A 0-cochain is a collection $s = (s_i)$ where $s_i \in \mathcal{F}(U_i)$. The differential is $(ds)_{ij} = s_i|_{U_i \cap U_j} - s_j|_{U_i \cap U_j}$

A 0-cochain $s$ is a cocycle if $ds = 0$, meaning $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$. By the sheaf axiom, this is equivalent to saying that the $s_i$ glue to give a global section of $\mathcal{F}$ on $X$.

Therefore, $\check{H}^0(\mathfrak{U}, \mathcal{F}) \cong \mathcal{F}(X)$.

A 1-cochain is a collection $(s_{ij})$ where $s_{ij} \in \mathcal{F}(U_i \cap U_j)$. The differential is $(ds)_{ijk} = s_{jk}|_{U_i \cap U_j \cap U_k} - s_{ik}|_{U_i \cap U_j \cap U_k} + s_{ij}|_{U_i \cap U_j \cap U_k}$

A 1-cocycle satisfies $s_{ik} = s_{ij} + s_{jk}$ on $U_i \cap U_j \cap U_k$. This is the cocycle condition for gluing data.

A 1-coboundary is of the form $s_{ij} = t_i|_{U_i \cap U_j} - t_j|_{U_i \cap U_j}$ for some 0-cochain $(t_i)$. Such cocycles represent trivial gluing obstructions.

Thus $\check{H}^1(\mathfrak{U}, \mathcal{F})$ measures the obstructions to gluing local data.

Let $X = \mathbb{A}^1 = \text{Spec } k[x]$ and consider the structure sheaf $\mathcal{O}_X$. Take the standard affine cover $\mathfrak{U} = \{U_0, U_1\}$ where $U_0 = D(x)$ and $U_1 = D(x-1)$.

We have:

• $\mathcal{O}_X(U_0) = k[x, x^{-1}]$
• $\mathcal{O}_X(U_1) = k[x, (x-1)^{-1}]$
• $\mathcal{O}_X(U_0 \cap U_1) = k[x, x^{-1}, (x-1)^{-1}]$

The Čech complex is: $0 \to k[x] \to k[x, x^{-1}] \oplus k[x, (x-1)^{-1}] \to k[x, x^{-1}, (x-1)^{-1}] \to 0$

For $p \geq 1$, we have $\check{H}^p(\mathfrak{U}, \mathcal{O}_X) = 0$ since any element of $k[x, x^{-1}, (x-1)^{-1}]$ can be written as a difference of elements from the two localizations using partial fractions.

Let $X = \mathbb{P}^1 = \text{Proj } k[x_0, x_1]$ with standard affine cover $\mathfrak{U} = \{U_0, U_1\}$ where $U_0 = D_+(x_0) \cong \text{Spec } k[x_1/x_0]$ and $U_1 = D_+(x_1) \cong \text{Spec } k[x_0/x_1]$.

The intersection is $U_0 \cap U_1 = D_+(x_0 x_1) \cong \text{Spec } k[x_0/x_1, x_1/x_0]$.

For the structure sheaf $\mathcal{O}_X$:

• $\mathcal{O}_X(U_0) = k[t]$ where $t = x_1/x_0$
• $\mathcal{O}_X(U_1) = k[s]$ where $s = x_0/x_1$
• $\mathcal{O}_X(U_0 \cap U_1) = k[t, t^{-1}]$

The Čech complex is: $0 \to \check{C}^0 \to \check{C}^1 \to 0$ where $\check{C}^0 = k[t] \oplus k[s]$ and $\check{C}^1 = k[t, t^{-1}]$.

The differential sends $(f, g) \mapsto f|_{U_0 \cap U_1} - g|_{U_0 \cap U_1}$.

We compute:

• $\check{H}^0(\mathfrak{U}, \mathcal{O}_X) = \ker(d^0) = k$ (constant polynomials)
• $\check{H}^1(\mathfrak{U}, \mathcal{O}_X) = \text{coker}(d^0) = 0$

Indeed, any Laurent polynomial $\sum_{i=m}^n a_i t^i \in k[t, t^{-1}]$ can be written as a difference: take $f = \sum_{i=0}^n a_i t^i$ and $g = -\sum_{i=m}^{-1} a_i t^{-i} s^{-i}$, noting that $s = t^{-1}$ on the overlap.

Thus $H^0(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}) = k$ and $H^1(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}) = 0$.

