more-algebra-lemma-factor-through-derived-tensor-product

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. There exists an integer $n > 0$ such that $I^nM \to M$ factors through the map $I \otimes_A^\mathbf{L} M \to M$ in $D(A)$.


more-algebra-lemma-ext-annihilated-into

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $K \in D(A)$ be pseudo-coherent. Let $a \in \mathbf{Z}$. Assume that for every finite $A$-module $M$ the modules $\Ext^i_A(K, M)$ are $I$-power torsion for $i \geq a$. Then for $i \geq a$ and $M$ finite the system $\Ext^i_A(K, M/I^nM)$ is essentially constant with value $$ \Ext^i_A(K, M) = \lim \Ext^i_A(K, M/I^nM) $$


more-algebra-lemma-tor-annihilated

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$. There exists an integer $n > 0$ such that $\text{Tor}^A_p(I^nM, N) \to \text{Tor}^A_p(M, N)$ is zero for all $p \geq 0$.


more-algebra-lemma-pseudo-coherent-tensor-limit

Let $R$ be a ring. Let $K \in D(R)$ be pseudo-coherent. Let $(M_n)$ be an inverse system of $R$-modules. Then $R\lim K \otimes_R^\mathbf{L} M_n = K \otimes_R^\mathbf{L} R\lim M_n$.


more-algebra-lemma-prod-qis-gives-qis

Let $$ (A_0^\bullet \to A_1^\bullet \to A_2^\bullet \to \ldots) \longrightarrow (B_0^\bullet \to B_1^\bullet \to B_2^\bullet \to \ldots) $$ be a map between two complexes of complexes of abelian groups. Set $A^{p, q} = A_p^q$, $B^{p, q} = B_p^q$ to obtain double complexes. Let $\text{Tot}_\pi(A^{\bullet, \bullet})$ and $\text{Tot}_\pi(B^{\bullet, \bullet})$ be the product total complexes associated to the double complexes. If each $A_p^\bullet \to B_p^\bullet$ is a quasi-isomorphism, then $\text{Tot}_\pi(A^{\bullet, \bullet}) \to \text{Tot}_\pi(B^{\bullet, \bullet})$ is a quasi-isomorphism.


more-algebra-lemma-key

Let $A \to B$ be a ring map such that $B \otimes_A B \to B$ is flat. Let $N$ be a $B$-module. If $N$ is flat as an $A$-module, then $N$ is flat as a $B$-module.


more-algebra-lemma-weak-dimension-goes-up

Let $A \to B$ be a weakly \'etale ring map. If $A$ has weak dimension at most $d$, then so does $B$.


more-algebra-lemma-absolutely-flat

Let $A$ be a ring. The following are equivalent \begin{enumerate} \item $A$ has weak dimension $\leq 0$, \item $A$ is absolutely flat, and \item $A$ is reduced and every prime is maximal. \end{enumerate} In this case every local ring of $A$ is a field.


more-algebra-lemma-product-fields-absolutely-flat

A product of fields is an absolutely flat ring.


more-algebra-lemma-base-change-weakly-etale

Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes_A A'$ be the base change of $B$. \begin{enumerate} \item If $B \otimes_A B \to B$ is flat, then $B' \otimes_{A'} B' \to B'$ is flat. \item If $A \to B$ is weakly \'etale, then $A' \to B'$ is weakly \'etale. \end{enumerate}


more-algebra-lemma-absolutely-flat-over-absolutely-flat

Let $A \to B$ be a ring map such that $B \otimes_A B \to B$ is flat. \begin{enumerate} \item If $A$ is an absolutely flat ring, then so is $B$. \item If $A$ is reduced and $A \to B$ is weakly \'etale, then $B$ is reduced. \end{enumerate}


more-algebra-lemma-composition-weakly-etale

Let $A \to B$ and $B \to C$ be ring maps. \begin{enumerate} \item If $B \otimes_A B \to B$ and $C \otimes_B C \to C$ are flat, then $C \otimes_A C \to C$ is flat. \item If $A \to B$ and $B \to C$ are weakly \'etale, then $A \to C$ is weakly \'etale. \end{enumerate}


more-algebra-lemma-go-down

Let $A \to B \to C$ be ring maps. \begin{enumerate} \item If $B \to C$ is faithfully flat and $C \otimes_A C \to C$ is flat, then $B \otimes_A B \to B$ is flat. \item If $B \to C$ is faithfully flat and $A \to C$ is weakly \'etale, then $A \to B$ is weakly \'etale. \end{enumerate}


more-algebra-lemma-weakly-etale-permanence

Let $A$ be a ring. Let $B \to C$ be an $A$-algebra map of weakly \'etale $A$-algebras. Then $B \to C$ is weakly \'etale.


more-algebra-lemma-formally-unramified

Let $A \to B$ be a ring map such that $B \otimes_A B \to B$ is flat. Then $\Omega_{B/A} = 0$, i.e., $B$ is formally unramified over $A$.


more-algebra-lemma-when-weakly-etale

Let $A \to B$ be a ring map. Then $A \to B$ is weakly \'etale in each of the following cases \begin{enumerate} \item $B = S^{-1}A$ is a localization of $A$, \item $A \to B$ is \'etale, \item $B$ is a filtered colimit of weakly \'etale $A$-algebras. \end{enumerate}


more-algebra-lemma-absolutely-flat-fields

Let $K \subset L$ be an extension of fields. If $L \otimes_K L \to L$ is flat, then $L$ is an algebraic separable extension of $K$.


more-algebra-lemma-absolutely-flat-over-field

Let $B$ be an algebra over a field $K$. The following are equivalent \begin{enumerate} \item $B \otimes_K B \to B$ is flat, \item $K \to B$ is weakly \'etale, and \item $B$ is a filtered colimit of \'etale $K$-algebras. \end{enumerate} Moreover, every finitely generated $K$-subalgebra of $B$ is \'etale over $K$.


more-algebra-lemma-weakly-etale-residue-field-extensions

Let $A \to B$ be a ring map. If $A \to B$ is weakly \'etale, then $A \to B$ induces separable algebraic residue field extensions.


more-algebra-lemma-weak-dimension-at-most-1

Let $A$ be a ring. The following are equivalent \begin{enumerate} \item $A$ has weak dimension $\leq 1$, \item every ideal of $A$ is flat, \item every finitely generated ideal of $A$ is flat, \item every submodule of a flat $A$-module is flat, and \item every local ring of $A$ is a valuation ring. \end{enumerate}


more-algebra-lemma-product-weak-dimension-at-most-1

Let $J$ be a set. For each $j \in J$ let $A_j$ be a valuation ring with fraction field $K_j$. Set $A = \prod A_j$ and $K = \prod K_j$. Then $A$ has weak dimension at most $1$ and $A \to K$ is a localization.


more-algebra-lemma-product-found-valuation-rings

Let $A$ be a normal domain with fraction field $K$. There exists a cartesian diagram $$ \xymatrix{ A \ar[d] \ar[r] & K \ar[d] \\ V \ar[r] & L } $$ of rings where $V$ has weak dimension at most $1$ and $V \to L$ is a flat, injective, epimorphism of rings.


more-algebra-lemma-weak-dimension-at-most-1-integrally-closed

Let $A$ be a ring of weak dimension at most $1$. If $A \to B$ is a flat, injective, epimorphism of rings, then $A$ is integrally closed in $B$.


more-algebra-lemma-normality-goes-up

Let $A$ be a normal domain with fraction field $K$. Let $A \to B$ be weakly \'etale. Then $B$ is integrally closed in $B \otimes_A K$.


more-algebra-lemma-integral-over-henselian

Let $A \to B$ be a ring homomorphism. Assume \begin{enumerate} \item $A$ is a henselian local ring, \item $A \to B$ is integral, \item $B$ is a domain. \end{enumerate} Then $B$ is a henselian local ring and $A \to B$ is a local homomorphism. If $A$ is strictly henselian, then $B$ is a strictly henselian local ring and the extension $\kappa(\mathfrak m_A) \subset \kappa(\mathfrak m_B)$ of residue fields is purely inseparable.


more-algebra-lemma-local-tensor-with-integral

Let $A \to B$ and $A \to C$ be local homomorphisms of local rings. If $A \to C$ is integral and either $\kappa(\mathfrak m_A) \subset \kappa(\mathfrak m_C)$ or $\kappa(\mathfrak m_A) \subset \kappa(\mathfrak m_B)$ is purely inseparable, then $D = B \otimes_A C$ is a local ring and $B \to D$ and $C \to D$ are local.


more-algebra-lemma-class-weakly-etale-over-field

Let $K$ be a field. If $B$ is weakly \'etale over $K$, then \begin{enumerate} \item $B$ is reduced, \item $B$ is integral over $K$, \item any finitely generated $K$-subalgebra of $B$ is a finite product of finite separable extensions of $K$, \item $B$ is a field if and only if $B$ does not have nontrivial idempotents and in this case it is a separable algebraic extension of $K$, \item any sub or quotient $K$-algebra of $B$ is weakly \'etale over $K$, \item if $B'$ is weakly \'etale over $K$, then $B \otimes_K B'$ is weakly \'etale over $K$. \end{enumerate}


more-algebra-lemma-max-weakly-etale-subalgebra

Let $K$ be a field. Let $A$ be a $K$-algebra. There exists a maximal weakly \'etale $K$-subalgebra $B_{max} \subset A$.


more-algebra-lemma-properties-of-max-weakly-etale-subalgebra

Let $K$ be a field. For a $K$-algebra $A$ denote $B_{max}(A)$ the maximal weakly \'etale $K$-subalgebra of $A$ as in Lemma \ref{lemma-max-weakly-etale-subalgebra}. Then \begin{enumerate} \item any $K$-algebra map $A' \to A$ induces a $K$-algebra map $B_{max}(A') \to B_{max}(A)$, \item if $A' \subset A$, then $B_{max}(A') = B_{max}(A) \cap A'$, \item if $A = \colim A_i$ is a filtered colimit, then $B_{max}(A) = \colim B_{max}(A_i)$, \item the map $B_{max}(A) \to B_{max}(A_{red})$ is an isomorphism, \item $B_{max}(A_1 \times \ldots \times A_n) = B_{max}(A_1) \times \ldots \times B_{max}(A_n)$, \item if $A$ has no nontrivial idempotents, then $B_{max}(A)$ is a field and a separable algebraic extension of $K$, \item add more here. \end{enumerate}


more-algebra-lemma-change-fields-max-weakly-etale-subalgebra

Let $L/K$ be an extension of fields. Let $A$ be a $K$-algebra. Let $B \subset A$ be the maximal weakly \'etale $K$-subalgebra of $A$ as in Lemma \ref{lemma-max-weakly-etale-subalgebra}. Then $B \otimes_K L$ is the maximal weakly \'etale $L$-subalgebra of $A \otimes_K L$.


more-algebra-lemma-branches

Let $A$ be a local ring. Assume $A$ has finitely many minimal prime ideals. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Let $A^h$ be the henselization of $A$. Consider the maps $$ \Spec(A') \leftarrow \Spec((A')^h) \rightarrow \Spec(A^h) $$ where $(A')^h = A' \otimes_A A^h$. Then \begin{enumerate} \item the left arrow is bijective on maximal ideals, \item the right arrow is bijective on minimal primes, \item every minimal prime of $(A')^h$ is contained in a unique maximal ideal and every maximal ideal contains exact one minimal prime. \end{enumerate}


more-algebra-lemma-unibranch

\begin{reference} \cite[Chapter IV Proposition 18.6.12]{EGA4} \end{reference} Let $A$ be a local ring. Let $A^h$ be the henselization of $A$. The following are equivalent \begin{enumerate} \item $A$ is unibranch, and \item $A^h$ has a unique minimal prime. \end{enumerate}


more-algebra-lemma-geometric-branches

Let $(A, \mathfrak m, \kappa)$ be a local ring. Assume $A$ has finitely many minimal prime ideals. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Choose an algebraic closure $\overline{\kappa}$ of $\kappa$ and denote $\kappa^{sep} \subset \overline{\kappa}$ the separable algebraic closure of $\kappa$. Let $A^{sh}$ be the strict henselization of $A$ with respect to $\kappa^{sep}$. Consider the maps $$ \Spec(A') \xleftarrow{c} \Spec((A')^{sh}) \xrightarrow{e} \Spec(A^{sh}) $$ where $(A')^{sh} = A' \otimes_A A^{sh}$. Then \begin{enumerate} \item for $\mathfrak m' \subset A'$ maximal the residue field $\kappa'$ is algebraic over $\kappa$ and the fibre of $c$ over $\mathfrak m'$ can be canonically identified with $\Hom_\kappa(\kappa', \overline{\kappa})$, \item the right arrow is bijective on minimal primes, \item every minimal prime of $(A')^{sh}$ is contained in a unique maximal ideal and every maximal ideal contains a unique minimal prime. \end{enumerate}


more-algebra-lemma-geometrically-unibranch

\begin{reference} \cite[Lemma 2.2]{Etale-coverings} and \cite[Chapter IV Proposition 18.8.15]{EGA4} \end{reference} Let $A$ be a local ring. Let $A^{sh}$ be a strict henselization of $A$. The following are equivalent \begin{enumerate} \item $A$ is geometrically unibranch, and \item $A^{sh}$ has a unique minimal prime. \end{enumerate}


more-algebra-lemma-number-of-branches-1

Let $(A, \mathfrak m, \kappa)$ be a local ring. \begin{enumerate} \item If $A$ has infinitely many minimal prime ideals, then the number of (geometric) branches of $A$ is $\infty$. \item The number of branches of $A$ is $1$ if and only if $A$ is unibranch. \item The number of geometric branches of $A$ is $1$ if and only if $A$ is geometrically unibranch. \end{enumerate} Assume $A$ has finitely many minimal primes and let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Then \begin{enumerate} \item[(4)] the number of branches of $A$ is the number of maximal ideals $\mathfrak m'$ of $A'$, \item[(5)] to get the number of geometric branches of $A$ we have to count each maximal ideal $\mathfrak m'$ of $A'$ with multiplicity given by the separable degree of $\kappa(\mathfrak m')/\kappa$. \end{enumerate}


more-algebra-lemma-invariance-number-branches-smooth

Let $A \to B$ be a local homomorphism of local rings which is the localization of a smooth ring map. \begin{enumerate} \item The number of geometric branches of $A$ is equal to the number of geometric branches of $B$. \item If $A \to B$ induces a purely inseparable extension of residue fields, then the number of branches of $A$ is the number of branches of $B$. \end{enumerate}


more-algebra-lemma-minimal-primes-tensor-strictly-henselian

Let $k$ be an algebraically closed field. Let $A$, $B$ be strictly henselian local $k$-algebras with residue field equal to $k$. Let $C$ be the strict henselization of $A \otimes_k B$ at the maximal ideal $\mathfrak m_A \otimes_k B + A \otimes_k \mathfrak m_B$. Then the minimal primes of $C$ correspond $1$-to-$1$ to pairs of minimal primes of $A$ and $B$.


