---
title: "Worksheet 27 - Cohomology on projective schemes"
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# Worksheet 27: Cohomology on projective schemes {#br-bgk-2019-w27}

At the frozen revision boundary, none of the fourteen exercises has a public solution. The candidate map records negative results for Exercises 27.1–27.14; this edition does not create new solutions.

<!-- upstream_entity: Polynomring/1/Nenneraufnahme an X/Cech-Kohomologie/Aufgabe -->

## Exercise 27.1 {#br-bgk-2019-w27-ex01}

Let $A=R[X]$ over a commutative ring $R$. Determine the Čech complex of the structure sheaf for the one-element cover consisting of $D(X)$ of the punctured line. What homology results?

<!-- upstream_entity: Polynomring/2/Cech-Komplex/Monom/Aufgabe -->

## Exercise 27.2 {#br-bgk-2019-w27-ex02}

Let $A=R[X,Y]$ over a commutative ring $R$. Determine the Čech complex of the structure sheaf for the standard cover of the punctured plane at the monomials

1. $X^2Y^3$,
2. $X^5Y^{-4}$,
3. $X^{-3}Y^{-6}$.

What homology results in each case?

<!-- upstream_entity: Polynomring/3/Cech-Komplex/Monom/1/Aufgabe -->

## Exercise 27.3 {#br-bgk-2019-w27-ex03}

Let $A=R[X,Y,Z]$ over a commutative ring $R$. Determine the Čech complex of the structure sheaf for the standard cover of punctured space at the monomial $X^3Y^{-2}Z^7$. What homology results?

<!-- upstream_entity: Polynomring/3/Cech-Komplex/Monom/2/Aufgabe -->

## Exercise 27.4 {#br-bgk-2019-w27-ex04}

Let $A=R[X,Y,Z]$ over a commutative ring $R$. Determine the Čech complex of the structure sheaf for the standard cover of punctured space at the monomial $X^{-5}YZ^{-4}$. What homology results?

<!-- upstream_entity: Polynomring/4/Cech-Komplex/Monom/Aufgabe -->

## Exercise 27.5 {#br-bgk-2019-w27-ex05}

Let $A=R[X,Y,Z,W]$ over a commutative ring $R$. Determine the Čech complex of the structure sheaf for the standard cover of punctured space at the monomial $X^{-3}YZ^3W^{-2}$. What homology results?

<!-- upstream_entity: Polynomring/Höchste lokale Kohomologie/Modulstruktur/Direkt/Aufgabe -->

## Exercise 27.6 {#br-bgk-2019-w27-ex06}

Let $K[X_1,\ldots,X_d]$ be the polynomial ring over a field $K$, and let $H$ be the $K$-vector space spanned by all monomials $X^\nu$ in the variables $X_1,\ldots,X_d$ with $\nu_j\leq-1$ for every $j$, that is,

$$
H=K\left\langle
X_1^{\nu_1}\cdots X_d^{\nu_d},\ \nu_j\leq-1
\right\rangle.
$$

Define a natural $K[X_1,\ldots,X_d]$-module structure on $H$.

<!-- upstream_entity: Polynomring/Höchste lokale Kohomologie/K-Isomorphie zu Polynomring/Aufgabe -->

## Exercise 27.7 {#br-bgk-2019-w27-ex07}

Let $K[Y_1,\ldots,Y_d]$ be the polynomial ring over a field $K$, and let $H$ be the $K$-vector space spanned by all monomials $X^\nu$ in the variables $X_1,\ldots,X_d$ with $\nu_j\leq-1$ for every $j$, that is,

$$
H=K\left\langle
X_1^{\nu_1}\cdots X_d^{\nu_d},\ \nu_j\leq-1
\right\rangle.
$$

Prove that the map

$$
K[Y_1,\ldots,Y_d]\longrightarrow H,
\qquad
Y^\mu\longmapsto X^{-\mu-(1,1,\ldots,1)},
$$

is a $K$-linear isomorphism of $K$-vector spaces.

<!-- upstream_entity: Polynomring/Höchste lokale Kohomologie/Fakultäten/K-Isomorphie zu Polynomring/Aufgabe -->

## Exercise 27.8 {#br-bgk-2019-w27-ex08}

Let $K[Y_1,\ldots,Y_d]$ be the polynomial ring over a field $K$ of characteristic $0$, and let $H$ be the $K$-vector space spanned by all monomials $X^\nu$ in the variables $X_1,\ldots,X_d$ with $\nu_j\leq-1$ for every $j$, that is,

$$
H=K\left\langle
X_1^{\nu_1}\cdots X_d^{\nu_d},\ \nu_j\leq-1
\right\rangle.
$$

Prove that the map

$$
\theta:K[Y_1,\ldots,Y_d]\longrightarrow H,
\qquad
Y^\mu\longmapsto\mu!X^{-\mu-(1,1,\ldots,1)},
$$

is a $K$-linear isomorphism of $K$-vector spaces.

<!-- upstream_entity: Polynomring/Höchste lokale Kohomologie/Differentialoperatoren/Isomorphe Situation/Aufgabe -->

## Exercise 27.9 {#br-bgk-2019-w27-ex09}

Let $K$ be a field of characteristic $0$, and let $K[X_1,\ldots,X_d]$, $K[Y_1,\ldots,Y_d]$, and $K[D_1,\ldots,D_d]$ be polynomial rings in $d$ variables. Let $H$ be the $K$-vector space described in Exercise 27.6, with its natural $K[X_1,\ldots,X_d]$-module structure.

