M)\) satisfies \(\mfm C=C\), and \(C=0\) by the first part with \(I=\mfm=J(R)\).
Nakayama's lemma for a local ring (Tag FF-45SK3)
Let \((R,\mfm)\) be a local ring with residue field \(k=R/\mfm\), and let \(M\) be a finitely generated \(R\)-module. Elements \(m_1,\ldots,m_n\in M\) generate \(M\) if and only if their images span the \(k\)-vector space \(M/\mfm M\). Consequently, a generating set of \(M\) is minimal under inclusion if and only if its image is a \(k\)-basis of \(M/\mfm M\), and every minimal generating set of \(M\) has \(\dim_k M/\mfm M\) elements.
If \((m_1,\ldots,m_n)\) and \((m_1',\ldots,m_n')\) are minimal generating sets of \(M\), then there is \(P\in\GL_n(R)\) with \(m_i'=\sum_jP_{ij}m_j\) for every \(i\).
If the images of \(m_1,\ldots,m_n\) span \(M/\mfm M\), let \(N=Rm_1+\cdots+Rm_n\). Then \(M=N+\mfm M\), so \(\mfm(M/N)=M/N\), and \(M/N=0\) by Nakayama's lemma, since \(M/N\) is finitely generated and \(\mfm=J(R)\). The converse is immediate.
If the image of a generating set is linearly dependent, some element has image in the span of the images of the others, and those others still generate \(M\) by the first part; if the image is a basis, no proper subset has spanning image.
For the last claim, write \(m_i'=\sum_jP_{ij}m_j\) and \(m_j=\sum_lQ_{jl}m_l'\) with \(P,Q\in\Mat_n(R)\). Reducing modulo \(\mfm\), the matrices \(\overline P\) and \(\overline Q\) are the change-of-basis matrices between two bases of \(M/\mfm M\), so \(\overline P\,\overline Q=I\). Hence \(\det P\notin\mfm\), so \(\det P\) is a unit of \(R\) and \(P\in\GL_n(R)\).
Finitely generated flat modules over a Noetherian local ring are free (Tag FF-QILDV)
Let \((R,\mfm)\) be a Noetherian local ring and let \(M\) be a finitely generated flat \(R\)-module. Then \(M\) is free.
Let \(k=R/\mfm\) and \(n=\dim_kM/\mfm M\). By Nakayama's lemma for a local ring, lifts of a \(k\)-basis of \(M/\mfm M\) give a surjection \(R^n\to M\); let \(K\) be its kernel, which is finitely generated because \(R\) is Noetherian. Since \(M\) is flat, \(\Tor_1^R(k,M)=0\), so applying \(k\otimes_R-\) to \(0\to K\to R^n\to M\to0\) gives an exact sequence \[ 0\to K/\mfm K\to k^n\to M/\mfm M\to0. \] The map \(k^n\to M/\mfm M\) is a surjection between \(k\)-vector spaces of dimension \(n\), hence an isomorphism, so \(K/\mfm K=0\). By Nakayama's lemma, \(K=0\), and \(M\cong R^n\).
Applied to stalks, this shows that a coherent sheaf \(\mathcal F\) on a Noetherian scheme \(X\) whose stalks \(\mathcal F_x\) are flat \(\OO_{X,x}\)-modules has free stalks, and is therefore locally free, that is, the sheaf of sections of a vector bundle.
Geometric interpretation of Nakayama's lemma (Tag FF-6K35J)
Let \(X\) be a Noetherian scheme, let \(\mathcal F\) be a coherent \(\OO_X\)-module, let \(p\in X\), and let \(\kappa(p)=\OO_{X,p}/\mfm_p\). The fiber \(\mathcal F(p)\coloneqq\mathcal F_p/\mfm_p\mathcal F_p\) of \(\mathcal F\) at \(p\) is a finite-dimensional \(\kappa(p)\)-vector space.
Every lift to the stalk \(\mathcal F_p\) of a basis of \(\mathcal F(p)\) is a minimal generating set of \(\mathcal F_p\).
Given such a lift, there are an open neighborhood \(U\) of \(p\) and sections \(s_1,\ldots,s_n\in\mathcal F(U)\) with these germs at \(p\) that generate \(\mathcal F_q\) for every \(q\in U\).
If \(\mathcal F\) is locally free near \(p\), the sections \(s_1,\ldots,s_n\) of (b) can be chosen so that \(\OO_U^n\to\mathcal F|_U\), \((f_i)\mapsto\sum_if_is_i\), is an isomorphism; that is, a basis of the fiber of a vector bundle at \(p\) extends to a local frame on a neighborhood of \(p\).
The stalk \(\mathcal F_p\) is a finitely generated module over the local ring \(\OO_{X,p}\), so (a) is Nakayama's lemma for a local ring.
For (b), choose an open neighborhood \(V\) of \(p\) on which all the germs are represented by sections \(s_i\in\mathcal F(V)\), and let \(\mathcal C\) be the cokernel of \(\OO_V^n\to\mathcal F|_V\). Then \(\mathcal C\) is coherent with \(\mathcal C_p=0\), and the support of a coherent sheaf is closed, so \(\mathcal C\) vanishes on an open neighborhood \(U\subseteq V\) of \(p\).