more-algebra-lemma-nr-branches-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring. \begin{enumerate} \item The map $A^h \to A^\wedge$ defines a surjective map from minimal primes of $A^\wedge$ to minimal primes of $A^h$. \item The number of branches of $A$ is at most the number of branches of $A^\wedge$. \item The number of geometric branches of $A$ is at most the number of geometric branches of $A^\wedge$. \end{enumerate}


more-algebra-lemma-equal-nr-branches-completion

Let $(A, \mathfrak m)$ be a Noetherian local ring. The number of branches of $A$ is the same as the number of branches of $A^\wedge$ if and only if $\sqrt{\mathfrak qA^\wedge}$ is prime for every minimal prime $\mathfrak q \subset A^h$ of the henselization.


more-algebra-lemma-glueing-sum-components-open

Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $A \to C$ be a ring map such that for all $f \in I$ the ring map $A_f \to C_f$ is localization at an idempotent. Then there exists a surjection $A \to C'$ such that $A_f \to (C \times C')_f$ is an isomorphism for all $f \in I$.


more-algebra-lemma-quotient-by-idempotent

Let $A$ be a Noetherian ring and $I$ an ideal. Let $B$ be a finite type $A$-algebra. Let $B^\wedge \to C$ be a surjective ring map with kernel $J$ where $B^\wedge$ is the $I$-adic completion. If $J/J^2$ is annihilated by $I^c$ for some $c \geq 0$, then $C$ is isomorphic to the completion of a finite type $A$-algebra.


more-algebra-lemma-one-dimensional-formal-branch

Let $(A, \mathfrak m)$ be a Noetherian local ring with henselization $A^h$. Let $\mathfrak q \subset A^\wedge$ be a minimal prime with $\dim(A^\wedge/\mathfrak q) = 1$. Then there exists a minimal prime $\mathfrak q^h$ of $A^h$ such that $\mathfrak q = \sqrt{\mathfrak q^hA^\wedge}$.


more-algebra-lemma-one-dimensional-number-of-branches

Let $(A, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Then the number of (geometric) branches of $A$ and $A^\wedge$ is the same.


more-algebra-lemma-geometrically-normal-formal-fibres-number-of-branches

\begin{reference} \cite[Theorem 2.3]{Beddani} \end{reference} Let $(A, \mathfrak m)$ be a Noetherian local ring. If the formal fibres of $A$ are geometrically normal (for example if $A$ is excellent or quasi-excellent), then $A$ is Nagata and the number of (geometric) branches of $A$ and $A^\wedge$ is the same.


more-algebra-lemma-not-formally-catenary

Let $(A, \mathfrak m)$ be a Noetherian local ring which is not formally catenary. Then $A$ is not universally catenary.


more-algebra-lemma-flat-under-catenary-equidimensional

Let $A \to B$ be a flat local ring map of local Noetherian rings. Assume $B$ is catenary and is $\Spec(B)$ equidimensional. Then \begin{enumerate} \item $\Spec(B/\mathfrak p B)$ is equidimensional for all $\mathfrak p \subset A$ and \item $A$ is catenary and $\Spec(A)$ is equidimensional. \end{enumerate}


more-algebra-lemma-formally-catenary

Let $A$ be a formally catenary Noetherian local ring. Then $A$ is universally catenary.


more-algebra-lemma-pol-lifting

Let $\varphi : A \to B$ be a surjection of rings. Let $G$ be a finite group of order $n$ acting on $\varphi : A \to B$. If $b \in B^G$, then there exists a monic polynomial $P \in A^G[T]$ which maps to $(T - b)^n$ in $B^G[T]$.


more-algebra-lemma-invariants-modulo

Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $I \subset R$ be an ideal such that $\sigma(I) \subset I$ for all $\sigma \in G$. Then $R^G/I^G \subset (R/I)^G$ is an integral extension of rings which induces homeomorphisms on spectra and purely inseparable extensions of residue fields.


more-algebra-lemma-functor-invariants-tensor

Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$. Let $A$ be an $R^G$-algebra. \begin{enumerate} \item for $b \in (A \otimes_{R^G} R)^G$ there exists a monic polynomial $P \in A[T]$ whose image in $(A \otimes_{R^G} R)^G[T]$ is $(T - b)^n$, \item for $a \in A$ mapping to zero in $(A \otimes_{R^G} R)^G$ we have $(T - a)^{n^2} = T^{n^2}$ in $A[T]$. \end{enumerate}


more-algebra-lemma-base-change-invariants

Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $R^G \to A$ be a ring map. The map $$ A \to (A \otimes_{R^G} R)^G $$ is an isomorphism if $R^G \to A$ is flat. In general the map is integral, induces a homeomorphism on spectra, and induces purely inseparable residue field extensions.


more-algebra-lemma-one-orbit

Let $G$ be a finite group acting on a ring $R$. For any two primes $\mathfrak q, \mathfrak q' \subset R$ lying over the same prime in $R^G$ there exists a $\sigma \in G$ with $\sigma(\mathfrak q) = \mathfrak q'$.


more-algebra-lemma-one-orbit-geometric

Let $G$ be a finite group acting on a ring $R$. Let $\mathfrak q \subset R$ be a prime lying over $\mathfrak p \subset R^G$. Then $\kappa(\mathfrak q)/\kappa(\mathfrak p)$ is an algebraic normal extension and the map $$ D = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q\} \longrightarrow \text{Aut}(\kappa(\mathfrak q)/\kappa(\mathfrak p)) $$ is surjective\footnote{Recall that we use the notation $\text{Gal}$ only in the case of Galois extensions.}.


more-algebra-lemma-one-orbit-geometric-galois

Let $A$ be a normal domain with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$. \begin{enumerate} \item For any two primes $\mathfrak q, \mathfrak q' \subset B$ lying over the same prime in $A$ there exists a $\sigma \in G$ with $\sigma(\mathfrak q) = \mathfrak q'$. \item Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak p \subset A$. Then $\kappa(\mathfrak q)/\kappa(\mathfrak p)$ is an algebraic normal extension and the map $$ D = \{\sigma \in G \mid \sigma(\mathfrak q) = \mathfrak q\} \longrightarrow \text{Aut}(\kappa(\mathfrak q)/\kappa(\mathfrak p)) $$ is surjective. \end{enumerate}


more-algebra-lemma-one-orbit-geometric-galois-compare

Let $A$ be a normal domain with fraction field $K$. Let $M/L/K$ be a tower of (possibly infinite) Galois extensions of $K$. Let $H = \text{Gal}(M/K)$ and $G = \text{Gal}(L/K)$ and let $C$ and $B$ be the integral closure of $A$ in $M$ and $L$. Let $\mathfrak r \subset C$ and $\mathfrak q = B \cap \mathfrak r$. Set $D_\mathfrak r = \{\tau \in H \mid \tau(\mathfrak r) = \mathfrak r\}$ and $I_\mathfrak r = \{\tau \in D_\mathfrak r \mid \tau \bmod \mathfrak r = \text{id}_{\kappa(\mathfrak r)}\}$ and similarly for $D_\mathfrak q$ and $I_\mathfrak q$. Under the map $H \to G$ the induced maps $D_\mathfrak r \to D_\mathfrak q$ and $I_\mathfrak r \to I_\mathfrak q$ are surjective.


more-algebra-lemma-inequality

Let $A \subset B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If the extension $L/K$ is finite, then the residue field extension is finite and we have $ef \leq [L : K]$.


more-algebra-lemma-multiplicative-e-f

Let $A \subset B \subset C$ be extensions of discrete valuation rings. Then the ramification indices of $B/A$ and $C/B$ multiply to give the ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$. Similarly for the residual degrees in case they are finite.


more-algebra-lemma-ramification-index-a-power-of-p

Let $A \subset B$ be an extension of discrete valuation rings inducing the field extension $K \subset L$. If the characteristic of $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then the ramification index $e$ is a power of $p$.


more-algebra-lemma-extension-dvrs-formally-smooth

Let $A \subset B$ be an extension of discrete valuation rings. The following are equivalent \begin{enumerate} \item $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology, and \item $A \to B$ is weakly unramified and $\kappa_A \subset \kappa_B$ is a separable field extension. \end{enumerate}


more-algebra-lemma-permanence-unramified

Let $A$ be a discrete valuation ring with fraction field $K$. \begin{enumerate} \item If $M/L/K$ are finite separable extensions and $M$ is unramified with respect to $A$, then $L$ is unramified with respect to $A$. \item If $L/K$ is a finite separable extension which is unramified with respect to $A$, then there exists a Galois extension $M/K$ containing $L$ which is unramified with respect to $A$. \item If $L_1/K$, $L_2/K$ are finite separable extensions which are unramified with respect to $A$, then there exists a a finite separable extension $L/K$ which is unramified with respect to $A$ containing $L_1$ and $L_2$. \end{enumerate}


more-algebra-lemma-galois

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Then $G$ acts on the ring $B$ of Remark \ref{remark-finite-separable-extension} and acts transitively on the set of maximal ideals of $B$.


more-algebra-lemma-galois-conclusion

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Then there are $e \geq 1$ and $f \geq 1$ such that $e_i = e$ and $f_i = f$ for all $i$ (notation as in Remark \ref{remark-finite-separable-extension}). In particular $[L : K] = n e f$.


more-algebra-lemma-galois-galois

Let $A$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m$ be a maximal ideal of $B$. Then \begin{enumerate} \item the field extension $\kappa(\mathfrak m)/\kappa$ is normal, and \item $D \to \text{Aut}(\kappa(\mathfrak m)/\kappa)$ is surjective. \end{enumerate} If for some (equivalently all) maximal ideal(s) $\mathfrak m \subset B$ the field extension $\kappa(\mathfrak m)/\kappa$ is separable, then \begin{enumerate} \item[(3)] $\kappa(\mathfrak m)/\kappa$ is Galois, and \item[(4)] $D \to \text{Gal}(\kappa(\mathfrak m)/\kappa)$ is surjective. \end{enumerate} Here $D \subset G$ is the decomposition group of $\mathfrak m$.


more-algebra-lemma-galois-inertia

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m \subset B$ be a maximal ideal. The inertia group $I$ of $\mathfrak m$ sits in a canonical exact sequence $$ 1 \to P \to I \to I_t \to 1 $$ such that \begin{enumerate} \item $P = \{\sigma \in D \mid \sigma|_{B/\mathfrak m^2} = \text{id}_{B/\mathfrak m^2}\}$ where $D$ is the decomposition group, \item $P$ is a normal subgroup of $D$, \item $P$ is a $p$-group if the characteristic of $\kappa_A$ is $p > 0$ and $P = \{1\}$ if the characteristic of $\kappa_A$ is zero, \item $I_t$ is cyclic of order the prime to $p$ part of the integer $e$, and \item there is a canonical isomorphism $\theta : I_t \to \mu_e(\kappa(\mathfrak m))$. \end{enumerate} Here $e$ is the integer of Lemma \ref{lemma-galois-conclusion}.


more-algebra-lemma-inertia-character

With assumptions and notation as in Lemma \ref{lemma-galois-inertia}. The inertia character $\theta : I \to \mu_e(\kappa(\mathfrak m))$ satisfies the following property $$ \theta(\tau \sigma \tau^{-1}) = \tau(\theta(\sigma)) $$ for $\tau \in D$ and $\sigma \in I$.


