On $\mathbb{P}^1_k$, the cohomology of $\mathcal{O}(m)$ is:

• $H^0(\mathbb{P}^1, \mathcal{O}(m)) = k[x_0, x_1]_m$ has dimension $m+1$ for $m \geq 0$, and is zero for $m < 0$.
• $H^1(\mathbb{P}^1, \mathcal{O}(m)) = 0$ for $m \geq -1$, and has dimension $-m-1$ for $m \leq -2$.

For instance:

• $H^0(\mathbb{P}^1, \mathcal{O}(2)) \cong k^3$ (spanned by $x_0^2, x_0x_1, x_1^2$)
• $H^1(\mathbb{P}^1, \mathcal{O}(-3)) \cong k^1$
• All cohomology vanishes for $\mathcal{O}(-1)$ and $\mathcal{O}(-2)$

This illustrates the vanishing theorem: for $m \geq -1$, we have $H^1(\mathbb{P}^1, \mathcal{O}(m)) = 0$.

On $\mathbb{P}^2_k$, we have:

• $H^0(\mathbb{P}^2, \mathcal{O}(m)) = k[x_0, x_1, x_2]_m$ has dimension $\binom{m+2}{2}$ for $m \geq 0$
• $H^1(\mathbb{P}^2, \mathcal{O}(m)) = 0$ for all $m$
• $H^2(\mathbb{P}^2, \mathcal{O}(m)) = 0$ for $m \geq -3$, and equals $k[x_0, x_1, x_2]_{-m-3}^*$ for $m \leq -3$

Notice that $H^1$ vanishes completely for all twists. The middle cohomology often behaves specially in this way. For $H^2$, we have:

• $H^2(\mathbb{P}^2, \mathcal{O}(-3)) \cong k$ (the canonical sheaf $\omega_{\mathbb{P}^2} = \mathcal{O}(-3)$)
• $H^2(\mathbb{P}^2, \mathcal{O}(-4)) \cong k^3$
• $H^2(\mathbb{P}^2, \mathcal{O}(-2)) = 0$ (vanishing threshold)

Consider $\mathbb{P}^n_k$ and the standard affine open subset $U_i = \{x_i \neq 0\} \cong \mathbb{A}^n_k$. By Serre's criterion, we can verify that $U_i$ is affine by checking that $H^j(U_i, \mathcal{F}) = 0$ for all $j > 0$ and all quasi-coherent $\mathcal{F}$.

However, $\mathbb{P}^n_k$ itself is not affine because $H^n(\mathbb{P}^n_k, \mathcal{O}(-n-1)) \neq 0$. This shows that the cohomology condition in Serre's criterion is sharp: we must check all quasi-coherent sheaves, not just structure sheaves.

A line bundle $\mathcal{L}$ on a projective scheme $X$ is called ample if some positive tensor power $\mathcal{L}^{\otimes n}$ is very ample (gives a closed embedding into projective space).

Serre's vanishing theorem implies that if $\mathcal{L}$ is ample, then for any coherent sheaf $\mathcal{F}$ and any $i > 0$, we have $H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for $n$ sufficiently large. This is a key property characterizing ample line bundles.

Conversely, Serre proved that if $X$ is projective and $\mathcal{L}$ is a line bundle such that for every coherent sheaf $\mathcal{F}$, we have $H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for $i > 0$ and $n \gg 0$, then $\mathcal{L}$ is ample.

For $X = \mathbb{P}^n_k$, we have $\text{reg}(\mathcal{O}_X) = 0$. This follows because:

• $H^i(\mathbb{P}^n, \mathcal{O}(-i)) = 0$ for all $i > 0$ (in fact, for $i < n$)
• $\mathcal{O}_{\mathbb{P}^n}$ is already generated by global sections

More generally, for a linear subspace $L \subset \mathbb{P}^n$ of dimension $d$, we have $\text{reg}(\mathcal{O}_L) = 0$ as well, since $L \cong \mathbb{P}^d$ and inherits the same regularity properties.

