The Stacks project

Proof. The equivalence of (1) and (2) is a restatement of Definition 88.8.1.

The equivalence of (2) and (3) follows from More on Algebra, Lemma 15.84.11.

The equivalence of (3) and (4) follows from the fact that the systems $\{ \mathop{N\! L}\nolimits _{B_ n/A_ n}\} $ and $\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B B_ n$ are strictly isomorphic, see Lemma 88.3.3. Some details omitted.

Assume (2). Let $a \in I$. Let $c$ be such that multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge $. By More on Algebra, Lemma 15.84.4 part (1) there exists a map $\alpha : \bigoplus B\text{d}x_ i \to J/J^2$ such that $\text{d} \circ \alpha $ and $\alpha \circ \text{d}$ are both multiplication by $a^ c$. Let $f_ i \in J$ be an element whose class modulo $J^2$ is equal to $\alpha (\text{d}x_ i)$. A simple calculation gives that (6)(a), (b) hold.

We omit the verification that (6) implies (5); it is just a statement on two term complexes over $B$ of the form $M \to B^{\oplus r}$.

Assume (5) holds. Say $I = (a_1, \ldots , a_ t)$. Let $c_ i \geq 0$ be the integer such that (5)(a), (b) hold for $a_ i^{c_ i}$. Then we see that $I^{\sum c_ i}$ annihilates $H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge )$. Let $f_{i, 1}, \ldots , f_{i, r} \in J$ be as in (5)(b) for $a_ i$. Consider the composition

where the $j$th basis vector is mapped to the class of $f_{i, j}$ in $J/J^2$. By (5)(a) and (b) the cokernel of the composition is annihilated by $a_ i^{2c_ i}$. Thus this map is surjective after inverting $a_ i^{c_ i}$, and hence an isomorphism (Algebra, Lemma 10.16.4). Thus the kernel of $B^{\oplus r} \to \bigoplus B\text{d}x_ i$ is $a_ i$-power torsion, and hence $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge ) = \mathop{\mathrm{Ker}}(J/J^2 \to \bigoplus B\text{d}x_ i)$ is $a_ i$-power torsion. Since $B$ is Noetherian (Lemma 88.2.2), all modules including $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge )$ are finite. Thus $a_ i^{d_ i}$ annihilates $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge )$ for some $d_ i \geq 0$. It follows that $I^{\sum d_ i}$ annihilates $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge )$ and we see that (3) holds.

Thus conditions (2), (3), (4), (5), and (6) are equivalent. Thus it remains to show that these conditions are equivalent with (7). Observe that the Cramer ideal $\text{Fit}_ r(J) B$ is equal to $\text{Fit}_ r(J/J^2)$ as $J/J^2 = J \otimes _{A[x_1, \ldots , x_ r]^\wedge } B$, see More on Algebra, Lemma 15.8.4 part (3). Also, observe that the Jacobian ideal is just $\text{Fit}_0(H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge ))$. Thus we see that the equivalence of (3) and (7) is a purely algebraic question which we discuss in the next paragraph.

Let $R$ be a Noetherian ring and let $I \subset R$ be an ideal. Let $M \xrightarrow {d} R^{\oplus r}$ be a two term complex. We have to show that the following are equivalent

1. the cohomology of $M \to R^{\oplus r}$ is annihilated by a power of $I$, and

2. the ideals $\text{Fit}_ r(M)$ and $\text{Fit}_0(\text{Coker}(d))$ contain a power of $I$.

Since $R$ is Noetherian, we can reformulate part (2) as an inclusion of the corresponding closed subschemes, see Algebra, Lemmas 10.17.2 and 10.32.5. On the other hand, over the complement of $V(\text{Fit}_0(\mathop{\mathrm{Coker}}(d)))$ the cokernel of $d$ vanishes and over the complement of $V(\text{Fit}_ r(M))$ the module $M$ is locally generated by $r$ elements, see More on Algebra, Lemma 15.8.7. Thus (B) is equivalent to

1. away from $V(I)$ the cokernel of $d$ vanishes and the module $M$ is locally generated by $\leq r$ elements.