more-algebra-lemma-inertial-invariants-unramified

Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Let $\mathfrak m \subset B$ be a maximal ideal of the integral closure of $A$ in $L$. Let $I \subset G$ be the inertia group of $\mathfrak m$. Then $B^I$ is the integral closure of $A$ in $L^I$ and $A \to (B^I)_{B^I \cap \mathfrak m}$ is \'etale.


more-algebra-lemma-compare-inertia

Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be a tower with $M/K$ and $L/K$ finite Galois. Let $C$, $B$ be the integral closure of $A$ in $M$, $L$. Let $\mathfrak m' \subset C$ be a maximal ideal and set $\mathfrak m = \mathfrak m' \cap B$. Let $$ P \subset I \subset D \subset \text{Gal}(L/K) \quad\text{and}\quad P' \subset I' \subset D' \subset \text{Gal}(M/K) $$ be the wild inertia, inertia, decomposition group of $\mathfrak m$ and $\mathfrak m'$. Then the canonical surjection $\text{Gal}(M/K) \to \text{Gal}(L/K)$ induces surjections $P' \to P$, $I' \to I$, and $D' \to D$. Moreover these fit into commutative diagrams $$ \vcenter{ \xymatrix{ D' \ar[r] \ar[d] & \text{Aut}(\kappa(\mathfrak m')/\kappa_A) \ar[d] \\ D \ar[r] & \text{Aut}(\kappa(\mathfrak m)/\kappa_A) } } \quad\text{and}\quad \vcenter{ \xymatrix{ I' \ar[r]_-{\theta'} \ar[d] & \mu_{e'}(\kappa(\mathfrak m')) \ar[d]^{(-)^{e'/e}} \\ I \ar[r]^-\theta & \mu_e(\kappa(\mathfrak m)) } } $$ where $e'$ and $e$ are the ramification indices of $A \to C_{\mathfrak m'}$ and $A \to B_\mathfrak m$.


more-algebra-lemma-krasner

Let $A$ be a complete local domain of dimension $1$. Let $P(t) \in A[t]$ be a polynomial with coefficients in $A$. Let $\alpha \in A$ be a root of $P$ but not a root of the derivative $P' = \text{d}P/\text{d}t$. For every $c \geq 0$ there exists an integer $n$ such that for any $Q \in A[t]$ whose coefficients are in $\mathfrak m_A^n$ the polynomial $P + Q$ has a root $\beta \in A$ with $\beta - \alpha \in \mathfrak m_A^c$.


more-algebra-lemma-approximate-separable-extension

Let $A$ be a discrete valuation ring with field of fractions $K$. Let $A^\wedge$ be the completion of $A$ with fraction field $K^\wedge$. If $M/K^\wedge$ is a finite separable extension, then there exists a finite separable extension $L/K$ such that $M = K^\wedge \otimes_K L$.


more-algebra-lemma-pull-root-uniformizer

Let $A$ be a discrete valuation ring with uniformizer $\pi$. Let $n \geq 2$. Then $K_1 = K[\pi^{1/n}]$ is a degree $n$ extension of $K$ and the integral closure $A_1$ of $A$ in $K_1$ is the ring $A[\pi^{1/n}]$ which is a discrete valuation ring with ramification index $n$ over $A$.


more-algebra-lemma-formally-smooth-goes-up

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume that $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. Then for any finite extension $K \subset K_1$ we have $L_1 = L \otimes_K K_1$, $B_1 = B \otimes_A A_1$, and each extension $(A_1)_{\mathfrak m_i} \subset (B_1)_{\mathfrak m_{ij}}$ (see Remark \ref{remark-construction}) is formally smooth in the $\mathfrak m_{ij}$-adic topology.


more-algebra-lemma-abhyankar

Let $A \subset B$ be an extension of discrete valuation rings. Assume that either the residue characteristic of $A$ is $0$ or it is $p$, the ramification index $e$ is prime to $p$, and $\kappa_B/\kappa_A$ is a separable field extension. Let $K_1/K$ be a finite extension. Using the notation of Remark \ref{remark-construction} assume $e$ divides the ramification index of $A \subset (A_1)_{\mathfrak m_i}$ for some $i$. Then $(A_1)_{\mathfrak m_i} \subset (B_1)_{\mathfrak m_{ij}}$ is formally smooth $\mathfrak m_{ij}$-adic topology for all $j = 1, \ldots, m_i$.


more-algebra-lemma-composition-tame

Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be finite separable extensions. Let $B$ be the integral closure of $A$ in $L$. If $L/K$ is tamely ramified with respect to $A$ and $M/L$ is tamely ramified with respect to $B_\mathfrak m$ for every maximal ideal $\mathfrak m$ of $B$, then $M/K$ is tamely ramified with respect to $A$.


more-algebra-lemma-subextension-tame

Let $A$ be a discrete valuation ring with fraction field $K$. If $M/L/K$ are finite separable extensions and $M$ is tamely ramified with respect to $A$, then $L$ is tamely ramified with respect to $A$.


more-algebra-lemma-characterize-tame

Let $A$ be a discrete valuation ring with fraction field $K$. Let $\pi \in A$ be a uniformizer. Let $L/K$ be a finite separable extension. The following are equivalent \begin{enumerate} \item $L$ is tamely ramified with respect to $A$, \item there exists an $e \geq 1$ invertible in $\kappa_A$ and an extension $L'/K' = K[\pi^{1/e}]$ unramified with respect to $A' = A[\pi^{1/e}]$ such that $L$ is contained in $L'$, and \item there exists an $e_0 \geq 1$ invertible in $\kappa_A$ such that for every $d \geq 1$ invertible in $\kappa_A$ (2) holds with $e = de_0$. \end{enumerate}


more-algebra-lemma-tame-goes-up

Let $A \subset B$ be an extension of discrete valuation rings. Denote $L/K$ the corresponding extension of fraction fields. Let $K'/K$ be a finite separable extension. Then $$ K' \otimes_K L = \prod L'_i $$ is a finite product of fields and the following is true \begin{enumerate} \item If $K'$ is unramified with respect to $A$, then each $L'_i$ is unramified with respect to $B$. \item If $K'$ is tamely ramified with respect to $A$, then each $L'_i$ is tamely ramified with respect to $B$. \end{enumerate}


more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume that $A \to B$ is weakly unramified. Then for any finite separable extension $K_1/K$ totally ramified with respect to $A$ we have that $L_1 = L \otimes_K K_1$ is a field, $A_1$ and $B_1 = B \otimes_A A_1$ are discrete valuation rings, and the extension $A_1 \subset B_1$ (see Remark \ref{remark-construction}) is weakly unramified.


more-algebra-lemma-solutions-go-down

Let $A \to B \to C$ be extensions of discrete valuation rings with fraction fields $K \subset L \subset M$. Let $K \subset K_1$ be a finite extension. \begin{enumerate} \item If $K_1$ is a (weak) solution for $A \to C$, then $K_1$ is a (weak) solution for $A \to B$. \item If $K_1$ is a (weak) solution for $A \to B$ and $L_1 = (L \otimes_K K_1)_{red}$ is a product of fields which are (weak) solutions for $B \to C$, then $K_1$ is a weak solution for $A \to C$. \end{enumerate}


more-algebra-lemma-separable-solution-separable-solution

Let $A \subset B$ be an extension of discrete valuation rings. Assume \begin{enumerate} \item the extension $K \subset L$ of fraction fields is separable, \item $B$ is Nagata, and \item there exists a solution for $A \subset B$. \end{enumerate} Then there exists a separable solution for $A \subset B$.


more-algebra-lemma-solution-after-strict-henselization

Let $A \to B$ be an extension of discrete valuation rings. There exists a commutative diagram $$ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u] & A' \ar[u] } $$ of extensions of discrete valuation rings such that \begin{enumerate} \item the extensions $K \subset K'$ and $L \subset L'$ of fraction fields are separable algebraic, \item the residue fields of $A'$ and $B'$ are separable algebraic closures of the residue fields of $A$ and $B$, and \item if a solution, weak solution, or separable solution exists for $A' \to B'$, then a solution, weak solution, or separable solution exists for $A \to B$. \end{enumerate}


more-algebra-lemma-galois-relative

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Let $K \subset K_1$ be a normal extension. Say $G = \text{Aut}(K_1/K)$. Then $G$ acts on the rings $K_1$, $L_1$, $A_1$ and $B_1$ of Remark \ref{remark-construction} and acts transitively on the set of maximal ideals of $B_1$.


more-algebra-lemma-make-degree-q-extension

Let $A$ be a discrete valuation ring with uniformizer $\pi$. If the residue characteristic of $A$ is $p > 0$, then for every $n > 1$ and $p$-power $q$ there exists a degree $q$ separable extension $L/K$ totally ramified with respect to $A$ such that the integral closure $B$ of $A$ in $L$ has ramification index $q$ and a uniformizer $\pi_B$ such that $\pi_B^q = \pi + \pi^n b$ and $\pi_B^q = \pi + (\pi_B)^{nq}b'$ for some $b, b' \in B$.


more-algebra-lemma-pre-purely-inseparable-case

Let $A$ be a discrete valuation ring. Assume the reside field $\kappa_A$ has characteristic $p > 0$ and that $a \in A$ is an element whose residue class in $\kappa_A$ is not a $p$th power. Then $a$ is not a $p$th power in $K$ and the integral closure of $A$ in $K[a^{1/p}]$ is the ring $A[a^{1/p}]$ which is a discrete valuation ring weakly unramified over $A$.


more-algebra-lemma-purely-inseparable-case

Let $A \subset B \subset C$ be extensions of discrete valuation rings with fractions fields $K \subset L \subset M$. Let $\pi \in A$ be a uniformizer. Assume \begin{enumerate} \item $B$ is a Nagata ring, \item $A \subset B$ is weakly unramified, \item $M$ is a degree $p$ purely inseparable extension of $L$. \end{enumerate} Then either \begin{enumerate} \item $A \to C$ is weakly unramified, or \item $C = B[\pi^{1/p}]$, or \item there exists a degree $p$ separable extension $K_1/K$ totally ramified with respect to $A$ such that $L_1 = L \otimes_K K_1$ and $M_1 = M \otimes_K K_1$ are fields and the maps of integral closures $A_1 \to B_1 \to C_1$ are weakly unramified extensions of discrete valuation rings. \end{enumerate}


more-algebra-lemma-cohen

Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is nilpotent. There exists a ring map $\sigma : \kappa_A \to A$ which is a section to the residue map $A \to \kappa_A$. If $A \to A'$ is a local homomorphism of local rings, then we can choose a similar ring map $\sigma' : \kappa_{A'} \to A'$ compatible with $\sigma$ provided that the extension $\kappa_A \subset \kappa_{A'}$ is separable.


more-algebra-lemma-pre-characteristic-p-case

Let $A$ be a discrete valuation ring with fraction field $K$ of characteristic $p > 0$. Let $\xi \in K$. Let $L$ be an extension of $K$ obtained by adjoining a root of $z^p - z = \xi$. Then $L/K$ is Galois and one of the following happens \begin{enumerate} \item $L = K$, \item $L/K$ is unramified with respect to $A$ of degree $p$, \item $L/K$ is totally ramified with respect to $A$ with ramification index $p$, and \item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring, $A \subset B$ is weakly unramified, and $A \to B$ induces a purely inseparable residue field extension of degree $p$. \end{enumerate} Let $\pi$ be a uniformizer of $A$. We have the following implications: \begin{enumerate} \item[(A)] If $\xi \in A$, then we are in case (1) or (2). \item[(B)] If $\xi = \pi^{-n}a$ where $n > 0$ is not divisible by $p$ and $a$ is a unit in $A$, then we are in case (3) \item[(C)] If $\xi = \pi^{-n} a$ where $n > 0$ is divisible by $p$ and the image of $a$ in $\kappa_A$ is not a $p$th power, then we are in case (4). \end{enumerate}


more-algebra-lemma-characteristic-p-case

Let $A \subset B \subset C$ be extensions of discrete valuation rings with fractions fields $K \subset L \subset M$. Assume \begin{enumerate} \item $A \subset B$ weakly unramified, \item the characteristic of $K$ is $p$, \item $M$ is a degree $p$ Galois extension of $L$, and \item $\kappa_A = \bigcap_{n \geq 1} \kappa_B^{p^n}$. \end{enumerate} Then there exists a finite Galois extension $K_1/K$ totally ramified with respect to $A$ which is a weak solution for $A \to C$.


more-algebra-lemma-prepare

Let $A$ be a ring which contains a primitive $p$th root of unity $\zeta$. Set $w = 1 - \zeta$. Then $$ P(z) = \frac{(1 + wz)^p - 1}{w^p} = z^p - z + \sum\nolimits_{0 < i < p} a_i z^i $$ is an element of $A[z]$ and in fact $a_i \in (w)$. Moreover, we have $$ P(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2) $$ in the polynomial ring $A[z_1, z_2]$.