For the twisted sheaf $\mathcal{O}_{\mathbb{P}^1}(n)$, we have:

• $\mathcal{O}_{\mathbb{P}^1}(n)(U_0) = k[t]$
• $\mathcal{O}_{\mathbb{P}^1}(n)(U_1) = k[s]$
• $\mathcal{O}_{\mathbb{P}^1}(n)(U_0 \cap U_1) = t^n k[t, t^{-1}]$

where sections on $U_0$ correspond to degree $n$ elements viewed in the $U_0$ coordinate, and similarly for $U_1$.

Case $n \geq 0$: A 1-cocycle is an element $h \in t^n k[t, t^{-1}]$, which we can write as $h = t^n p(t)$ for some Laurent polynomial $p$. Writing $p = p_+ + p_-$ where $p_+$ has non-negative powers and $p_-$ has negative powers, we have: $h = t^n p_+ - (- t^n p_-)$ where $t^n p_+ \in k[t]$ and $-t^n p_- = -s^{-n} p_-(s^{-1}) \in k[s]$ (using $s = t^{-1}$).

Therefore $\check{H}^1(\mathfrak{U}, \mathcal{O}_{\mathbb{P}^1}(n)) = 0$ for $n \geq 0$.

For $\check{H}^0$, we need $(f, g) \in k[t] \oplus k[s]$ such that $t^n f(t) = s^{-n} g(s^{-1})$ on the overlap. For $n \geq 0$, this forces $f$ to have degree at most $n$ and $g = 0$ up to a constant multiple issue. More carefully, the space of global sections has dimension $n+1$, spanned by $x_0^i x_1^{n-i}$ for $i = 0, \ldots, n$.

Thus $H^0(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(n)) = k^{n+1}$ for $n \geq 0$.

Case $n < 0$: For a 0-cocycle $(f, g) \in k[t] \oplus k[s]$ to satisfy the gluing condition $t^n f(t) = s^{-n} g(s^{-1})$, we need $f = 0$ and $g = 0$ since $n < 0$ forces negative powers. Thus $H^0(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(n)) = 0$ for $n < 0$.

For $H^1$, the space of 1-cocycles is $t^n k[t, t^{-1}]$. A 1-coboundary is of the form $t^n f(t) - s^{-n} g(s^{-1})$ where $f \in k[t]$ and $g \in k[s]$. Since $n < 0$, we have $t^n k[t] = t^n k[t^{-1}]_{+}$ (polynomials in $t$ of degree $\geq n$). The quotient is $\check{H}^1(\mathfrak{U}, \mathcal{O}_{\mathbb{P}^1}(n)) = t^n k[t, t^{-1}] / (t^n k[t] + s^{-n} k[s]) = \text{span}\{t^n, t^{n+1}, \ldots, t^{-1}\}$ which has dimension $-n$.

Thus $H^1(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(n)) = k^{-n}$ for $n < 0$.

Let $X = \mathbb{P}^n = \text{Proj } k[x_0, \ldots, x_n]$ with standard affine cover $\mathfrak{U} = \{U_0, \ldots, U_n\}$ where $U_i = D_+(x_i)$.

For $p \leq n$, a $p$-fold intersection $U_{i_0} \cap \cdots \cap U_{i_p}$ is the distinguished open set $D_+(x_{i_0} \cdots x_{i_p})$, which is affine. Since affine schemes have vanishing higher cohomology for quasi-coherent sheaves, Leray's theorem applies.

The Čech complex for $\mathcal{O}_X$ has: $\check{C}^p(\mathfrak{U}, \mathcal{O}_X) = \prod_{i_0 < \cdots < i_p} k[x_0/x_{i_0}, \ldots, x_n/x_{i_0}, 1/x_{i_1}, \ldots, 1/x_{i_p}]$

For $p = 0$, global sections must be regular on all $U_i$. Such a function is a polynomial in $x_j/x_i$ for each $i$, which forces it to be constant by homogeneity.

For $p > 0$, we can show inductively that $\check{H}^p(\mathfrak{U}, \mathcal{O}_X) = 0$ for $0 < p < n$ and $\check{H}^n(\mathfrak{U}, \mathcal{O}_X) = 0$ as well.

Thus $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}) = k$ and $H^p(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}) = 0$ for $0 < p < n$.

For $H^n(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n})$, the computation is more delicate and also gives zero, though $H^n(\mathbb{P}^n, \omega_{\mathbb{P}^n}) \neq 0$ where $\omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1)$ is the canonical sheaf.