For (c), shrink \(U\) to an affine open \(\Spec A\) containing \(p\) on which \(\mathcal F\) is free; its rank is \(\dim_{\kappa(p)}\mathcal F(p)=n\). Since \(\OO_U^n\to\mathcal F|_U\) is surjective and \(U\) is affine, the map \(A^n\to\mathcal F(\Spec A)\cong A^n\) is a surjective endomorphism of a finitely generated \(A\)-module, hence an isomorphism.
Closed immersion criterion for morphisms of smooth projective varieties (Tag FF-X6C7Z)
Let \(k\) be an algebraically closed field and \(f\colon X\to Y\) a morphism of smooth projective varieties over \(k\). Then \(f\) is a closed immersion if and only if \(f\) is injective on closed points and the differential \(df_p\colon T_pX\to T_{f(p)}Y\) is injective for every closed point \(p\in X\) [@Har10a].
The application of Nakayama’s lemma is in the local step: a local homomorphism \(\mathcal O_{Y,f(p)}\to\mathcal O_{X,p}\) of Noetherian local rings that is an isomorphism on residue fields, surjective on cotangent spaces and makes \(\mathcal O_{X,p}\) a finitely generated module is surjective [@Har10a].
Noetherian rings and Krull dimension
A ring is Noetherian if every ascending chain of ideals stabilizes, equivalently if every ideal is finitely generated. Let \(R\) be a Noetherian ring and \(I\subseteq R\) an ideal.
By Krull’s principal ideal theorem, a prime ideal minimal over a principal ideal has height at most \(1\).
By the Artin–Rees lemma, for a finitely generated \(R\)-module \(M\) and a submodule \(N\subseteq M\) there is \(c\geq 0\) with \(I^nM\intersect N = I^{n-c}(I^cM\intersect N)\) for all \(n\geq c\).
By Krull’s intersection theorem, if \(R\) is local and \(I\) is proper, then \(\Intersect_{n\geq 1}I^nM=0\) for every finitely generated \(R\)-module \(M\).
Krull's Hauptidealsatz (Tag FF-ESIOA)
Let \(R\) be a commutative Noetherian ring and let \(a\in R\) be neither a zero divisor nor a unit. Then every prime ideal \(\mfp\) minimal among the prime ideals containing \(a\) has height \(1\).
Krull's intersection theorem (Tag FF-3K36R)
Let \(R\) be a commutative Noetherian ring and let \(I\subseteq R\) be an ideal contained in the Jacobson radical \(J(R)\). Then \[ \bigcap_{n\ge0}I^n=0. \] In particular, \(\bigcap_{n\ge0}\mfm^n=0\) for every Noetherian local ring \((R,\mfm)\).
Let \(N=\bigcap_{n\ge0}I^n\). By the Artin--Rees lemma applied to \(N\subseteq R\), there is \(c\ge0\) with \(I^n\cap N=I^{n-c}(I^c\cap N)\) for all \(n\ge c\). Since \(N\subseteq I^n\) for every \(n\), taking \(n=c+1\) gives \(N=I(I^c\cap N)=IN\). The ideal \(N\) is finitely generated because \(R\) is Noetherian, so Nakayama's lemma with \(I\subseteq J(R)\) gives \(N=0\).
Geometric interpretation of Krull's intersection theorem (Tag FF-HJGPV)
Let \(X\) be a locally Noetherian scheme, for example a variety, let \(p\in X\), and let \((\OO_{X,p},\mfm_p)\) be the local ring at \(p\). For \(f\in\OO_{X,p}\), define the order of vanishing of \(f\) at \(p\) as \[ \ord_p(f)\coloneqq\sup\theset{n\ge0\suchthat f\in\mfm_p^n}\in\ZZ_{\ge0}\cup\theset{\infty}. \] Then \(\ord_p(f)=\infty\) if and only if \(f=0\): a nonzero function vanishes to finite order at \(p\).
The local ring \(\OO_{X,p}\) is Noetherian, and \(\ord_p(f)=\infty\) means \(f\in\bigcap_{n\ge0}\mfm_p^n\), which is \(0\) by Krull's intersection theorem.
Nakayama's lemma for a local ring gives the companion statement for generators. Let \(k\) be a field and let \(\mfm_0\subseteq\OO_{\AA^n_k,0}\) be the maximal ideal at the origin. Functions \(f_1,\ldots,f_n\in\mfm_0\) generate \(\mfm_0\), that is, form a local coordinate system at \(0\), if and only if their classes form a basis of the cotangent space \(\mfm_0/\mfm_0^2\). The class of \(f\) in \(\mfm_0/\mfm_0^2\) is its linear part, the differential \(df_0=\sum_i\frac{\partial f}{\partial x_i}(0)\,dx_i\), so the condition is that the Jacobian matrix \(\qty{\frac{\partial f_i}{\partial x_j}(0)}\) is invertible, as in the inverse function theorem.