more-algebra-lemma-extension-defined-by-nice-polynial

Let $A$ be a discrete valuation ring of mixed characteristic $(0, p)$ which contains a primitive $p$th root of $1$. Let $P(t) \in A[t]$ be the polynomial of Lemma \ref{lemma-prepare}. Let $\xi \in K$. Let $L$ be an extension of $K$ obtained by adjoining a root of $P(z) = \xi$. Then $L/K$ is Galois and one of the following happens \begin{enumerate} \item $L = K$, \item $L/K$ is unramified with respect to $A$ of degree $p$, \item $L/K$ is totally ramified with respect to $A$ with ramification index $p$, and \item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring, $A \subset B$ is weakly unramified, and $A \to B$ induces a purely inseparable residue field extension of degree $p$. \end{enumerate} Let $\pi$ be a uniformizer of $A$. We have the following implications: \begin{enumerate} \item[(A)] If $\xi \in A$, then we are in case (1) or (2). \item[(B)] If $\xi = \pi^{-n}a$ where $n > 0$ is not divisible by $p$ and $a$ is a unit in $A$, then we are in case (3) \item[(C)] If $\xi = \pi^{-n} a$ where $n > 0$ is divisible by $p$ and the image of $a$ in $\kappa_A$ is not a $p$th power, then we are in case (4). \end{enumerate}


more-algebra-lemma-make-finite-level

Let $A \subset B \subset C$ be extensions of discrete valuation rings with fractions fields $K \subset L \subset M$. Assume that \begin{enumerate} \item $A$ has mixed characteristic $(0, p)$, \item $A \subset B$ is weakly unramified, \item $B$ contains a primitive $p$th root of $1$, and \item $M/L$ is Galois of degree $p$. \end{enumerate} Then there exists a finite Galois extension $K_1/K$ totally ramified with respect to $A$ which is either a weak solution for $A \to C$ or is such that $M_1/L_1$ is a degree $p$ extension of finite level.


more-algebra-lemma-lowering-the-level

Let $A \subset B \subset C$ be extensions of discrete valuation rings with fractions fields $K \subset L \subset M$. Assume \begin{enumerate} \item $A$ has mixed characteristic $(0, p)$, \item $A \subset B$ weakly unramified, \item $B$ contains a primitive $p$th root of $1$, \item $M/L$ is a degree $p$ extension of finite level $l > 0$, \item $\kappa_A = \bigcap_{n \geq 1} \kappa_B^{p^n}$. \end{enumerate} Then there exists a finite separable extension $K_1$ of $K$ totally ramified with respect to $A$ such that either $K_1$ is a weak solution for $A \to C$, or the extension $M_1/L_1$ is a degree $p$ extension of finite level $\leq \max(0, l - 1, 2l - p)$.


more-algebra-lemma-special-case

Let $A \subset B \subset C$ be extensions of discrete valuation rings with fraction fields $K \subset L \subset M$. Assume \begin{enumerate} \item the residue field $k$ of $A$ is algebraically closed of characteristic $p > 0$, \item $A$ and $B$ are complete, \item $A \to B$ is weakly unramified, \item $M$ is a finite extension of $L$, \item $k = \bigcap\nolimits_{n \geq 1} \kappa_B^{p^n}$ \end{enumerate} Then there exists a finite extension $K \subset K_1$ which is a weak solution for $A \to C$.


more-algebra-lemma-big-extension-is-ok

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume $B$ is essentially of finite type over $A$. Let $K \subset K'$ be an algebraic extension of fields such that the integral closure $A'$ of $A$ in $K'$ is Noetherian. Then the integral closure $B'$ of $B$ in $L' = (L \otimes_K K')_{red}$ is Noetherian as well. Moreover, the map $\Spec(B') \to \Spec(A')$ is surjective and the corresponding residue field extensions are finitely generated field extensions.


more-algebra-lemma-epp-essentially-finite-type-separable

Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume \begin{enumerate} \item $B$ is essentially of finite type over $A$, \item either $A$ or $B$ is a Nagata ring, and \item $L/K$ is separable. \end{enumerate} Then there exists a separable solution for $A \to B$ (Definition \ref{definition-solution}).


more-algebra-lemma-invertible

Let $R$ be a ring. Let $M$ be an $R$-module. Equivalent are \begin{enumerate} \item $M$ is finite locally free module of rank $1$, \item $M$ is invertible, and \item there exists an $R$-module $N$ such that $M \otimes_R N \cong R$. \end{enumerate} Moreover, in this case the module $N$ is (3) is isomorphic to $\Hom_R(M, R)$.


more-algebra-lemma-UFD-Pic-trivial

Let $R$ be a UFD. Then $\Pic(R)$ is trivial.


more-algebra-lemma-det-ses

Let $R$ be a ring. Let $$ 0 \to M' \to M \to M'' \to 0 $$ be a short exact sequence of finite projective $R$-modules. Then there is a canonical isomorphism $$ \gamma : \det(M') \otimes \det(M'') \longrightarrow \det(M) $$


more-algebra-lemma-det-ses-functorial

Let $R$ be a ring. Let $$ \xymatrix{ 0 \ar[r] & M' \ar[r] \ar[d]^u & M \ar[r] \ar[d]^v & M'' \ar[r] \ar[d]^w & 0 \\ 0 \ar[r] & K' \ar[r] & K \ar[r] & K'' \ar[r] & 0 } $$ be a commutative diagram of finite projective $R$-modules whose vertical arrows are isomorphisms. Then we get a commutative diagram of isomorphisms $$ \xymatrix{ \det(M') \otimes \det(M'') \ar[r]_-\gamma \ar[d]_{\det(u) \otimes \det(w)} & \det(M) \ar[d]^{\det(v)} \\ \det(K') \otimes \det(K'') \ar[r]^-\gamma & \det(K) } $$ where the horizontal arrows are the ones constructed in Lemma \ref{lemma-det-ses}.


more-algebra-lemma-det-filtration

Let $R$ be a ring. Let $$ K \subset L \subset M $$ be $R$-modules such that $K$, $L/K$, and $M/L$ are finite projective $R$-modules. Then the diagram $$ \xymatrix{ \det(K) \otimes \det(L/K) \otimes \det(M/L) \ar[r] \ar[d] & \det(L) \otimes \det(M/L) \ar[d] \\ \det(K) \otimes \det(M/K) \ar[r] & \det(M) } $$ commutes where the maps are those of Lemma \ref{lemma-det-ses}.


more-algebra-lemma-det-direct-sum

Let $R$ be a ring. Let $M'$ and $M''$ be two finite projective $R$-modules. Then the diagram $$ \xymatrix{ \det(M') \otimes \det(M'') \ar[r] \ar[d]_{\epsilon \cdot (\text{switch tensors})} & \det(M' \oplus M'') \ar[d]^{\det(\text{swith summands})} \\ \det(M'') \otimes \det(M') \ar[r] & \det(M'' \oplus M') } $$ commutes where $\epsilon = \det( -\text{id}_{M' \otimes M''}) \in R^*$ and the horizontal arrows are those of Lemma \ref{lemma-det-ses}.


more-algebra-lemma-det-switch

Let $R$ be a ring. Let $M$, $N$ be finite projective $R$-modules. Let $a : M \to N$ and $b : N \to M$ be $R$-linear maps. Then $$ \det(\text{id} + a \circ b) = \det(\text{id} + b \circ a) $$ as elements of $R$.


more-algebra-lemma-det

Let $R$ be a ring. There is a map $$ \det : K_0(R) \longrightarrow \Pic(R) $$ which maps $[M]$ to the class of the invertible module $\wedge^n(M)$ if $M$ is a finite locally free module of rank $n$.


more-algebra-lemma-perfect-to-K-group

Let $R$ be a ring. There is a map $$ c : \text{perfect complexes over }R \longrightarrow K_0(R) $$ with the following properties \begin{enumerate} \item $c(K[n]) = (-1)^nc(K)$ for a perfect complex $K$, \item if $K \to L \to M \to K[1]$ is a distinguished triangle of perfect complexes, then $c(L) = c(K) + c(M)$, \item if $K$ is represented by a finite complex $M^\bullet$ consisting of finite projective modules, then $c(K) = \sum (-1)^i[M_i]$. \end{enumerate}


more-algebra-lemma-perfect-to-K-group-universal

Let $R$ be a ring. Let $D_{perf}(R)$ be the derived category of perfect objects, see Lemma \ref{lemma-perfect-ring-classical-generator}. The map $c$ of Lemma \ref{lemma-perfect-to-K-group} gives an isomorphism $K_0(D_{perf}(R)) = K_0(R)$.


more-algebra-lemma-regular-local-Pic-zero

Let $R$ be a regular local ring. Let $f \in R$. Then $\Pic(R_f) = 0$.


more-algebra-lemma-regular-local-UFD

A regular local ring is a UFD.


more-algebra-lemma-picard-group-generic-fibre-regular

Let $R$ be a valuation ring with fraction field $K$ and residue field $\kappa$. Let $R \to A$ be a homomorphism of rings such that \begin{enumerate} \item $A$ is local and $R \to A$ is local, \item $A$ is flat and essentially of finite type over $R$, \item $A \otimes_R \kappa$ regular. \end{enumerate} Then $\Pic(A \otimes_R K) = 0$.


more-algebra-lemma-canonical-element-well-defined

Let $R$ be a ring. Let $a^\bullet : K^\bullet \to L^\bullet$ be a map of complexes of $R$-modules satisfying (1), (2), (3) above. If $L^\bullet$ has rank $0$, then $\det(a^\bullet)$ maps the canonical element $\delta(L^\bullet)$ to $\delta(K^\bullet)$.


more-algebra-lemma-homotopic-surjections

Let $R$ be a ring. Let $a^\bullet : K^\bullet \to L^\bullet$ be a map of complexes of $R$-modules satisfying (1), (2), (3) above. Let $h : K^0 \to L^{-1}$ be a map such that $b^0 = a^0 + d \circ h$ and $b^{-1} = a^{-1} + h \circ d$ are surjective. Then $\det(a^\bullet) = \det(b^\bullet)$ as maps $\det(L^\bullet) \to \det(K^\bullet)$.


more-algebra-lemma-compose-surjections

Let $R$ be a ring. Let $a^\bullet : K^\bullet \to L^\bullet$ and $b^\bullet : L^\bullet \to M^\bullet$ be maps of complexes of $R$-modules satisfying (1), (2), (3) above. Then we have $\det(a^\bullet) \circ \det(b^\bullet) =  \det(b^\bullet \circ a^\bullet)$ as maps $\det(M^\bullet) \to \det(K^\bullet)$.


more-algebra-lemma-determinant-two-term-complexes

Let $R$ be a ring. The constructions above determine a functor $$ \det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{with tor amplitude in }[-1, 0] \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} $$ Moreover, given a rank $0$ perfect object $L$ of $D(R)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\delta(L) \in \det(L)$ such that for any isomorphism $a : L \to K$ in $D(R)$ we have $\det(a)(\delta(L)) = \delta(K)$.


more-algebra-lemma-inequality-general

Let $A \subset B$ be an extension of valuation rings with fraction fields $K \subset L$. If the extension $K \subset L$ is finite, then the residue field extension is finite, the index of $\Gamma_A$ in $\Gamma_B$ is finite, and $$ [\Gamma_B : \Gamma_A] [\kappa_B : \kappa_A] \leq [L : K]. $$


more-algebra-lemma-extension-normal-domains-and-roots

Let $A \to B$ be a flat local homomorphism of Noetherian local normal domains. Let $f \in A$ and $h \in B$ such that $f = w h^n$ for some $n > 1$ and some unit $w$ of $B$. Assume that for every height $1$ prime $\mathfrak p \subset A$ there is a height $1$ prime $\mathfrak q \subset B$ lying over $\mathfrak p$ such that the extension $A_\mathfrak p \subset B_\mathfrak q$ is weakly unramified. Then $f = u g^n$ for some $g \in A$ and unit $u$ of $A$.


more-algebra-lemma-etale-extension-valuation-ring

Let $A$ be a valuation ring. Let $A \to B$ be an \'etale ring map and let $\mathfrak m \subset B$ be a prime lying over the maximal ideal of $A$. Then $A \subset B_\mathfrak m$ is an extension of valuation rings which is weakly unramified.


more-algebra-lemma-henselization-valuation-ring

Let $A$ be a valuation ring. Let $A^h$, resp.\ $A^{sh}$ be its henselization, resp.\ strict henselization. Then $$ A \subset A^h \subset A^{sh} $$ are extensions of valuation rings which induce bijections on value groups, i.e., which are weakly unramified.


more-algebra-lemma-characterize-PD-modules

\begin{reference} \cite[Corollary 1]{Warfield-Purity} \end{reference} Let $P$ be a module over a ring $R$. The following are equivalent \begin{enumerate} \item $P$ is a direct summand of a direct sum of modules of the form $R/fR$, for $f \in R$ varying. \item for every short exact sequence $0 \to A \to B \to C \to 0$ of $R$-modules such that $fA = A \cap fB$ for all $f \in R$ the map $\Hom_R(P, B) \to \Hom_R(P, C)$ is surjective. \end{enumerate}


more-algebra-lemma-generalized-valuation-ring

\begin{reference} \cite{Warfield-Decomposition} \end{reference} Let $R$ be a nonzero ring. The following are equivalent \begin{enumerate} \item For $a, b \in R$ either $a$ divides $b$ or $b$ divides $a$. \item Every finitely generated ideal is principal and $R$ is local. \item The set of ideals of $R$ are linearly ordered by inclusion. \end{enumerate} This holds in particular if $R$ is a valuation ring.


