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 \(\Omega _{B/A}\) 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 \(d \colon A \to M\)

\(X\) 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 \(\Omega _{X / S}\) which globalizes the construction \(\Omega _{A / R}\).

Let \(X\) be a scheme. We say \(X\) is locally noetherian if there exists an open affine cover \(\{ U_{i} = \operatorname {Spec}A_{i}\} \) 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 \(f \in A \smallsetminus \mathfrak {p}\) such that \(M_{f}\) is a free \(A_{f}\)-module.

An algebraically closed field is perfect.

\(\operatorname {Der}_{R}(A,M)\) is an \(R\)-module.

The following module satisfies the universal property of \(\Omega _{R / A}\) : 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. \(d(a + a^{\prime }) = da + da^{\prime }\)

3. \(d(aa^{\prime }) = ada^{\prime } + a^{\prime }da\) .

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

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 \(\phi \), 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 \(\phi (T) \subset S\). Assume the following diagram commutes

We have a (canonical) isomorphism

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

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

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

\(\Omega _{X / S}\) 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 \(\mathscr {I} / \mathscr {I}^{2}\) 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 \(C \colonequals B / I\). 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 \(X = \operatorname {Spec}R\) 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 \(\mathfrak {m} / \mathfrak {m}^{2} \to \Omega _{B / k} \otimes _{B} k\) 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 \(\Omega _{B / k}\) 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 \(\Omega _{X / k}\) 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.