Temkin's definition of weight function

Posted on January 17, 2018 by Dima

tags: Berkovich spaces, weight, log structures

Let \(k\) be a non-Archimedean valued field. We want to define a natural metric on the module of Kähler differentials \(\Omega_{Y/X}\) for a morphism of \(k\)-analytic spaces \(X \to Y\). This is presented in the utmost generality in this paper, see also the survey at the NATG’15 conference.

A morphism of normed modules \(f: B \to A\) is called non-expansive if \(||f(b)|| \leq ||b||\).

Given a morphism of normed rings \(A \to B\) one can define a norm on the module of differentials \(\Omega_{B/A}\) so that it is the maxmial norm making the morphism \(d: B \to \Omega_{B/A}\) non-expansive.

In particular, given a \(k\)-ring \(A\), the module of Kähler differentials \(\Omega_{A/k}\) is equipped with the Kähler norm as follows: \[ ||x||_\Omega = \inf_{x = \sum c_i db_i} \max |c_i| |b_i| \] The Kährler semi-norm on \(\Omega_B\) is characterized by the fact that it is the maximal semi-norm making the differential a non-expansive map.

Let \(A\) be an Banach algebra over \(k\), and let \(v \in {\mathcal{M}}(A)\) be a point of its Banach spectrum. Then an element \(f \in A\) is a function on \(A\) with values in residue fields of each point \(v\), \({\operatorname{\mathcal{H}}}(v)\), which is naturally normed, and \(|f|\) is a real-valued function, defined as \(|f|(v) = ||f||_{{\operatorname{\mathcal{H}}}(v)}\).

If \(v \in X\) is a point of a Berkovich analytic space, we want to look at the Kähler semi-norm on the complete residue fields \({\operatorname{\mathcal{H}}}(v)\), so that we can define a real-valued function \(|\omega|\) by putting it to be \(||\omega||_{{\operatorname{\mathcal{H}}}(v)}\) at a point \(v\).

A log structure on a ring \(A\) is a morphism of multiplictive monoids \(\alpha: M \to A\) that induces an isomorphism \(M^\times \to A^\times\).

If \(A\) is a log \(k\)-ring then the module of log differentials is defined as \(\Omega_{A/k} \oplus (A \otimes M^{gp})\) (where \(M^{gp}\) is a groupification of the monoid) module the relations \[ \begin{array}{c} (0, 1 \otimes c)\\ (da, -a \otimes a) = 0 \\ \end{array} \]

Along with the usual derivative \(d\) we have a log derivative \(\delta\), \(\delta a = (0, 1 \otimes a)\), so that elements \(\delta a\) should be thought of as \(d\log a\), with the above relation meaning \[ \begin{array}{c} d\log c = 0, c \in k\\ a d\log a = da, a \in M^{gp}\\ \end{array} \]

If \(K\) is a valued field with a valued ring \(K^\circ\), then \(K^\circ {\setminus}\{0\} \to K^\circ\) is a log structure on \(K^\circ\). We denote \(\Omega^{\log}_{K^\circ/A}\) the module of log differentials, where the log structure on \(K^\circ\) is as above, and the log structure on \(A\) is \(A {\setminus}{\mathrm{Ker}}(A \to K)\).

Adic seminorm: given a \(K^\circ\)-submodule \(M\) is defined on \(V=M\otimes K\) as: \[ ||v|| = \inf {\{ |a| \mid a \in K^\circ, v \in aM \}} \]

One checks that \(\Omega^{\log}_{K^\circ/A^\circ} \otimes_{K^\circ} K = \Omega_{K/A}\), so the image \(\Omega^{\log}_{K^\circ/A^\circ} \to \Omega^{\log}_{K^\circ/A^\circ} \otimes K=\Omega_{K/A}\) is a lattice.

Theorem. The adic semi-norm on \(\Omega^{log}_{K^\circ/A^\circ}\) is the maximal norm making the differential non-expansive.