more-algebra-lemma-generalized-valuation-ring-modules

\begin{reference} \cite[Theorem 1]{Warfield-Decomposition} \end{reference} Let $R$ be a ring satisfying the equivalent conditions of Lemma \ref{lemma-generalized-valuation-ring}. Then every finitely presented $R$-module is isomorphic to a finite direct sum of modules of the form $R/fR$.


more-algebra-lemma-warfield

\begin{reference} \cite[Theorem 3]{Warfield-Decomposition} \end{reference} Let $R$ be a ring such that every local ring of $R$ at a maximal ideal satisfies the equivalent conditions of Lemma \ref{lemma-generalized-valuation-ring}. Then every finitely presented $R$-module is a summand of a  finite direct sum of modules of the form $R/fR$ for $f$ in $R$ varying.


more-algebra-lemma-elementary-divisor-is-bezout

An elementary divisor domain is B\'ezout.


more-algebra-lemma-localize-bezout

The localization of a B\'ezout domain is B\'ezout. Every local ring of a B\'ezout domain is a valuation ring. A local domain is B\'ezout if and only if it is a valuation ring.


more-algebra-lemma-split-off-free-part

Let $R$ be a B\'ezout domain. \begin{enumerate} \item Every finite submodule of a free module is finite free. \item Every finitely presented $R$-module $M$ is a direct sum of a finite free module and a torsion module $M_{tors}$ which is a summand of a module of the form $\bigoplus_{i = 1, \ldots, n} R/f_iR$ with $f_1, \ldots, f_n \in R$ nonzero. \end{enumerate}


more-algebra-lemma-modules-PID

Let $R$ be a PID. Every finite $R$-module $M$ is of isomorphic to a module of the form $$ R^{\oplus r} \oplus \bigoplus\nolimits_{i = 1, \ldots, n} R/f_iR $$ for some $r, n \geq 0$ and $f_1, \ldots, f_n \in R$ nonzero.


more-algebra-lemma-polypoly

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension one, and let $x\in\mathfrak m$ be an element not contained in any minimal prime of $R$. Then \begin{enumerate} \item the function $P : n \mapsto \text{length}_R(R/x^n R)$ satisfies $P(n) \leq n P(1)$ for $n \geq 0$, \item if $x$ is a nonzerodivisor, then $P(n) = nP(1)$ for $n \geq 0$. \end{enumerate}


more-algebra-lemma-minprimespoly

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $x \in \mathfrak m$ be an element not contained in any minimal prime of $R$. Let $t$ be the number of minimal prime ideals of $R$. Then $t \leq \text{length}_R(R/xR)$.


more-algebra-lemma-sopexists

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d > 1$, let $f \in \mathfrak m$ be an element not contained in any minimal prime ideal of $R$, and let $k\in\mathbf{N}$. Then there exist elements $g_1, \ldots, g_{d - 1} \in \mathfrak m^k$ such that $f, g_1, \ldots, g_{d - 1}$ is a system of parameters.


more-algebra-lemma-syspar

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension two, and let $f \in \mathfrak m$ be an element not contained in any minimal prime ideal of $R$. Then there exist $g \in \mathfrak m$ and $N \in \mathbf{N}$ such that \begin{enumerate} \item[(a)] $f,g$ form a system of parameters for $R$. \item[(b)] If $h \in \mathfrak m^N$, then $f + h, g$ is a system of parameters and $\text{length}_R (R/(f, g)) = \text{length}_R(R/(f + h, g))$. \end{enumerate}


more-algebra-lemma-radical-element

Let $R$ be a Noetherian local normal domain of dimension $2$. Let $\mathfrak p_1, \ldots, \mathfrak p_r$ be pairwise distinct primes of height $1$. There exists a nonzero element $f \in \mathfrak p_1 \cap \ldots \cap \mathfrak p_r$ such that $R/fR$ is reduced.


more-algebra-lemma-divides-radical

Let $(A, \mathfrak m, \kappa)$ be a Noetherian normal local domain of dimension $2$. If $a \in \mathfrak m$ is nonzero, then there exists an element $c \in A$ such that $A/cA$ is reduced and such that $a$ divides $c^n$ for some $n$.


more-algebra-lemma-multiplicity

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d$, let $g_1, \ldots, g_d$ be a system of parameters, and let $I = (g_1, \ldots, g_d)$. If $e_I/d!$ is the leading coefficient of the numerical polynomial $n \mapsto \text{length}_R(R/I^{n+1})$, then $e_I \leq \text{length}_R(R/I)$.


more-algebra-lemma-minprimespolyhigher

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d$, let $t$ be the number of minimal prime ideals of $R$ of dimension $d$, and let $(g_1,\ldots,g_d)$ be a system of parameters. Then $t \leq \text{length}_R(R/(g_1,\ldots,g_n))$.


more-algebra-lemma-sysparhigher

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d$, and let $f \in \mathfrak m$ be an element not contained in any minimal prime ideal of $R$. Then there exist elements $g_1, \ldots, g_{d - 1} \in \mathfrak m$ and $N \in \mathbf{N}$ such that \begin{enumerate} \item $f, g_1, \ldots, g_{d - 1}$ form a system of parameters for $R$ \item If $h \in \mathfrak m^N$, then $f + h, g_1, \ldots, g_{d - 1}$ is a system of parameters and we have $\text{length}_R R/(f, g_1, \ldots, g_{d-1}) = \text{length}_R R/(f + h, g_1, \ldots, g_{d-1})$. \end{enumerate}


more-algebra-lemma-complex-bounded-above-free-colim-bounded-finite-free

Let $R$ be a ring. Let $F^\bullet$ be a bounded above complex of free $R$-modules. Given pairs $(n_i, f_i)$, $i = 1, \ldots, N$ with $n_i \in \mathbf{Z}$ and $f_i \in F^{n_i}$ there exists a subcomplex $G^\bullet \subset F^\bullet$ containing all $f_i$ which is bounded and consists of finite free $R$-modules.


more-algebra-lemma-have-dual-derived

Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following are equivalent \begin{enumerate} \item $M$ has a left dual in $D(R)$ as in Categories, Definition \ref{categories-definition-dual}, \item $M$ is a perfect object of $D(R)$. \end{enumerate} Moreover, in this case the left dual of $M$ is the object $M^\vee$ of Lemma \ref{lemma-dual-perfect-complex}.


more-algebra-lemma-invertible-derived

Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following are equivalent \begin{enumerate} \item $M$ is invertible in $D(R)$, see Categories, Definition \ref{categories-definition-invertible}, and \item for every prime ideal $\mathfrak p \subset R$ there exists an $f \in R$, $f \not \in \mathfrak p$ such that $M_f \cong R_f[-n]$ for some $n \in \mathbf{Z}$. \end{enumerate} Moreover, in this case \begin{enumerate} \item[(a)] $M$ is a perfect object of $D(R)$, \item[(b)] $M = \bigoplus H^n(M)[-n]$ in $D(R)$, \item[(c)] each $H^n(M)$ is a finite projective $R$-module, \item[(d)] we can write $R = \prod_{a \leq n \leq b} R_n$ such that $H^n(M)$ corresponds to an invertible $R_n$-module. \end{enumerate}


more-algebra-proposition-characterization-geometrically-regular

Let $k$ be a field of characteristic $p > 0$. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. The following are equivalent \begin{enumerate} \item $A$ is geometrically regular over $k$, \item for all $k \subset k' \subset k^{1/p}$ finite over $k$ the ring $A \otimes_k k'$ is regular, \item $A$ is regular and the canonical map $H_1(L_{K/k}) \to \mathfrak m/\mathfrak m^2$ is injective, and \item $A$ is regular and the map $\Omega_{k/\mathbf{F}_p} \otimes_k K \to \Omega_{A/\mathbf{F}_p} \otimes_A K$ is injective. \end{enumerate}


more-algebra-proposition-fs-flat-fibre-fs

Let $A \to B$ be a local homomorphism of Noetherian local rings. Let $k$ be the residue field of $A$ and $\overline{B} = B \otimes_A k$ the special fibre. The following are equivalent \begin{enumerate} \item $A \to B$ is flat and $\overline{B}$ is geometrically regular over $k$, \item $A \to B$ is flat and $k \to \overline{B}$ is formally smooth in the $\mathfrak m_{\overline{B}}$-adic topology, and \item $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. \end{enumerate}


more-algebra-proposition-ubiquity-J-2

The following types of rings are J-2: \begin{enumerate} \item fields, \item Noetherian complete local rings, \item $\mathbf{Z}$, \item Noetherian local rings of dimension $1$, \item Nagata rings of dimension $1$, \item Dedekind domains with fraction field of characteristic zero, \item finite type ring extensions of any of the above. \end{enumerate}


more-algebra-proposition-fs-regular

Let $A \to B$ be a local homomorphism of Noetherian complete local rings. The following are equivalent \begin{enumerate} \item $A \to B$ is regular, \item $A \to B$ is flat and $\overline{B}$ is geometrically regular over $k$, \item $A \to B$ is flat and $k \to \overline{B}$ is formally smooth in the $\mathfrak m_{\overline{B}}$-adic topology, and \item $A \to B$ is formally smooth in the $\mathfrak m_B$-adic topology. \end{enumerate}


more-algebra-proposition-Noetherian-complete-G-ring

A Noetherian complete local ring is a G-ring.


more-algebra-proposition-finite-type-over-G-ring

Let $R$ be a G-ring. If $R \to S$ is essentially of finite type then $S$ is a G-ring.


more-algebra-proposition-ubiquity-G-ring

The following types of rings are G-rings: \begin{enumerate} \item fields, \item Noetherian complete local rings, \item $\mathbf{Z}$, \item Dedekind domains with fraction field of characteristic zero, \item finite type ring extensions of any of the above. \end{enumerate}


more-algebra-proposition-finite-type-over-P-ring

Let $R$ be a $P$-ring where $P$ satisfies (A), (B), (C), and (D). If $R \to S$ is essentially of finite type then $S$ is a $P$-ring.


more-algebra-proposition-ubiquity-excellent

The following types of rings are excellent: \begin{enumerate} \item fields, \item Noetherian complete local rings, \item $\mathbf{Z}$, \item Dedekind domains with fraction field of characteristic zero, \item finite type ring extensions of any of the above. \end{enumerate}


more-algebra-proposition-perfect-is-compact

Let $R$ be a ring. For an object $K$ of $D(R)$ the following are equivalent \begin{enumerate} \item $K$ is perfect, and \item $K$ is a compact object of $D(R)$. \end{enumerate}


more-algebra-proposition-equivalence

Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots, f_t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Then $\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories $$ \text{Mod}_R = \text{Glue}(R \to S, f_1, \ldots, f_t) $$


more-algebra-proposition-formal-glueing

Let $R$ be a Noetherian ring. Let $f \in R$ be an element. Let $R^\wedge$ be the $f$-adic completion of $R$. Then the functor $M \mapsto (M^\wedge, M_f, \text{can})$ defines an equivalence $$ \text{Mod}^{fg}_R \longrightarrow \text{Mod}^{fg}_{R^\wedge} \times_{\text{Mod}^{fg}_{(R^\wedge)_f}} \text{Mod}^{fg}_{R_f} $$


more-algebra-proposition-derived-complete-modules

Let $I \subset A$ be a finitely generated ideal of a ring $A$. Let $M$ be an $A$-module. The following are equivalent \begin{enumerate} \item $M$ is $I$-adically complete, and \item $M$ is derived complete with respect to $I$ and $\bigcap I^nM = 0$. \end{enumerate}


more-algebra-proposition-noetherian-naive-completion-is-completion

Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. The functor which sends $K \in D(A)$ to the derived limit $K' = R\lim( K \otimes_A^\mathbf{L} A/I^n )$ is the left adjoint to the inclusion functor $D_{comp}(A) \to D(A)$ constructed in Lemma \ref{lemma-derived-completion}.


more-algebra-proposition-ratliff

\begin{reference} \cite{Ratliff} \end{reference} A Noetherian local ring is universally catenary if and only if it is formally catenary.


more-algebra-proposition-epp-essentially-finite-type

\begin{reference} See \cite[Lemma 2.13]{alterations} for a special case. \end{reference} Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If $B$ is essentially of finite type over $A$, then there exists a finite extension $K \subset K_1$ which is a solution for $A \to B$ as defined in Definition \ref{definition-solution}.