If $\mathcal{F}$ has regularity $m$, then $\mathcal{F}(k)$ has regularity $m - k$. This is immediate from the definition: $H^i(X, \mathcal{F}(k)(m-k-i)) = H^i(X, \mathcal{F}(m-i)) = 0$

For example, if $\text{reg}(\mathcal{O}_{\mathbb{P}^2}) = 0$, then:

• $\text{reg}(\mathcal{O}_{\mathbb{P}^2}(1)) = -1$
• $\text{reg}(\mathcal{O}_{\mathbb{P}^2}(-1)) = 1$
• $\text{reg}(\mathcal{O}_{\mathbb{P}^2}(-2)) = 2$

This shows how negative twists increase the regularity bound.

Let $C \subset \mathbb{P}^3$ be a smooth curve of degree $d$ and genus $g$. The ideal sheaf $\mathcal{I}_C$ of $C$ in $\mathbb{P}^3$ has regularity bounded by $\text{reg}(\mathcal{I}_C) \leq d - g + 2$

For instance:

• A line ($d = 1$, $g = 0$) has $\text{reg}(\mathcal{I}_L) \leq 3$, but actually $\text{reg}(\mathcal{I}_L) = 2$
• A plane conic ($d = 2$, $g = 0$) has $\text{reg}(\mathcal{I}_C) \leq 4$, but actually $\text{reg}(\mathcal{I}_C) = 3$
• A twisted cubic ($d = 3$, $g = 0$) has $\text{reg}(\mathcal{I}_C) \leq 5$, but actually $\text{reg}(\mathcal{I}_C) = 3$

These bounds are useful for understanding the syzygies and free resolutions of curves in projective space.

For $\mathbb{P}^n_k$ and a coherent sheaf $\mathcal{F}$, we can give an explicit bound for when $H^i(\mathbb{P}^n, \mathcal{F}(m)) = 0$.

If $\mathcal{F}$ is $m_0$-regular, then $H^i(\mathbb{P}^n, \mathcal{F}(m)) = 0$ for all $i > 0$ and $m \geq m_0 - i$.

For the structure sheaf $\mathcal{O}_X$ of a subscheme $X \subset \mathbb{P}^n$ of dimension $d$ and degree $\delta$, we have $\text{reg}(\mathcal{O}_X) \leq \delta - d + 1$

This gives explicit vanishing: $H^i(\mathbb{P}^n, \mathcal{O}_X(m)) = 0$ for $i > 0$ and $m \geq \delta - d + 1 - i$.

For $X = \mathbb{P}^n_k$, there is a close connection between coherent sheaves and finitely generated graded modules over $S = k[x_0, \ldots, x_n]$.

Given a graded $S$-module $M$, let $\widetilde{M}$ be the associated coherent sheaf on $\mathbb{P}^n$. Then: $H^i(\mathbb{P}^n, \widetilde{M}(n)) = H^i_{\mathfrak{m}}(M)_n$ for $i > 0$, where $H^i_{\mathfrak{m}}(M)$ denotes local cohomology with support in the maximal ideal $\mathfrak{m} = (x_0, \ldots, x_n)$.

Serre's vanishing theorem translates to the statement that the local cohomology modules $H^i_{\mathfrak{m}}(M)$ are finitely generated $S$-modules with zero Hilbert function in large degrees.

Let $S$ be a smooth projective surface over $\mathbb{C}$, and let $H$ be an ample divisor on $S$. Consider the sheaf $\omega_S \otimes \mathcal{O}_S(H)$.