Consider $\mathbb{P}^1 = U_0 \cup U_1$ where $U_0, U_1$ are the standard affine opens. For $\mathcal{O}_{\mathbb{P}^1}(n)$, the Mayer-Vietoris sequence is: $0 \to H^0(\mathbb{P}^1, \mathcal{O}(n)) \to k[t] \oplus k[s] \to k[t, t^{-1}] \cdot t^n$ $\to H^1(\mathbb{P}^1, \mathcal{O}(n)) \to 0$

where the last two terms vanish since affines have no higher cohomology for quasi-coherent sheaves.

The map $k[t] \oplus k[s] \to k[t, t^{-1}] \cdot t^n$ sends $(f, g) \mapsto t^n f(t) - s^{-n} g(s^{-1})$.

For $n \geq 0$: The image consists of all elements of $t^n k[t, t^{-1}]$, so the cokernel vanishes and $H^1 = 0$. The kernel consists of pairs $(f, g)$ with $t^n f = s^{-n} g$ on the overlap, giving $\dim H^0 = n + 1$.

For $n < 0$: The kernel is zero (no global sections). The cokernel has dimension $-n$, giving $\dim H^1 = -n$.

This recovers our earlier computation.

Let $C \subseteq \mathbb{P}^2$ be a smooth plane curve of degree $d$, and consider $\mathbb{P}^2 = U_0 \cup U_1 \cup U_2$ where $U_i = D_+(x_i)$.

Rather than using all three opens at once, we can apply Mayer-Vietoris repeatedly. Set $U = U_0 \cup U_1$ and $V = U_2$. Then $\mathbb{P}^2 = U \cup V$ and $U \cap V = (U_0 \cap U_2) \cup (U_1 \cap U_2)$.

For the structure sheaf $\mathcal{O}_C$ of the curve: $H^0(U, \mathcal{O}_C) \oplus H^0(V, \mathcal{O}_C) \to H^0(U \cap V, \mathcal{O}_C) \to H^1(C, \mathcal{O}_C) \to 0$

Since $U \cap V$ is a union of two affine opens whose intersection is also affine, we can compute $H^0(U \cap V, \mathcal{O}_C)$ using another Mayer-Vietoris sequence.

This nested approach allows computation of $H^1(C, \mathcal{O}_C)$, which equals the genus $g$ of the curve by Riemann-Roch.

Let $X$ be a noetherian separated scheme, $\mathcal{F}$ quasi-coherent, and $\mathfrak{U}$ an affine cover. Then:

• Each $U_{i_0} \cap \cdots \cap U_{i_p}$ is affine (by separatedness)
• $H^q(U_{i_0} \cap \cdots \cap U_{i_p}, \mathcal{F}) = 0$ for $q > 0$ (quasi-coherent sheaves on affines)

Therefore $E_2^{p,q} = 0$ for $q > 0$, and the spectral sequence gives $\check{H}^p(\mathfrak{U}, \mathcal{F}) \cong H^p(X, \mathcal{F})$

Let $X$ be a noetherian scheme and $\mathcal{E}$ a locally free sheaf of rank $r+1$ on $X$. Consider the projective bundle $\pi: \mathbb{P}(\mathcal{E}) \to X$ with tautological sheaf $\mathcal{O}(1)$.

For an affine cover $\{U_i\}$ of $X$, the sets $\pi^{-1}(U_i) \cong \mathbb{P}^r_{U_i}$ form an affine-like cover of $\mathbb{P}(\mathcal{E})$ in the sense that they satisfy Leray's condition.

The Čech complex for $\mathcal{O}_{\mathbb{P}(\mathcal{E})}$ with respect to this cover can be computed using the known cohomology of projective space fibers. The projection formula gives: $H^p(\mathbb{P}(\mathcal{E}), \mathcal{O}(n)) \cong \bigoplus_{i=0}^r H^{p-i}(X, S^{n+i}(\mathcal{E}) \otimes \det(\mathcal{E})^{-i})$ for appropriate $n$ and $p$, where $S^k$ denotes the $k$-th symmetric power.

Let $X$ be a scheme and $\mathcal{L}$ an invertible sheaf. The first Čech cohomology $\check{H}^1(\mathfrak{U}, \mathcal{O}_X^*)$ classifies invertible sheaves via gluing data.

A 1-cocycle is a collection $\{g_{ij}\} \in \mathcal{O}_X^*(U_i \cap U_j)$ satisfying $g_{ik} = g_{ij} \cdot g_{jk}$ on triple intersections. This is precisely the data needed to glue local trivializations of an invertible sheaf.

Two cocycles are cohomologous if they differ by $g_{ij} = f_i / f_j$ for some $f_i \in \mathcal{O}_X^*(U_i)$, which corresponds to isomorphic invertible sheaves.