etale-theorem-unramified-equivalence

Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $x$ be a point of $X$. The following are equivalent \begin{enumerate} \item $f$ is unramified at $x$, \item the stalk $\Omega_{X/Y, x}$ of the module of relative differentials at $x$ is trivial, \item there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a commutative diagram $$ \xymatrix{ U \ar[rr]_i \ar[rd] & & \mathbf{A}^n_V \ar[ld] \\ & V } $$ where $i$ is a closed immersion defined by a quasi-coherent sheaf of ideals $\mathcal{I}$ such that the differentials $\text{d}g$ for $g \in \mathcal{I}_{i(x)}$ generate $\Omega_{\mathbf{A}^n_V/V, i(x)}$, and \item the diagonal $\Delta_{X/Y} : X \to X \times_Y X$ is a local isomorphism at $x$. \end{enumerate}


etale-theorem-sections-unramified-maps

Let $Y$ be a connected scheme. Let $f : X \to Y$ be unramified and separated. Every section of $f$ is an isomorphism onto a connected component. There exists a bijective correspondence $$ \text{sections of }f \leftrightarrow \left\{ \begin{matrix} \text{connected components }X'\text{ of }X\text{ such that}\\ \text{the induced map }X' \to Y\text{ is an isomorphism} \end{matrix} \right\} $$ In particular, given $x \in X$ there is at most one section passing through $x$.


etale-theorem-etale-radicial-open

Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is an open immersion, \item $f$ is universally injective and \'etale, and \item $f$ is a flat monomorphism, locally of finite presentation. \end{enumerate}


etale-theorem-etale-topological

Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S_0$ be a closed subscheme of $S$ with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). Denote $X_0$ (resp.\ $Y_0$) the base change $S_0 \times_S X$ (resp.\ $S_0 \times_S Y$). If $X$ is \'etale over $S$, then the map $$ \Mor_S(Y, X) \longrightarrow \Mor_{S_0}(Y_0, X_0) $$ is bijective.


etale-theorem-remarkable-equivalence

\begin{reference} \cite[IV, Theorem 18.1.2]{EGA} \end{reference} Let $S$ be a scheme. Let $S_0 \subset S$ be a closed subscheme with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). The functor $$ X \longmapsto X_0 = S_0 \times_S X $$ defines an equivalence of categories $$ \{ \text{schemes }X\text{ \'etale over }S \} \leftrightarrow \{ \text{schemes }X_0\text{ \'etale over }S_0 \} $$


etale-lemma-characterize-unramified-Noetherian

\begin{slogan} Unramifiedness is a stalk local condition. \end{slogan} Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is unramified at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an unramified homomorphism of local rings.


etale-lemma-unramified-completions

Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism. \begin{enumerate} \item if $A \to B$ is an unramified homomorphism of local rings, then $B^\wedge$ is a finite $A^\wedge$ module, \item if $A \to B$ is an unramified homomorphism of local rings and $\kappa(\mathfrak m_A) = \kappa(\mathfrak m_B)$, then $A^\wedge \to B^\wedge$ is surjective, \item if $A \to B$ is an unramified homomorphism of local rings and $\kappa(\mathfrak m_A)$ is separably closed, then $A^\wedge \to B^\wedge$ is surjective, \item if $A$ and $B$ are complete discrete valuation rings, then $A \to B$ is an unramified homomorphism of local rings if and only if the uniformizer for $A$ maps to a uniformizer for $B$, and the residue field extension is finite separable (and $B$ is essentially of finite type over $A$). \end{enumerate}


etale-lemma-characterize-unramified-completions

Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \begin{enumerate} \item $A \to B$ is an unramified homomorphism of local rings \item $A^\wedge \to B^\wedge$ is an unramified homomorphism of local rings, and \item $A^\wedge \to B^\wedge$ is unramified. \end{enumerate}


etale-lemma-unramified-definition

Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. The morphism $f$ is unramified at $x$ in the sense of Definition \ref{definition-unramified-schemes} if and only if it is unramified in the sense of Morphisms, Definition \ref{morphisms-definition-unramified}.


etale-lemma-universally-injective-unramified

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is unramified and a monomorphism, \item $f$ is unramified and universally injective, \item $f$ is locally of finite type and a monomorphism, \item $f$ is universally injective, locally of finite type, and formally unramified, \item $f$ is locally of finite type and $X_s$ is either empty or $X_s \to s$ is an isomorphism for all $s \in S$. \end{enumerate}


etale-lemma-characterize-closed-immersion

Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is a closed immersion, \item $f$ is a proper monomorphism, \item $f$ is proper, unramified, and universally injective, \item $f$ is universally closed, unramified, and a monomorphism, \item $f$ is universally closed, unramified, and universally injective, \item $f$ is universally closed, locally of finite type, and a monomorphism, \item $f$ is universally closed, universally injective, locally of finite type, and formally unramified. \end{enumerate}


etale-lemma-finite-unramified-one-point

Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $\pi$ is finite, \item $\pi$ is unramified, \item $\pi^{-1}(\{s\}) = \{x\}$, and \item $\kappa(s) \subset \kappa(x)$ is purely inseparable\footnote{In view of condition (2) this is equivalent to $\kappa(s) = \kappa(x)$.}. \end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is a closed immersion.


etale-lemma-characterize-etale-Noetherian

Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is \'etale at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an \'etale homomorphism of local rings.


etale-lemma-characterize-etale-completions

Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \begin{enumerate} \item $A \to B$ is an \'etale homomorphism of local rings \item $A^\wedge \to B^\wedge$ is an \'etale homomorphism of local rings, and \item $A^\wedge \to B^\wedge$ is \'etale. \end{enumerate} Moreover, in this case $B^\wedge \cong (A^\wedge)^{\oplus n}$ as $A^\wedge$-modules for some $n \geq 1$.


etale-lemma-finite-etale-one-point

Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $\pi$ is finite, \item $\pi$ is \'etale, \item $\pi^{-1}(\{s\}) = \{x\}$, and \item $\kappa(s) \subset \kappa(x)$ is purely inseparable\footnote{In view of condition (2) this is equivalent to $\kappa(s) = \kappa(x)$.}. \end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is an isomorphism.


etale-lemma-relative-frobenius-etale

Let $U \to X$ be an \'etale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \to U \times_{X, F_X} X$ is an isomorphism.


etale-lemma-unramified-etale-local

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is unramified at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_U$, $i = 1, \ldots, n$, $j = 1, \ldots, m_i$ such that \begin{enumerate} \item $V_{i, j} \to U$ is a closed immersion passing through $u$, \item $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \end{enumerate}


etale-lemma-unramified-etale-local-technical

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is unramified at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$ X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j} $$ such that \begin{enumerate} \item $V_{i, j} \to U$ is a closed immersion passing through $u$, \item the fibre $W_u$ contains no point mapping to any $x_i$. \end{enumerate} In particular, if $f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}$, then the fibre $W_u$ is empty.


etale-lemma-finite-unramified-etale-local

Let $f : X \to S$ be a finite unramified morphism of schemes. Let $s \in S$. There exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition $$ X_U = \coprod\nolimits_j V_j $$ such that each $V_j \to U$ is a closed immersion.


etale-lemma-etale-etale-local

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is \'etale at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_U$, $i = 1, \ldots, n$, $j = 1, \ldots, m_i$ such that \begin{enumerate} \item $V_{i, j} \to U$ is an isomorphism, \item $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \end{enumerate}


etale-lemma-etale-etale-local-technical

Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is \'etale at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$ X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j} $$ such that \begin{enumerate} \item $V_{i, j} \to U$ is an isomorphism, \item the fibre $W_u$ contains no point mapping to any $x_i$. \end{enumerate} In particular, if $f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}$, then the fibre $W_u$ is empty.


etale-lemma-finite-etale-etale-local

Let $f : X \to S$ be a finite \'etale morphism of schemes. Let $s \in S$. There exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$ X_U = \coprod\nolimits_j V_j $$ such that each $V_j \to U$ is an isomorphism.


etale-lemma-etale-dimension

Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a \'etale homomorphism of local rings. Then $\dim(A) = \dim(B)$.


etale-lemma-faithful

If $f : X \to S$ is surjective, then the functor (\ref{equation-descent-etale}) is faithful.


etale-lemma-fully-faithful

Assume $f : X \to S$ is submersive and any \'etale base change of $f$ is submersive. Then the functor (\ref{equation-descent-etale}) is fully faithful.


etale-lemma-fully-faithful-cases

Let $f : X \to S$ be a morphism of schemes. In the following cases the functor (\ref{equation-descent-etale}) is fully faithful: \begin{enumerate} \item $f$ is surjective and universally closed (e.g., finite, integral, or proper), \item $f$ is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale), \item $f$ is surjective, quasi-compact, and flat. \end{enumerate}


etale-lemma-reduce-to-affine

Let $f : X \to S$ be a morphism of schemes. Let $(V, \varphi)$ be a descent datum relative to $X/S$ with $V \to X$ \'etale. Let $S = \bigcup S_i$ be an open covering. Assume that \begin{enumerate} \item the pullback of the descent datum $(V, \varphi)$ to $X \times_S S_i/S_i$ is effective, \item the functor (\ref{equation-descent-etale}) for $X \times_S (S_i \cap S_j) \to (S_i \cap S_j)$ is fully faithful, and \item the functor (\ref{equation-descent-etale}) for $X \times_S (S_i \cap S_j \cap S_k) \to (S_i \cap S_j \cap S_k)$ is faithful. \end{enumerate} Then $(V, \varphi)$ is effective.


etale-lemma-split-henselian

Let $(A, I)$ be a henselian pair. Let $U \to \Spec(A)$ be a quasi-compact, separated, \'etale morphism such that $U \times_{\Spec(A)} \Spec(A/I) \to \Spec(A/I)$ is finite. Then $$ U = U_{fin} \amalg U_{away} $$ where $U_{fin} \to \Spec(A)$ is finite and $U_{away}$ has no points lying over $Z$.


etale-lemma-strict-normal-crossings

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor. Let $D_i \subset D$, $i \in I$ be its irreducible components viewed as reduced closed subschemes of $X$. The following are equivalent \begin{enumerate} \item $D$ is a strict normal crossings divisor, and \item $D$ is reduced, each $D_i$ is an effective Cartier divisor, and for $J \subset I$ finite the scheme theoretic intersection $D_J = \bigcap_{j \in J} D_j$ is a regular scheme each of whose irreducible components has codimension $|J|$ in $X$. \end{enumerate}


etale-lemma-smooth-pullback-strict-normal-crossings

\begin{slogan} Pullback of a strict normal crossings divisor by a smooth morphism is a strict normal crossings divisor. \end{slogan} Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a strict normal crossings divisor. If $f : Y \to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a strict normal crossings divisor on $Y$.


etale-lemma-characterize-normal-crossings-normalization

Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme. The following are equivalent \begin{enumerate} \item $D$ is a normal crossings divisor in $X$, \item $D$ is reduced, the normalization $\nu : D^\nu \to D$ is unramified, and for any $n \geq 1$ the scheme $$ Z_n = D^\nu \times_D \ldots \times_D D^\nu \setminus \{(p_1, \ldots, p_n) \mid p_i = p_j\text{ for some }i\not = j\} $$ is regular, the morphism $Z_n \to X$ is a local complete intersection morphism whose conormal sheaf is locally free of rank $n$. \end{enumerate}


etale-proposition-properties-sections

Sections of unramified morphisms. \begin{enumerate} \item Any section of an unramified morphism is an open immersion. \item Any section of a separated morphism is a closed immersion. \item Any section of an unramified separated morphism is open and closed. \end{enumerate}


etale-proposition-equality

Let $S$ is be a scheme. Let $\pi : X \to S$ be unramified and separated. Let $Y$ be an $S$-scheme and $y \in Y$ a point. Let $f, g : Y \to X$ be two $S$-morphisms. Assume \begin{enumerate} \item $Y$ is connected \item $x = f(y) = g(y)$, and \item the induced maps $f^\sharp, g^\sharp : \kappa(x) \to \kappa(y)$ on residue fields are equal. \end{enumerate} Then $f = g$.


etale-proposition-etale-CM

\begin{slogan} Being Cohen-Macaulay ascends and descends along \'etale maps. \end{slogan} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is Cohen-Macaulay if and only if $B$ is so.


etale-proposition-effective

Let $f : X \to S$ be a surjective integral morphism. The functor (\ref{equation-descent-etale}) induces an equivalence $$ \begin{matrix} \text{schemes quasi-compact,}\\ \text{separated, \'etale over }S \end{matrix} \longrightarrow \begin{matrix} \text{descent data }(V, \varphi)\text{ relative to }X/S\text{ with}\\ V\text{ quasi-compact, separated, \'etale over }X \end{matrix} $$


crystalline-lemma-divided-power-envelope

Let $(A, I, \gamma)$ be a divided power ring. Let $A \to B$ be a ring map. Let $J \subset B$ be an ideal with $IB \subset J$. There exists a homomorphism of divided power rings $$ (A, I, \gamma) \longrightarrow (D, \bar J, \bar \gamma) $$ such that $$ \Hom_{(A, I, \gamma)}((D, \bar J, \bar \gamma), (C, K, \delta)) = \Hom_{(A, I)}((B, J), (C, K)) $$ functorially in the divided power algebra $(C, K, \delta)$ over $(A, I, \gamma)$. Here the LHS is morphisms of divided power rings over $(A, I, \gamma)$ and the RHS is morphisms of (ring, ideal) pairs over $(A, I)$.