Let \((V, ||\cdot||)\) be a finite-dimensional normed vector space. Then a basis \(e_1, \ldots, e_n\) is called \(r\)-orthogonal if for any \(v=\sum a_i e_i\) the inequality \(||v|| \geq r\max (|a_i|\cdot||e_i||)\) holds. If \(r=1\) and \(||e_i||=1\) then the basis is called orthogonal.

The valuation \[ v(\sum a_i t^i) = \max |a_i| \] on \(T_1\) is called Gauss valuation (there is a more general expression for more variables, and with radii \(\neq 1\), but let us forfeit it here).

Let \(K/k\) be an extension of valued fields. Let \(t=(t_1, \ldots, t_n)\) be a tuple of elements of \(L\). We call a valuation on \(K\) \(t\)-monomial if its restriction to \(k[t_1, \ldots, t_n]\) is a Gauss valuation.

Lemma. Let \(\xi\) be the Gauss point on \({\mathcal{M}}(T_1)\), the unit disc. Then \(||dt/t||_{\xi}\) is 1.

Proof. From the definition one immediately gets \[ ||dt/t||_{\xi} = ||df/(f' t)||_\xi = \inf_f |1/f' t|\cdot|f| = 1 \] where \(f=\sum a_i t^i\), \(f \neq const\). Indeed, \[ f't=\sum i a_i t^i \] so its norm coincides with that of \(f\).

Corollary. The function \(x \mapsto ||dt||_{x}: {\mathcal{M}}(T_1) \to {\mathbb{R}}\) is the radius function \(r(x)\), i.e. the infimum of a ball containing the point \(x\).

Proof. Indeed, any point in the unit disc is the gauss point of a disc of radius \(r(x)\) and coordinate function of the form \(T-a\), and \(dt=d(t-a)\), so from the previous lemma \(1=||dt/t||_x \leq ||dt||_x \cdot |1/t|\).

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Local cohomology and cohomology of line bundles on projective spaces

Posted on December 4, 2017 by Dima

tags: likbez, local rings, cohomology, modules, injective hull

all this is terribly unpolished, and arguments are ridden with holes, as is customary on this blog

Flatness

Frist things first: Nakayama’s lemma.

Statement 1: Let \(R\) be a ring, \(I\) an ideal Let \(M\) be a finitely generated module, and assume \(IM=M\). Then there exists \(x \in 1 + I\) such that \(xM=0\).

Proof: let \(x_1, \ldots, x_n\) be the generators of \(M\). Then there exists a matrix \(A=(a_{ij}) \in I^{n \times n}\) such that \(x_i = \sum a_{ij} x_j\), or, \((I-a)x=0\). Multiply by the matrix formed of minors on the left and get \(\det (Id-A) \cdot I \cdot x = 0\) where \(x=(x_i)\). Note that \(\det (Id-A) \in 1 + I\)

Statement 2: Let \(R\) be a ring, \(I\) an ideal that annihilates all simple \(R\)-modules (equivalently: if it lies in the intersection of all maximal ideals, i.e. in a local ring, in the maximal ideal), then \(M=0\).

Proof. Follows from Statement 1, since \(x\) is invertible. Indeed, otherwise \((x)\) would be contained in a maxmial ideal ideal that intersects trivially with \(I\) or otherwise \(1 \in I\), but \(I \cap {\mathfrak{m}}\neq 0\) for any maxmial \({\mathfrak{m}}\).

But now there is also a more direct proof. Let \(x_1, \ldots, x_n\) be the minimal set of generators of \(M\) and assume \({\mathfrak{m}}M = M\). Then \[ (1-a_n)x_n = \sum_{i=1}^{n-1} a_i x_i, \] and since \(1-a_n\) is invertible, \(x_1, \ldots, x_{n-1}\) generate \(M\), which contradicts the minimality of the set of generators.

A module \(M\) is flat iff \(-- \otimes M\) is an exact functor.