The polynomial ring $K[D_1,\ldots,D_d]$ acts on $K[Y_1,\ldots,Y_d]$ by letting $D_i$ act as the $i$th partial derivative, namely

$$
D_i=\partial_i=\frac{\partial}{\partial Y_i}.
$$

Prove that the correspondence $D_i\mapsto X_i$, together with the map

$$
\theta:K[Y_1,\ldots,Y_d]\longrightarrow H
$$

from Exercise 27.8, gives an isomorphism of modules.

For the next exercise, note that in the smooth case, by Corollary 19.12, one is computing the dimension of the space of global differential forms.

<!-- upstream_entity: Projektive Ebene/Kurve/Getwistete Strukturgabe zu d-3/Globale Schnitte/Aufgabe -->

## Exercise 27.10 {#br-bgk-2019-w27-ex10}

Let

$$
C=V_+(f)\subset\mathbb P_K^2
$$

be a projective plane curve of degree $d$ over a field $K$. Using the long exact cohomology sequence arising from the short exact sequence of sheaves (compare Exercise 13.23)

$$
0\longrightarrow\mathcal O_{\mathbb P_K^2}(-3)
\xrightarrow{f}\mathcal O_{\mathbb P_K^2}(d-3)
\longrightarrow\mathcal O_C(d-3)\longrightarrow0
$$

on the projective plane and Theorem 27.4, prove that the dimension of

$$
H^0(C,\mathcal O_C(d-3))
$$

is

$$
\frac{(d-1)(d-2)}2.
$$

For the next two exercises, compare Theorem 22.12.

<!-- upstream_entity: Projektive Gerade/Einheitengarbe/Cech-Komplex/Erste Kohomologie/Aufgabe -->

## Exercise 27.11 {#br-bgk-2019-w27-ex11}

Compute the Čech complex of the sheaf of units $\mathcal O_{\mathbb P_K^1}^{\times}$ for the standard affine cover of the projective line $\mathbb P_K^1$ over a field $K$, and its first Čech cohomology

$$
\check H^1\!\left(
D_+(X),D_+(Y),\mathcal O_{\mathbb P_K^1}^{\times}
\right).
$$

<!-- upstream_entity: Projektive Ebene/Einheitengarbe/Cech-Komplex/Erste Kohomologie/Aufgabe -->

## Exercise 27.12 {#br-bgk-2019-w27-ex12}

Compute the Čech complex of the sheaf of units $\mathcal O_{\mathbb P_K^2}^{\times}$ for the standard affine cover of the projective plane $\mathbb P_K^2$ over a field $K$, and its first Čech cohomology

$$
\check H^1\!\left(
D_+(X),D_+(Y),D_+(Z),\mathcal O_{\mathbb P_K^2}^{\times}
\right).
$$

<!-- upstream_entity: Modultheorie/Exakte Komplexe/Kurze exakte Sequenzen/Aufgabe -->

## Exercise 27.13 {#br-bgk-2019-w27-ex13}

Let $R$ be a commutative ring and let $M_i$, $i\in\mathbb N$, be $R$-modules with fixed $R$-module homomorphisms

$$
\varphi_i:M_i\longrightarrow M_{i+1}.
$$

The sequence

$$
\cdots\longrightarrow M_i\longrightarrow M_{i+1}
\longrightarrow M_{i+2}\longrightarrow M_{i+3}\longrightarrow\cdots
$$

is called exact if, for every $i\geq1$,

$$
\ker(\varphi_i)=\operatorname{im}(\varphi_{i-1}).
$$

1. Prove that for a short exact sequence, this definition agrees with the [definition of a short exact sequence referenced in the source](https://de.wikiversity.org/wiki/Modultheorie_(kommutative_Algebra)/Kurze_exakte_Sequenz/Definition).

2. Now let $R=K$ be a field, assume that the displayed sequence is exact, suppose all $M_i$ are finitely generated, $M_0=0$, and $M_i=0$ for every $i\geq n$, for some $n$. Prove that

   $$
   \sum_{i=0}^n(-1)^i\dim_KM_i=0.
   $$

> **Edition note—incomplete reference label.** In the comparison sentence above, the frozen source displays the link text “Definition .” without a number, but the link target is available. This edition preserves the target and supplies a descriptive English label without inventing a definition number.

> **Edition note—missing exactness hypothesis.** Part 2 of the frozen source does not explicitly say that the displayed sequence is exact. Without that hypothesis the asserted alternating-dimension formula is false in general, so the necessary hypothesis has been supplied above.

> **Edition note—index range.** The frozen source requires the equality above “for every $i$”, although the modules and maps are indexed by $\mathbb N$ and $\varphi_{-1}$ is therefore not defined at $i=0$. The condition has been stated for $i\geq1$, the indices for which both sides are defined.

<!-- upstream_entity: Projektiver Raum/Getwistete Strukturgarben/Euler-Charakteristik/Aufgabe -->

## Exercise 27.14 {#br-bgk-2019-w27-ex14}

Compute the Euler characteristic of the twisted structure sheaves $\mathcal O_{\mathbb P_K^d}(n)$ on projective space $\mathbb P_K^d$ over an algebraically closed field $K$.