crystalline-lemma-describe-divided-power-envelope

Let $(B, I, \gamma)$ be a divided power algebra. Let $I \subset J \subset B$ be an ideal. Let $(D, \bar J, \bar \gamma)$ be the divided power envelope of $J$ relative to $\gamma$. Choose elements $f_t \in J$, $t \in T$ such that $J = I + (f_t)$. Then there exists a surjection $$ \Psi : B\langle x_t \rangle \longrightarrow D $$ of divided power rings mapping $x_t$ to the image of $f_t$ in $D$. The kernel of $\Psi$ is generated by the elements $x_t - f_t$ and all $$ \delta_n\left(\sum r_t x_t - r_0\right) $$ whenever $\sum r_t f_t = r_0$ in $B$ for some $r_t \in B$, $r_0 \in I$.


crystalline-lemma-divided-power-envelope-add-variables

Let $(A, I, \gamma)$ be a divided power ring. Let $B$ be an $A$-algebra and $IB \subset J \subset B$ an ideal. Let $x_i$ be a set of variables. Then $$ D_{B[x_i], \gamma}(JB[x_i] + (x_i)) = D_{B, \gamma}(J) \langle x_i \rangle $$


crystalline-lemma-divided-power-first-order-thickening

Let $(A, I, \gamma)$ be a divided power ring. Let $M$ be an $A$-module. Let $B = A \oplus M$ as an $A$-algebra where $M$ is an ideal of square zero and set $J = I \oplus M$. Set $$ \delta_n(x + z) = \gamma_n(x) + \gamma_{n - 1}(x)z $$ for $x \in I$ and $z \in M$. Then $\delta$ is a divided power structure and $A \to B$ is a homomorphism of divided power rings from $(A, I, \gamma)$ to $(B, J, \delta)$.


crystalline-lemma-divided-power-second-order-thickening

Let $(A, I, \gamma)$ be a divided power ring. Let $M$, $N$ be $A$-modules. Let $q : M \times M \to N$ be an $A$-bilinear map. Let $B = A \oplus M \oplus N$ as an $A$-algebra with multiplication $$ (x, z, w)\cdot (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z)) $$ and set $J = I \oplus M \oplus N$. Set $$ \delta_n(x, z, w) = (\gamma_n(x), \gamma_{n - 1}(x)z, \gamma_{n - 1}(x)w + \gamma_{n - 2}(x)q(z, z)) $$ for $(x, z, w) \in J$. Then $\delta$ is a divided power structure and $A \to B$ is a homomorphism of divided power rings from $(A, I, \gamma)$ to $(B, J, \delta)$.


crystalline-lemma-affine-thickenings-colimits

In Situation \ref{situation-affine}. \begin{enumerate} \item $\text{CRIS}(C/A)$ has products, \item $\text{CRIS}(C/A)$ has all finite nonempty colimits and (\ref{equation-forget-affine}) commutes with these, and \item $\text{Cris}(C/A)$ has all finite nonempty colimits and $\text{Cris}(C/A) \to \text{CRIS}(C/A)$ commutes with them. \end{enumerate}


crystalline-lemma-list-properties

In Situation \ref{situation-affine}. Let $P \to C$ be a surjection of $A$-algebras with kernel $J$. Write $D_{P, \gamma}(J) = (D, \bar J, \bar\gamma)$. Let $(D^\wedge, J^\wedge, \bar\gamma^\wedge)$ be the $p$-adic completion of $D$, see Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}. For every $e \geq 1$ set $P_e = P/p^eP$ and $J_e \subset P_e$ the image of $J$ and write $D_{P_e, \gamma}(J_e) = (D_e, \bar J_e, \bar\gamma)$. Then for all $e$ large enough we have \begin{enumerate} \item $p^eD \subset \bar J$ and $p^eD^\wedge \subset \bar J^\wedge$ are preserved by divided powers, \item $D^\wedge/p^eD^\wedge = D/p^eD = D_e$ as divided power rings, \item $(D_e, \bar J_e, \bar\gamma)$ is an object of $\text{Cris}(C/A)$, \item $(D^\wedge, \bar J^\wedge, \bar\gamma^\wedge)$ is equal to $\lim_e (D_e, \bar J_e, \bar\gamma)$, and \item $(D^\wedge, \bar J^\wedge, \bar\gamma^\wedge)$ is an object of $\text{Cris}^\wedge(C/A)$. \end{enumerate}


crystalline-lemma-set-generators

In Situation \ref{situation-affine}. Let $P$ be a polynomial algebra over $A$ and let $P \to C$ be a surjection of $A$-algebras with kernel $J$. With $(D_e, \bar J_e, \bar\gamma)$ as in Lemma \ref{lemma-list-properties}: for every object $(B, J_B, \delta)$ of $\text{CRIS}(C/A)$ there exists an $e$ and a morphism $D_e \to B$ of $\text{CRIS}(C/A)$.


crystalline-lemma-generator-completion

In Situation \ref{situation-affine}. Let $P$ be a polynomial algebra over $A$ and let $P \to C$ be a surjection of $A$-algebras with kernel $J$. Let $(D, \bar J, \bar\gamma)$ be the $p$-adic completion of $D_{P, \gamma}(J)$. For every object $(B \to C, \delta)$ of $\text{Cris}^\wedge(C/A)$ there exists a morphism $D \to B$ of $\text{Cris}^\wedge(C/A)$.


crystalline-lemma-omega

Let $A$ be a ring. Let $(B, J, \delta)$ be a divided power ring and $A \to B$ a ring map.  \begin{enumerate} \item Consider $B[x]$ with divided power ideal $(JB[x], \delta')$ where $\delta'$ is the extension of $\delta$ to $B[x]$. Then $$ \Omega_{B[x]/A, \delta'} = \Omega_{B/A, \delta} \otimes_B B[x] \oplus B[x]\text{d}x. $$ \item Consider $B\langle x \rangle$ with divided power ideal $(JB\langle x \rangle + B\langle x \rangle_{+}, \delta')$. Then $$ \Omega_{B\langle x\rangle/A, \delta'} = \Omega_{B/A, \delta} \otimes_B B\langle x \rangle \oplus B\langle x\rangle \text{d}x. $$ \item Let $K \subset J$ be an ideal preserved by $\delta_n$ for all $n > 0$. Set $B' = B/K$ and denote $\delta'$ the induced divided power on $J/K$. Then $\Omega_{B'/A, \delta'}$ is the quotient of $\Omega_{B/A, \delta} \otimes_B B'$ by the $B'$-submodule generated by $\text{d}k$ for $k \in K$. \end{enumerate}


crystalline-lemma-diagonal-and-differentials

Let $(A, I, \gamma) \to (B, J, \delta)$ be a homomorphism of divided power rings. Let $(B(1), J(1), \delta(1))$ be the coproduct of $(B, J, \delta)$ with itself over $(A, I, \gamma)$, i.e., such that $$ \xymatrix{ (B, J, \delta) \ar[r] & (B(1), J(1), \delta(1)) \\ (A, I, \gamma) \ar[r] \ar[u] & (B, J, \delta) \ar[u] } $$ is cocartesian. Denote $K = \Ker(B(1) \to B)$. Then $K \cap J(1) \subset J(1)$ is preserved by the divided power structure and $$ \Omega_{B/A, \delta} = K/ \left(K^2 + (K \cap J(1))^{[2]}\right) $$ canonically.


crystalline-lemma-diagonal-and-differentials-affine-site

In Situation \ref{situation-affine}. Let $(B, J, \delta)$ be an object of $\text{CRIS}(C/A)$. Let $(B(1), J(1), \delta(1))$ be the coproduct of $(B, J, \delta)$ with itself in $\text{CRIS}(C/A)$. Denote $K = \Ker(B(1) \to B)$. Then $K \cap J(1) \subset J(1)$ is preserved by the divided power structure and $$ \Omega_{B/A, \delta} = K/ \left(K^2 + (K \cap J(1))^{[2]}\right) $$ canonically.


crystalline-lemma-module-differentials-divided-power-envelope

Let $(A, I, \gamma)$ be a divided power ring. Let $A \to B$ be a ring map and let $IB \subset J \subset B$ be an ideal. Let $D_{B, \gamma}(J) = (D, \bar J, \bar \gamma)$ be the divided power envelope. Then we have $$ \Omega_{D/A, \bar\gamma} = \Omega_{B/A} \otimes_B D $$


crystalline-lemma-fibre-product

Let $(U', T', \delta') \to (S'_0, S', \gamma')$ and $(S_0, S, \gamma) \to (S'_0, S', \gamma')$ be morphisms of divided power schemes. If $(U', T', \delta')$ is a divided power thickening, then there exists a divided power scheme $(T_0, T, \delta)$ and $$ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } $$ which is a cartesian diagram in the category of divided power schemes.


crystalline-lemma-divided-power-thickening-fibre-products

In Situation \ref{situation-global}. The category $\text{CRIS}(X/S)$ has all finite nonempty limits, in particular products of pairs and fibre products. The functor (\ref{equation-forget}) commutes with limits.


crystalline-lemma-compare-big-small

Assumptions as in Definition \ref{definition-divided-power-thickening-X}. The inclusion functor $$ \text{Cris}(X/S) \to \text{CRIS}(X/S) $$ commutes with finite nonempty limits, is fully faithful, continuous, and cocontinuous. There are morphisms of topoi $$ (X/S)_{\text{cris}} \xrightarrow{i} (X/S)_{\text{CRIS}} \xrightarrow{\pi} (X/S)_{\text{cris}} $$ whose composition is the identity and of which the first is induced by the inclusion functor. Moreover, $\pi_* = i^{-1}$.


crystalline-lemma-localize

In Situation \ref{situation-global}. Let $X' \subset X$ and $S' \subset S$ be open subschemes such that $X'$ maps into $S'$. Then there is a fully faithful functor $\text{Cris}(X'/S') \to \text{Cris}(X/S)$ which gives rise to a morphism of topoi fitting into the commutative diagram $$ \xymatrix{ (X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} & (X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\ \Sh(X'_{Zar}) \ar[r] & \Sh(X_{Zar}) } $$ Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma \ref{sites-lemma-localize-morphism-topoi}.


crystalline-lemma-crystal-quasi-coherent-modules

With notation $X/S, \mathcal{I}, \gamma, \mathcal{C}, \mathcal{F}$ as in Definition \ref{definition-modules}. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is quasi-coherent, and \item $\mathcal{F}$ is locally quasi-coherent and a crystal in $\mathcal{O}_{X/S}$-modules. \end{enumerate}


crystalline-lemma-module-differentials-divided-power-scheme

Let $(T, \mathcal{J}, \delta)$ be a divided power scheme. Let $T \to S$ be a morphism of schemes. The quotient $\Omega_{T/S} \to \Omega_{T/S, \delta}$ described above is a quasi-coherent $\mathcal{O}_T$-module. For $W \subset T$ affine open mapping into $V \subset S$ affine open we have $$ \Gamma(W, \Omega_{T/S, \delta}) = \Omega_{\Gamma(W, \mathcal{O}_W)/\Gamma(V, \mathcal{O}_V), \delta} $$ where the right hand side is as constructed in Section \ref{section-differentials}.


crystalline-lemma-module-of-differentials

In Situation \ref{situation-global}. For $(U, T, \delta)$ in $\text{Cris}(X/S)$ the restriction $(\Omega_{X/S})_T$ to $T$ is $\Omega_{T/S, \delta}$ and the restriction $\text{d}_{X/S}|_T$ is equal to $\text{d}_{T/S, \delta}$.


crystalline-lemma-module-of-differentials-on-affine

In Situation \ref{situation-global}. For any affine object $(U, T, \delta)$ of $\text{Cris}(X/S)$ mapping into an affine open $V \subset S$ we have $$ \Gamma((U, T, \delta), \Omega_{X/S}) = \Omega_{\Gamma(T, \mathcal{O}_T)/\Gamma(V, \mathcal{O}_V), \delta} $$ where the right hand side is as constructed in Section \ref{section-differentials}.


crystalline-lemma-omega-locally-quasi-coherent

In Situation \ref{situation-global}. The sheaf of differentials $\Omega_{X/S}$ has the following two properties: \begin{enumerate} \item $\Omega_{X/S}$ is locally quasi-coherent, and \item for any morphism $(U, T, \delta) \to (U', T', \delta')$ of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion the map $c_f : f^*(\Omega_{X/S})_{T'} \to (\Omega_{X/S})_T$ is surjective. \end{enumerate}


crystalline-lemma-automatic-connection

In Situation \ref{situation-global}. Let $\mathcal{F}$ be a crystal in $\mathcal{O}_{X/S}$-modules on $\text{Cris}(X/S)$. Then $\mathcal{F}$ comes equipped with a canonical integrable connection.