By Kodaira vanishing: $H^1(S, \omega_S \otimes \mathcal{O}_S(H)) = 0 \quad \text{and} \quad H^2(S, \omega_S \otimes \mathcal{O}_S(H)) = 0$

By Serre vanishing: $H^i(S, \omega_S \otimes \mathcal{O}_S(nH)) = 0 \quad \text{for } i > 0 \text{ and } n \gg 0$

Kodaira gives immediate vanishing for $n = 1$, while Serre only guarantees vanishing for large $n$. However, if $S$ is in characteristic $p > 0$ or singular, only Serre's theorem applies.

For instance, on $\mathbb{P}^2$, we have $\omega_{\mathbb{P}^2} = \mathcal{O}(-3)$ and $\mathcal{O}(1)$ is ample. Kodaira tells us: $H^i(\mathbb{P}^2, \mathcal{O}(-2)) = 0 \quad \text{for } i > 0$ which can be verified directly. Serre only guarantees vanishing for $\mathcal{O}(n)$ with $n \gg 0$.

The Euler characteristic of a coherent sheaf $\mathcal{F}$ on a projective scheme $X$ is defined as $\chi(\mathcal{F}) = \sum_{i=0}^{\dim X} (-1)^i \dim H^i(X, \mathcal{F})$

For large twists $n \gg 0$, Serre's theorem tells us that $H^i(X, \mathcal{F}(n)) = 0$ for $i > 0$, so: $\chi(\mathcal{F}(n)) = \dim H^0(X, \mathcal{F}(n))$

On $\mathbb{P}^n$, the Hilbert polynomial of $\mathcal{F}$ is defined by $P(n) = \chi(\mathcal{F}(n))$ for $n \gg 0$. Serre's theorem guarantees that this is indeed a polynomial, since the alternating sum stabilizes to a polynomial in $n$.

For example, on a smooth curve $C \subset \mathbb{P}^n$ of degree $d$ and genus $g$, the Riemann-Roch theorem gives: $\chi(\mathcal{O}_C(n)) = d \cdot n + 1 - g$

For $n$ large enough (specifically $n \geq 2g - 1$), Serre's theorem ensures $H^1(C, \mathcal{O}_C(n)) = 0$, so: $\dim H^0(C, \mathcal{O}_C(n)) = d \cdot n + 1 - g$

Serre vanishing and Serre duality work together to provide complete information about cohomology.

For a smooth projective variety $X$ of dimension $n$ over a field $k$, Serre duality states: $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^* \otimes \omega_X)^*$

Combined with vanishing theorems, this often determines all cohomology groups. For example, on a smooth curve $C$ of genus $g$:

• For $\mathcal{L} = \mathcal{O}_C(D)$ with $\deg(D) \geq 2g - 1$: Serre vanishing gives $H^1(C, \mathcal{L}) = 0$
• For $\mathcal{L} = \mathcal{O}_C(D)$ with $\deg(D) \leq 0$: Serre duality gives $H^0(C, \mathcal{L}) \cong H^1(C, \omega_C \otimes \mathcal{L}^{-1})^*$ and if $\deg(\omega_C \otimes \mathcal{L}^{-1}) = 2g - 2 - \deg(D) \geq 2g - 1$, then $H^1(C, \omega_C \otimes \mathcal{L}^{-1}) = 0$, so $H^0(C, \mathcal{L}) = 0$.

This interplay between vanishing and duality is fundamental in the theory of algebraic curves.

On $\mathbb{P}^n_k$, we have $H^n(\mathbb{P}^n, \mathcal{O}(-n-1)) \neq 0$, showing that cohomological dimension exactly equals the geometric dimension.

More generally, for any $n$-dimensional projective variety $X$ over $k$, the dualizing sheaf $\omega_X$ (when it exists) satisfies $H^n(X, \omega_X) \cong k$. This is the highest non-vanishing cohomology group.

For a smooth complete intersection of multidegree $(d_1, \ldots, d_r)$ in $\mathbb{P}^n$, the dimension is $n - r$, and the cohomological dimension is also $n - r$. The top cohomology $H^{n-r}(X, \omega_X)$ is non-zero and one-dimensional.