Lemma. \(M\) is flat if for all ideals \({\mathfrak{p}}\subset R\) \[ 0 \to {\mathfrak{p}}\otimes M \to M \to M/{\mathfrak{p}}M \to 0 \] is exact (and for that it suffices to check that the first arrow is inclusion).

Proposition. Assume \(R\) is a Dedekind domain (i.e. every ideal is a product of prime ideals; equivalently, 1-dimensional integrally closed). Then an (integral) \(R\)-algebra \(S\) is flat iff \(R \hookrightarrow S\) is injective.

Proposition. Let \(f: X={\mathrm{Spec}}S \to Y={\mathrm{Spec}}R\) be a morphism. Then \(S\) is a flat \(R\)-algebraic iff \({\mathcal{O}}_{X,x}\) is flat over \({\mathcal{O}}_{Y,y}\) for all \(x \in X\), all \(y = f(x)\).

Lemma. Let \(R\) be a local ring withe the maximal ideal \({\mathfrak{m}}\). Let \(M\) be a finitely generated \(R\)-module. Then \(M\) is free iff \({\mathrm{Tor}}_1(M,k) = 0\) (and moreover both conditions are equivalent to being projective and flat).

Proof. Assume \({\mathrm{Tor}}_1(M,k)\) vanishes. Since \(M\) is finitely generated, there is a surjective morphism \(f: R^n \to M\) where \(n = \dim k\). Tensoring the short exact sequence \({\mathrm{Ker}}f \to R^n \to M\) with \(k\) we get that \({\mathrm{Ker}}f \otimes k = 0\) which by Nakayama lemma implies \({\mathrm{Ker}}f = 0\).

Lemma. Setting as above. \(M\) has a projective resolution of length \(n\) if and only if \(Tor^n(M,k) = 0\).

Proof. The non-trivial direction is right-to-left. Assume \[ \ldots \to P_{n+1} \to P_n \to P_{n-1} \to \ldots \to P_0 \to M \to 0 \] is a resolution. Let \(Z_i = {\mathrm{Ker}}d_i \hookrightarrow P_i\). Then we have short exact sequences \[ 0 Z_i \to P_i \to Z_{i-1} \to 0 \] Let \(i=0\) and tensor with \(k\), then \[ \ldots{\mathrm{Tor}}_n(M,k)=0 \to {\mathrm{Tor}}_{n-1}(Z_0,k) \to {\mathrm{Tor}}_{n-1}(P_0, k)=0 \to \ldots \] So \({\mathrm{Tor}}_{n-1}(Z_0,k)=0\). Arguing inductively, we get \({\mathrm{Tor}}_{n-i-1}(Z_i,k) = 0\). In particular, \({\mathrm{Tor}}_{1}(Z_{n-2}, k)=0\) and by the previous lemma it’s free and hence projective. Then \[ Z_{n-2} \to P_{n_2}\to P_{n-1} \to \ldots \] is a length \(n\) projective resolution.

Koszul complex

Let \(R\) be a local ring with the maximal ideal \({\mathfrak{m}}\) and residue field \(k=R/{\mathfrak{m}}\). Then there exists a particularly interesting projective (in fact, free) resolution of \(k\). Let \(x_1, \ldots, x_n\) be a set of generators of \({\mathfrak{m}}\). Let \(K{\langle}\xi_1, \ldots, xi_n {\rangle}\) be the ring of grassamanian polynomials in \(\xi_1, \ldots, x_n\) graded by degree, where \(n\)-th graded piece is isomorphic to \(\wedge^n R\). Define the differential map to be \[ d(\xi_i) = x_i \] on the 1-homogeneaus piece, and extend it linearly and applying Leibniz’s rule \[ d(a \wedge b) = da \wedge b + (-1)^{|a||b|}a \wedge db \] The graded pieces of \(K{\langle}\xi_1, \ldots, \xi_n {\rangle}\) with this differential are element of the Koszul complex.

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