Therefore, $\check{H}^1(X, \mathcal{O}_X^*)$ is the Picard group $\text{Pic}(X)$ of isomorphism classes of invertible sheaves.

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ a quasi-coherent sheaf on $X$. The higher direct image $R^q f_* \mathcal{F}$ is the sheaf on $Y$ defined by $(R^q f_* \mathcal{F})(V) = H^q(f^{-1}(V), \mathcal{F})$

If $\mathfrak{V} = \{V_j\}$ is an affine cover of $Y$ and $\{f^{-1}(V_j)\}$ satisfies Leray's condition, we can compute $R^q f_* \mathcal{F}$ using Čech cohomology: $(R^q f_* \mathcal{F})(V_j) = \check{H}^q(f^{-1}(V_j), \mathcal{F})$

This is particularly useful for projections like $\pi: \mathbb{P}^n_Y \to Y$ where fibers are projective spaces.

Let $G = \text{Gr}(k, n)$ be the Grassmannian of $k$-dimensional subspaces of an $n$-dimensional vector space. The Grassmannian has a standard affine cover by Schubert cells.

For the structure sheaf $\mathcal{O}_G$, using the affine cover and Leray's theorem: $H^0(G, \mathcal{O}_G) = k, \quad H^p(G, \mathcal{O}_G) = 0 \text{ for } p > 0$

For the tautological bundle $\mathcal{S}$ and quotient bundle $\mathcal{Q}$ on $G$, Čech cohomology with respect to the Schubert cell cover can be used to compute their cohomology. The Borel-Weil-Bott theorem provides a complete description of $H^p(G, \mathcal{L})$ for any line bundle $\mathcal{L}$ on $G$.

The Čech complex provides a concrete realization of the local-to-global principle in algebraic geometry.

Consider a property $P$ that holds locally on opens $U_i$ of a cover. To check if $P$ globalizes, we:

1. Check compatibility on double intersections $U_i \cap U_j$ (captured by 1-cocycles)
2. Check higher coherence conditions on triple intersections (2-cocycles, etc.)

For example, lifting a section from $X/I^2$ to $X/I^3$ (deformation theory) involves checking obstructions in $\check{H}^2$.

The vanishing of higher Čech cohomology groups ensures that local data with compatible transition functions can be glued globally.

A sheaf $\mathcal{F}$ is flasque (or flabby) if all restriction maps $\mathcal{F}(U) \to \mathcal{F}(V)$ are surjective for $V \subseteq U$.

For a flasque sheaf, the Čech complex for any cover has vanishing higher cohomology: $\check{H}^p(\mathfrak{U}, \mathcal{F}) = 0$ for $p > 0$.

To see this, note that any 1-cocycle $(s_{ij}) \in \mathcal{F}(U_i \cap U_j)$ can be written as a coboundary: since $\mathcal{F}$ is flasque, we can extend sections from intersections to individual opens, allowing us to construct $(t_i)$ such that $s_{ij} = t_i|_{U_i \cap U_j} - t_j|_{U_i \cap U_j}$.

This proves that flasque sheaves are acyclic for the global sections functor, justifying their use in computing derived functor cohomology.

Given a scheme $X$, a quasi-coherent sheaf $\mathcal{F}$, and an affine cover $\mathfrak{U} = \{U_i\}_{i=0}^n$:

Step 1: Compute sections on opens and intersections.

• For each $U_i$, compute $\mathcal{F}(U_i)$
• For each $U_i \cap U_j$, compute $\mathcal{F}(U_i \cap U_j)$
• Continue for higher intersections up to the full dimension

Step 2: Write down the Čech complex.

• $\check{C}^0 = \prod_i \mathcal{F}(U_i)$
• $\check{C}^1 = \prod_{i < j} \mathcal{F}(U_i \cap U_j)$
• Continue as needed

Step 3: Compute kernels and images.

• Find $\ker(d^p: \check{C}^p \to \check{C}^{p+1})$
• Find $\text{im}(d^{p-1}: \check{C}^{p-1} \to \check{C}^p)$

Step 4: Take quotients.

• $\check{H}^p(\mathfrak{U}, \mathcal{F}) = \ker(d^p) / \text{im}(d^{p-1})$

Step 5: Verify Leray's condition.

• Check that all finite intersections are affine
• Conclude that $\check{H}^p(\mathfrak{U}, \mathcal{F}) = H^p(X, \mathcal{F})$

This algorithm is particularly effective when the number of opens is small and the rings involved have a nice structure (e.g., polynomial or localized polynomial rings).