crystalline-lemma-homotopy-tensor

Let $A_*$ be a cosimplicial ring. Let $\varphi_*, \psi_* : K_* \to M_*$ be homomorphisms of cosimplicial $A_*$-modules. \begin{enumerate} \item \label{item-tensor} If $\varphi_*$ and $\psi_*$ are homotopic, then $$ \varphi_* \otimes 1, \psi_* \otimes 1 : K_* \otimes_{A_*} L_* \longrightarrow M_* \otimes_{A_*} L_* $$ are homotopic for any cosimplicial $A_*$-module $L_*$. \item \label{item-wedge} If $\varphi_*$ and $\psi_*$ are homotopic, then $$ \wedge^i(\varphi_*), \wedge^i(\psi_*) : \wedge^i(K_*) \longrightarrow \wedge^i(M_*) $$ are homotopic. \item \label{item-base-change} If $\varphi_*$ and $\psi_*$ are homotopic, and $A_* \to B_*$ is a homomorphism of cosimplicial rings, then $$ \varphi_* \otimes 1, \psi_* \otimes 1 : K_* \otimes_{A_*} B_* \longrightarrow M_* \otimes_{A_*} B_* $$ are homotopic as homomorphisms of cosimplicial $B_*$-modules. \item \label{item-completion} If $I_* \subset A_*$ is a cosimplicial ideal, then the induced maps $$ \varphi^\wedge_*, \psi^\wedge_* : K_*^\wedge \longrightarrow M_*^\wedge $$ between completions are homotopic. \item Add more here as needed, for example symmetric powers. \end{enumerate}


crystalline-lemma-structure-Dn

Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}). The coprojection $P \to P \otimes_A \ldots \otimes_A P$, $f \mapsto f \otimes 1 \otimes \ldots \otimes 1$ induces an isomorphism \begin{equation} \label{equation-structure-Dn} D(n) = \lim_e D\langle \xi_i(j) \rangle/p^eD\langle \xi_i(j) \rangle \end{equation} of algebras over $D$ with $$ \xi_i(j) = x_i \otimes 1 \otimes \ldots \otimes 1 - 1 \otimes \ldots \otimes 1 \otimes x_i \otimes 1 \otimes \ldots \otimes 1 $$ for $j = 1, \ldots, n$ where the second $x_i$ is placed in the $j + 1$st slot; recall that $D(n)$ is constructed starting with the $n + 1$-fold tensor product of $P$ over $A$.


crystalline-lemma-property-Dn

Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}). Then $(D, \bar J, \bar\gamma)$ and $(D(n), \bar J(n), \bar\gamma(n))$ are objects of $\text{Cris}^\wedge(C/A)$, see Remark \ref{remark-completed-affine-site}, and $$ D(n) = \coprod\nolimits_{j = 0, \ldots, n} D $$ in $\text{Cris}^\wedge(C/A)$.


crystalline-lemma-crystals-on-affine

In the situation above there is a functor $$ \begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla)\text{ satisfying} \\ \text{(\ref{item-complete}), (\ref{item-connection}), (\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})} \end{matrix} $$


crystalline-lemma-crystals-on-affine-smooth

In Situation \ref{situation-affine}. Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \gamma}(J')$. There are homomorphisms of divided power $A$-algebras $$ a : D \longrightarrow D',\quad b : D' \longrightarrow D $$ compatible with the maps $D \to C$ and $D' \to C$ such that $a \circ b = \text{id}_{D'}$. These maps induce an equivalence of categories of pairs $(M, \nabla)$ satisfying (\ref{item-complete}), (\ref{item-connection}), (\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent}) over $D$ and pairs $(M', \nabla')$  satisfying (\ref{item-complete}), (\ref{item-connection}), (\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent}) over $D'$. In particular, the equivalence of categories of Proposition \ref{proposition-crystals-on-affine} also holds for the corresponding functor towards pairs over $D'$.


crystalline-lemma-vanishing-lqc

In Situation \ref{situation-global}. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then we have $$ H^p((U, T, \delta), \mathcal{F}) = 0 $$ for all $p > 0$ and all $(U, T, \delta)$ with $T$ or $U$ affine.


crystalline-lemma-compare

In Situation \ref{situation-global}. Assume moreover $X$ and $S$ are affine schemes. Consider the full subcategory $\mathcal{C} \subset \text{Cris}(X/S)$ consisting of divided power thickenings $(X, T, \delta)$ endowed with the chaotic topology (see Sites, Example \ref{sites-example-indiscrete}). For any locally quasi-coherent $\mathcal{O}_{X/S}$-module $\mathcal{F}$ we have $$ R\Gamma(\mathcal{C}, \mathcal{F}|_\mathcal{C}) = R\Gamma(\text{Cris}(X/S), \mathcal{F}) $$


crystalline-lemma-complete

In Situation \ref{situation-affine}. Set $\mathcal{C} = (\text{Cris}(C/A))^{opp}$ and $\mathcal{C}^\wedge = (\text{Cris}^\wedge(C/A))^{opp}$ endowed with the chaotic topology, see Remark \ref{remark-completed-affine-site} for notation. There is a morphism of topoi $$ g : \Sh(\mathcal{C}) \longrightarrow \Sh(\mathcal{C}^\wedge) $$ such that if $\mathcal{F}$ is a sheaf of abelian groups on $\mathcal{C}$, then $$ R^pg_*\mathcal{F}(B \to C, \delta) = \left\{ \begin{matrix} \lim_e \mathcal{F}(B_e \to C, \delta) & \text{if }p = 0 \\ R^1\lim_e \mathcal{F}(B_e \to C, \delta) & \text{if }p = 1 \\ 0 & \text{else} \end{matrix} \right. $$ where $B_e = B/p^eB$ for $e \gg 0$.


crystalline-lemma-category-with-covering

Let $\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\mathcal{C}$ such that every object of $\mathcal{C}$ has a morphism towards $X$. Assume that $\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ the total cohomology $R\Gamma(\mathcal{C}, \mathcal{F})$ is represented by the complex $$ \mathcal{F}(X) \to \mathcal{F}(X \times X) \to \mathcal{F}(X \times X \times X) \to \ldots $$ associated to the cosimplicial abelian group $[n] \mapsto \mathcal{F}(X^n)$.


crystalline-lemma-vanishing-omega-1

With notation as in (\ref{equation-omega-Dn}) the complex $$ \Omega_{D(0)} \to \Omega_{D(1)} \to \Omega_{D(2)} \to \ldots $$ is homotopic to zero as a $D(*)$-cosimplicial module.


crystalline-lemma-vanishing

With notation as in (\ref{equation-Dn}) and (\ref{equation-omega-Dn}), given any cosimplicial module $M_*$ over $D(*)$ and $i > 0$ the cosimplicial module $$ M_0 \otimes^\wedge_{D(0)} \Omega^i_{D(0)} \to M_1 \otimes^\wedge_{D(1)} \Omega^i_{D(1)} \to M_2 \otimes^\wedge_{D(2)} \Omega^i_{D(2)} \to \ldots $$ is homotopic to zero, where $\Omega^i_{D(n)}$ is the $p$-adic completion of the $i$th exterior power of $\Omega_{D(n)}$.


crystalline-lemma-poincare

Let $A$ be a ring. Let $P = A\langle x_i \rangle$ be a divided power polynomial ring over $A$. For any $A$-module $M$ the complex $$ 0 \to M \to M \otimes_A P \to M \otimes_A \Omega^1_{P/A, \delta} \to M \otimes_A \Omega^2_{P/A, \delta} \to \ldots $$ is exact. Let $D$ be the $p$-adic completion of $P$. Let $\Omega^i_D$ be the $p$-adic completion of the $i$th exterior power of $\Omega_{D/A, \delta}$. For any $p$-adically complete $A$-module $M$ the complex $$ 0 \to M \to M \otimes^\wedge_A D \to M \otimes^\wedge_A \Omega^1_D \to M \otimes^\wedge_A \Omega^2_D \to \ldots $$ is exact.


crystalline-lemma-relative-poincare

Let $A$ be a ring. Let $(B, J, \delta)$ be a divided power ring. Let $P = B\langle x_i \rangle$ be a divided power polynomial ring over $B$ with divided power ideal $J = IP + B\langle x_i \rangle_{+}$ as usual. Let $M$ be a $B$-module endowed with an integrable connection $\nabla : M \to M \otimes_B \Omega^1_{B/A, \delta}$. Then the map of de Rham complexes $$ M \otimes_B \Omega^*_{B/A, \delta} \longrightarrow M \otimes_P \Omega^*_{P/A, \delta} $$ is a quasi-isomorphism. Let $D$, resp.\ $D'$ be the $p$-adic completion of $B$, resp.\ $P$ and let $\Omega^i_D$, resp.\ $\Omega^i_{D'}$ be the $p$-adic completion of $\Omega^i_{B/A, \delta}$, resp.\ $\Omega^i_{P/A, \delta}$. Let $M$ be a $p$-adically complete $D$-module endowed with an integral connection $\nabla : M \to M \otimes^\wedge_D \Omega^1_D$. Then the map of de Rham complexes $$ M \otimes^\wedge_D \Omega^*_D \longrightarrow M \otimes^\wedge_D \Omega^*_{D'} $$ is a quasi-isomorphism.


crystalline-lemma-cohomology-is-zero

Assumptions and notation as in Proposition \ref{proposition-compute-cohomology}. Then $$ H^j(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}) = 0 $$ for all $i > 0$ and all $j \geq 0$.


crystalline-lemma-compute-cohomology-crystal-smooth

Assumptions as in Proposition \ref{proposition-compute-cohomology-crystal}. Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \gamma}(J')$. Let $(M', \nabla')$ be the pair over $D'$ corresponding to $\mathcal{F}$, see Lemma \ref{lemma-crystals-on-affine-smooth}. Then the complex $$ M' \otimes^\wedge_{D'} \Omega^*_{D'} $$ computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.


crystalline-lemma-find-homotopy

In the situation above there exists a map of complexes $$ e_M^\bullet : M \otimes_B (\Omega')^\bullet \longrightarrow M \otimes_B \Omega^\bullet $$ such that $c_M^\bullet \circ e_M^\bullet$ and $e_M^\bullet \circ c_M^\bullet$ are homotopic to multiplication by $a$.


crystalline-lemma-computation

In Situation \ref{situation-affine}. Assume $D$ and $\Omega_D$ are as in (\ref{equation-D}) and (\ref{equation-omega-D}). Let $\lambda \in D$. Let $D'$ be the $p$-adic completion of $$ D[z]\langle \xi \rangle/(\xi - (z^p - \lambda)) $$ and let $\Omega_{D'}$ be the $p$-adic completion of the module of divided power differentials of $D'$ over $A$. For any pair $(M, \nabla)$ over $D$ satisfying (\ref{item-complete}), (\ref{item-connection}), (\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent}) the canonical map of complexes (\ref{equation-base-change-map-complexes}) $$ c_M^\bullet : M \otimes_D^\wedge \Omega^\bullet_D \longrightarrow M \otimes_D^\wedge \Omega^\bullet_{D'} $$ has the following property: There exists a map $e_M^\bullet$ in the opposite direction such that both $c_M^\bullet \circ e_M^\bullet$ and $e_M^\bullet \circ c_M^\bullet$ are homotopic to multiplication by $p$.


crystalline-lemma-pullback-along-p-power-cover

Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma)$ be a divided power scheme over $\mathbf{Z}_{(p)}$ with $p \in \mathcal{I}$. We set $S_0 = V(\mathcal{I}) \subset S$. Let $f : X' \to X$ be an iterated $\alpha_p$-cover of schemes over $S_0$ with constant degree $q$. Let $\mathcal{F}$ be any crystal in quasi-coherent sheaves on $X$ and set $\mathcal{F}' = f_{\text{cris}}^*\mathcal{F}$. In the distinguished triangle $$ Ru_{X/S, *}\mathcal{F} \longrightarrow f_*Ru_{X'/S, *}\mathcal{F}' \longrightarrow E \longrightarrow Ru_{X/S, *}\mathcal{F}[1] $$ the object $E$ has cohomology sheaves annihilated by $q$.


crystalline-lemma-pullback-along-p-power-cover-cohomology

With notations and assumptions as in Lemma \ref{lemma-pullback-along-p-power-cover} the map $$ f^* : H^i(\text{Cris}(X/S), \mathcal{F}) \longrightarrow H^i(\text{Cris}(X'/S), \mathcal{F}') $$ has kernel and cokernel annihilated by $q^{i + 1}$.


crystalline-lemma-pullback-relative-frobenius

In the situation above, assume that $X \to S_0$ is smooth of relative dimension $d$. Then $F_{X/S_0}$ is an iterated $\alpha_p$-cover of degree $p^d$. Hence Lemmas \ref{lemma-pullback-along-p-power-cover} and \ref{lemma-pullback-along-p-power-cover-cohomology} apply to this situation. In particular, for any crystal in quasi-coherent modules $\mathcal{G}$ on $\text{Cris}(X^{(1)}/S)$ the map $$ F_{X/S_0}^* : H^i(\text{Cris}(X^{(1)}/S), \mathcal{G}) \longrightarrow H^i(\text{Cris}(X/S), F_{X/S_0, \text{cris}}^*\mathcal{G}) $$ has kernel and cokernel annihilated by $p^{d(i + 1)}$.


crystalline-proposition-crystals-on-affine

The functor $$ \begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla)\text{ satisfying} \\ \text{(\ref{item-complete}), (\ref{item-connection}), (\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})} \end{matrix} $$ of Lemma \ref{lemma-crystals-on-affine} is an equivalence of categories.