The Artin--Rees lemma (Tag FF-CSABG)
Let \(R\) be a commutative Noetherian ring, let \(I\subseteq R\) be an ideal, let \(M\) be a finitely generated \(R\)-module, and let \(N\subseteq M\) be a submodule. Then there is an integer \(c\ge0\) such that for every \(n\ge c\), \[ I^nM\cap N=I^{n-c}\qty{I^cM\cap N}. \]
Integral extensions
For an integral extension of rings \(A\subseteq B\), every prime of \(A\) is the contraction of a prime of \(B\), and by the going-up theorem a chain of primes \(\mfp_1\subseteq\cdots\subseteq\mfp_n\) of \(A\) together with a prime \(\mathfrak q_1\) of \(B\) satisfying \(\mathfrak q_1\intersect A=\mfp_1\) extends to a chain \(\mathfrak q_1\subseteq\cdots\subseteq\mathfrak q_n\) of primes of \(B\) with \(\mathfrak q_i\intersect A=\mfp_i\).
Lying over and going up for integral extensions (Tag FF-IBDAT)
Let \(R\subseteq S\) be an integral extension of commutative rings.
Lying over: for every prime ideal \(\mfp\subseteq R\) there is a prime ideal \(\mathfrak q\subseteq S\) with \(\mathfrak q\cap R=\mfp\). Equivalently, the map \(\Spec S\to\Spec R\), \(\mathfrak q\mapsto\mathfrak q\cap R\), is surjective.
Going up: if \(\mfp_1\subseteq\mfp_2\) are prime ideals of \(R\) and \(\mathfrak q_1\) is a prime ideal of \(S\) with \(\mathfrak q_1\cap R=\mfp_1\), then there is a prime ideal \(\mathfrak q_2\supseteq\mathfrak q_1\) of \(S\) with \(\mathfrak q_2\cap R=\mfp_2\).
Localization
Field of fractions (Tag D-OXIVT)
Let \(R\) be an integral domain. On \(R \times (R\setminus\theset{0})\) define \((a,s)\sim (b, t)\) if \(at = bs\), and write \(a/s\) for the class of \((a,s)\). The field of fractions of \(R\) is the set of classes \[ \ff(R) \coloneqq \bigl(R \times (R\setminus\theset{0})\bigr)/{\sim}, \] with operations \(\frac as+\frac bt=\frac{at+bs}{st}\) and \(\frac as\cdot\frac bt=\frac{ab}{st}\).
Transitivity of \(\sim\) uses that \(R\) has no nonzero zero divisors: if \(at=bs\) and \(bu=ct\), then \(aut=bsu=cst\), so \(t(au-cs)=0\), and \(t\neq0\) gives \(au=cs\). The same hypothesis gives \(st\neq 0\), so the operations are defined on \(R\times(R\setminus\theset{0})\).
For a submonoid \(S\leq (R,\cdot)\), write \(S^{-1}R\) for the localization of \(R\) at \(S\); its construction and universal property are in the Stacks Project, Tag 00CM.
The canonical map \[ \begin{aligned} R &\to R\localize{S} \\ x &\mapsto {x\over 1} \end{aligned} \] need not be injective: its kernel is \(\ts{x\in R \st sx=0 \text{ for some } s\in S}\).
For an integral domain \(R\), \[ \ff(R) \cong R\localize{ (\nonzeroof{R}) }. \]
Hilbert’s basis theorem and primary ideals
Hilbert basis theorem (Tag T-YYLPH)
If \(R\) is a commutative Noetherian ring, then the polynomial ring \(R[x]\) is Noetherian. By induction, \(R[x_1, \ldots, x_n]\) is Noetherian for every \(n\); in particular every ideal of \(k[x_1, \ldots, x_n]\) over a field \(k\) is finitely generated, and every ideal of \(\ZZ[x_1,\ldots,x_n]\) is finitely generated.
[@DF04].
Primary ideal (Tag D-JGYK4)
Let \(R\) be a commutative ring. A proper ideal \(I\subsetneq R\) is primary if for all \(p,q\in R\) with \(pq\in I\), either \(p\in I\) or there exists \(n\geq 1\) such that \(q^n\in I\).
In a commutative ring, every maximal ideal is prime, and every prime ideal is both radical and primary. [@AM18]
In \(k[x,y]\) the ideal \((xy)\) is radical but not primary, since \(x \cdot y \in (xy)\) while \(x \notin (xy)\) and no power of \(y\) lies in \((xy)\). In \(k[x]\) the ideal \((x^2)\) is primary but not radical, since \(x \notin (x^2)\) and \(x^2 \in (x^2)\).
If \(R\) is Noetherian, then so is \(R[x_1,\ldots,x_n]\), by Hilbert’s basis theorem and induction on \(n\). An ideal \(\mathfrak q\subsetneq R\) is primary if every zero divisor of \(R/\mathfrak q\) is nilpotent; every prime ideal is primary, and by the Lasker–Noether theorem every ideal of a Noetherian ring is a finite intersection of primary ideals.