Dependencies

Let $X$ be a variety over $k$. Same comclusion as above.

If $B$ is a finitely generated $A$-algebra or a localization thereof, then ${\mathrm{\Omega }}_{B/A}$ is finitely generated as a $B$-module.

The sheaf of kahler differentials is coherent.

A stalk of a quotient of ideal sheaves is isomoprhic to the quotient of the stalks of the ideal sheaves.

An $R$-linear derivation of $A$ into $M$ is a map of $R$-modules $d:A\to M$

$X$ is irreducible if $|X|$ is.

$|X|$ is irreducible if it cannot be written as a union of two closed subsets $Z_1\cup {\mathbf{Z}}_2$.

Given a map of schemes $X\to S$, we have a sheaf ${\mathrm{\Omega }}_{X/S}$ which globalizes the construction ${\mathrm{\Omega }}_{A/R}$.

Let $X$ be a scheme. We say $X$ is locally noetherian if there exists an open affine cover $\{U_i=\mathrm{Spec}{A}_i\}$ such that all $A_i$ are noetherian rings.

A field extension $K/k$ is separably generated if there exists a trancendence basis $\{x_i\}$ for $K/k$ such that $K$ is a separable algebraic extension of $k(\{x_i\})$.

Let $X$ be a scheme, $U$ an affine open neighborhood and $Y$ a closed (irreducible?) subscheme. Then $U\cap Y$ is an affine open neighborhood of $Y$

Let $A$ be a ring, and $M$ an $A$-module. Suppose that, for some prime ideal $\mathfrak{p}$, $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module. Then there exists an element $f\in A\setminus \mathfrak{p}$ such that $M_f$ is a free $A_f$-module.

An algebraically closed field is perfect.

${\mathrm{Der}}_R(A,M)$ is an $R$-module.

The following module satisfies the universal property of ${\mathrm{\Omega }}_{R/A}$ : Take the free $R$-module on the symbols $da$ for $a\in A$, and quotient out by the relations

1. $dr=0$ for $r\in R$

2. $d(a+a^{\prime })=da+d{a}^{\prime }$

3. $d(a{a}^{\prime })=ad{a}^{\prime }+a^{\prime }da$ .

Let $I$ be the kernel of the multiplication map $A{\otimes }_{R}A\to A$. Then $I/I^2$ satisfies the universal property of ${\mathrm{\Omega }}_{A/R}$

A surjection of finite free modules splits

Let $X$ be a suitable (noetherian? irred? zariski?) topological space. Then the intersection of an open with an irreducible is irreducible.

By $\varphi$, we mean the ring map $R\to A$ given by the algebra structure Let $S$ a multiplicative subset of $A$, and let $T$ be a multiplicative subset of $R$ with $\varphi (T)\subset S$. Assume the following diagram commutes

We have a (canonical) isomorphism

Let $K$ be a finitely generated (as an algebra) field extension of $k$. Then $\mathrm{trdeg}K/k\le \dim {\mathrm{\Omega }}_{K/k}$, with equality if (and only if) $K$ is separably generated over $k$.

The module of Kahler differentials ${\mathrm{\Omega }}_{A/R}$ is the $A$-module that represents the functor $M\mapsto {\mathrm{Der}}_R(A,M)$ from $A$-modules to $R$-modules.

The module of Kahler differentials has the following universal property: The map $d:A\to {\mathrm{\Omega }}_{A/R}$ defined by $a\mapsto da$ is initial in the category whose objects are derivations $\delta :A\to M$ and morphisms are diagrams

${\mathrm{\Omega }}_{X/S}$ is quasi-coherent

Let $A$ be a noetherian local integral domain, with residue field $k$ and quotient field $K$. If $M$ is a finitely generated $A$-module and ${\dim }_{k}M{\otimes }_{A}k={\dim }_{K}M{\otimes }_{A}K=r$, then $M$ is free of rank $r$.

Let $K/k$ be a perfect field extension. Then $K$ is separably generated over $k$.

Rank is additive on short exact sequences

Let $X$ be a regular variety, and let $Y$ be a regular subvariety. Let $\mathcal{I}$ be the ideal sheaf of $Y$. Then $\mathcal{I}/{\mathcal{I}}^2$ is a locally free sheaf of rank $\dim X-\dim Y$.

The functor on sheaves of abelian groups (and in particular, quasi-coherent sheaves) on a scheme $X$ which takes a sheaf $\mathcal{F}$ to its stalk at the point $x$, ${\mathcal{F}}_x$, is a functor that preserves colimits.

Let $B$ be an $A$-algebra, and $I$ some ideal of $B$. Let $C\text{colonequals}B/I$. We have the following right-exact sequence:

There is a sheafy exact sequence globalizing 2.3.10

Let $(R,\mathfrak{m},k)$ be a noetherian ring. The following are equivalent:

1. Let $n$ be the minimal number of generators of $\mathfrak{m}$. Then $n=\dim R$.

2. $\dim \mathfrak{m}/{\mathfrak{m}}^2=\dim R$

Let $X=\mathrm{Spec}R$ be an affine scheme. Then the irreducible subsets of the topological space $|X|$ are in one-to-one correspondence with the prime ideals of $R$ on the association $\mathfrak{p}\mapsto V(\mathfrak{p})$.

Let $X$ be a smooth variety and $D$ a (smooth?) divisor. Then

Theorem statement here

Theorem statement here.

Let $R$ be a noetherian ring. Then $R[x]$ is noetherian.

Let $(B,\mathfrak{m},k)$ be a local ring which contains a field $k$ isomorphic to its residue field Then the map $\mathfrak{m}/{\mathfrak{m}}^2\to {\mathrm{\Omega }}_{B/k}{\otimes }_{B}k$ which is the first map in the conormal right-exact sequence is an isomorphism.

Let $(B,\mathfrak{m},k)$ be a local ring of equal characteristic. In addition, assume that $k$ is a perfect field, and that $B$ is a localization of a finitely generatd $k$-algebra. Then ${\mathrm{\Omega }}_{B/k}$ is a free $B$-module of rank equal to the dimension of $B$ if and only if $B$ is a regular local ring.

The conormal sequence is exact on the left.

Let $X$ be an irreducible separated scheme of finite type over an algebraically closed field $k$. Then ${\mathrm{\Omega }}_{X/k}$ is a locally free sheaf of rank $\dim X$ if and only if $X$ is regular.

The localization of a regular local ring at (read: away from) a prime ideal is a regular local ring.

Consider an exact sequence of abelian (coherent) sheaves on a scheme. Can be left-exact, right-exact, exact in the middle, short exact, longer, anything. The seqeunce is exact iff it exact on all stalks.

Let $X$ be a locally noetherian scheme, and let $\mathcal{F}$ be a coherent sheaf. If the stalk ${\mathcal{F}}_x$ is a free ${\mathcal{O}}_{X,x}$-module for some point $x\in X$, then there exists a neighborhood $U$ containing $x$ such that $\mathcal{F}{|}_U$ is free.