Of course this is equivalent to the condition that $M \to R^{\oplus r}$ has vanishing cohomology over $\mathop{\mathrm{Spec}}(R) \setminus V(I)$ which in turn is equivalent to (A). This finishes the proof. $\square$

Proof. Immediate from Definitions 88.4.1 and 88.8.1. $\square$

Proof. Assume $B^\wedge $ satisfies the equivalent conditions of Lemma 88.8.2. The naive cotangent complex $\mathop{N\! L}\nolimits _{B/A}$ is a complex of finite type $B$-modules and hence $H^{-1}$ and $H^0$ are finite $B$-modules. Completion is an exact functor on finite $B$-modules (Algebra, Lemma 10.97.2) and $\mathop{N\! L}\nolimits _{B^\wedge /A}^\wedge $ is the completion of the complex $\mathop{N\! L}\nolimits _{B/A}$ (this is easy to see by choosing presentations). Hence the assumption implies there exists a $c \geq 0$ such that $H^{-1}/I^ nH^{-1}$ and $H^0/I^ nH^0$ are annihilated by $I^ c$ for all $n$. By Nakayama's lemma (Algebra, Lemma 10.20.1) this means that $I^ cH^{-1}$ and $I^ cH^0$ are annihilated by an element of the form $g = 1 + x$ with $x \in IB$. After inverting $g$ (which does not change the quotients $B/I^ nB$) we see that $\mathop{N\! L}\nolimits _{B/A}$ has cohomology annihilated by $I^ c$. Thus $A \to B$ is étale at any prime of $B$ not lying over $V(I)$ by the definition of étale ring maps, see Algebra, Definition 10.143.1.

Conversely, assume that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is étale over $\mathop{\mathrm{Spec}}(A) \setminus V(I)$. Then for every $a \in I$ there exists a $c \geq 0$ such that multiplication by $a^ c$ is zero $\mathop{N\! L}\nolimits _{B/A}$. Since $\mathop{N\! L}\nolimits _{B^\wedge /A}^\wedge $ is the derived completion of $\mathop{N\! L}\nolimits _{B/A}$ (see Lemma 88.3.3) it follows that $B^\wedge $ satisfies the equivalent conditions of Lemma 88.8.2. $\square$

Proof. By Lemma 88.3.4 there is a map

which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$. Thus the result by More on Algebra, Lemma 15.84.8. $\square$

Proof. Follows from Lemma 88.8.5 and Definition 88.8.1 and the fact that $I_2^ c \subset I_1A_2$ for some $c \geq 0$ as $A_2$ is Noetherian. $\square$

Proof. Uniqueness follows from the fact that $C^ h$ is a subring of $C^\wedge $, see for example More on Algebra, Lemma 15.12.4. The final assertion follows from the fact that $C^ h$ is the filtered colimit of these $C$-algebras $C'$, see proof of More on Algebra, Lemma 15.12.1. Having said this we now turn to the proof of existence.

Let $\varphi : B^\wedge \to C^\wedge $ be the given map. This defines a section

of the completion of the map $C \to B \otimes _ A C$. We may replace $(A, I, B, C, \varphi )$ by $(C, IC, B \otimes _ A C, C, \sigma )$. In this way we see that we may assume that $A = C$.

Proof of existence in the case $A = C$. In this case the map $\varphi : B^\wedge \to A^\wedge $ is necessarily surjective. By Lemmas 88.8.4 and 88.3.5 we see that the cohomology groups of $\mathop{N\! L}\nolimits _{A^\wedge /\! _\varphi B^\wedge }^\wedge $ are annihilated by a power of $I$. Since $\varphi $ is surjective, this implies that $\mathop{\mathrm{Ker}}(\varphi )/\mathop{\mathrm{Ker}}(\varphi )^2$ is annihilated by a power of $I$. Hence $\varphi : B^\wedge \to A^\wedge $ is the completion of a finite type $B$-algebra $B \to D$, see More on Algebra, Lemma 15.108.4. Hence $A \to D$ is a finite type algebra map which induces an isomorphism $A^\wedge \to D^\wedge $. By Lemma 88.8.4 we may replace $D$ by a localization and assume that $A \to D$ is étale away from $V(I)$. Since $A^\wedge \to D^\wedge $ is an isomorphism, we see that $\mathop{\mathrm{Spec}}(D) \to \mathop{\mathrm{Spec}}(A)$ is also étale in a neighbourhood of $V(ID)$ (for example by More on Morphisms, Lemma 37.12.3). Thus $\mathop{\mathrm{Spec}}(D) \to \mathop{\mathrm{Spec}}(A)$ is étale. Therefore $D$ maps to $A^ h$ and the lemma is proved. $\square$