Let $R = \mathbb{Z}$ and consider $M = \mathbb{Z}/n\mathbb{Z}$ and $N = \mathbb{Z}/m\mathbb{Z}$.

The projective resolution of $M$ is: $0 \to \mathbb{Z} \xrightarrow{n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0$

Applying $\text{Hom}_\mathbb{Z}(-, \mathbb{Z}/m\mathbb{Z})$: $0 \to \text{Hom}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) \to \text{Hom}(\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) \xrightarrow{n} \text{Hom}(\mathbb{Z}, \mathbb{Z}/m\mathbb{Z})$

This gives: $0 \to \mathbb{Z}/\gcd(n,m)\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z} \xrightarrow{n} \mathbb{Z}/m\mathbb{Z}$

Therefore:

• $\text{Ext}^0_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) = \mathbb{Z}/\gcd(n,m)\mathbb{Z}$
• $\text{Ext}^1_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) = \mathbb{Z}/\gcd(n,m)\mathbb{Z}$
• $\text{Ext}^i_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) = 0$ for $i \geq 2$

Let $X$ be a scheme and let $\mathcal{E}$ be a locally free sheaf of finite rank on $X$. For any $\mathcal{O}_X$-module $\mathcal{F}$: $\mathcal{E}xt^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) = 0 \quad \text{for } i > 0$

This is because locally free sheaves are locally isomorphic to $\mathcal{O}_X^{\oplus n}$, which has projective dimension zero. Moreover: $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) \cong \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{F}$ where $\mathcal{E}^\vee = \mathcal{H}om_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)$ is the dual sheaf.

Let $X = \text{Spec } A$ be an affine scheme and $Z \subset X$ a closed subscheme defined by an ideal $I \subset A$. Let $\mathcal{I} = \tilde{I}$ be the corresponding ideal sheaf.

For the structure sheaf $\mathcal{O}_Z = \mathcal{O}_X/\mathcal{I}$: $\mathcal{E}xt^0_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_X) = \mathcal{I}$ $\mathcal{E}xt^1_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_X) = \mathcal{O}_X/\mathcal{I} = \mathcal{O}_Z$

These come from the short exact sequence: $0 \to \mathcal{I} \to \mathcal{O}_X \to \mathcal{O}_Z \to 0$

Let $X$ be a smooth projective variety and $\mathcal{E}$ a locally free sheaf. For any coherent sheaf $\mathcal{F}$: $\mathcal{E}xt^q_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) = 0 \quad \text{for } q > 0$

The spectral sequence degenerates at $E_2$:

• $E_2^{p,0} = H^p(X, \mathcal{E}^\vee \otimes \mathcal{F})$
• $E_2^{p,q} = 0$ for $q > 0$

Therefore: $\text{Ext}^n_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) = H^n(X, \mathcal{E}^\vee \otimes \mathcal{F})$

This is a fundamental computation technique.

Let $X = \mathbb{P}^1_k$ and consider extensions: $0 \to \mathcal{O}(a) \to \mathcal{E} \to \mathcal{O}(b) \to 0$

Using the spectral sequence: $\text{Ext}^1_{\mathcal{O}_X}(\mathcal{O}(b), \mathcal{O}(a)) = H^1(\mathbb{P}^1, \mathcal{O}(a-b))$

For $a - b \geq -1$, we have $H^1(\mathbb{P}^1, \mathcal{O}(a-b)) = 0$, so all extensions split.

For $a - b \leq -2$, we have: $\dim \text{Ext}^1(\mathcal{O}(b), \mathcal{O}(a)) = b - a - 1$

These non-split extensions give rank 2 vector bundles on $\mathbb{P}^1$.

On $\mathbb{P}^n$, the tangent bundle $T_{\mathbb{P}^n}$ fits in the Euler sequence: $0 \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus(n+1)} \to T_{\mathbb{P}^n} \to 0$

This represents an element of: $\text{Ext}^1_{\mathcal{O}}(T_{\mathbb{P}^n}, \mathcal{O}_{\mathbb{P}^n}) = H^1(\mathbb{P}^n, T_{\mathbb{P}^n}^\vee)$

Using $T_{\mathbb{P}^n}^\vee \cong \Omega^1_{\mathbb{P}^n} \cong \mathcal{O}(-2)^{\oplus n}$ (from the cotangent sequence), this shows: $\text{Ext}^1(T_{\mathbb{P}^n}, \mathcal{O}) = H^1(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n})$ which vanishes for $n \geq 2$ by cohomology computations.

Let $C$ be a smooth projective curve of genus $g$ and let $p \in C$ be a closed point. Consider: $\text{Ext}^i_{\mathcal{O}_C}(k(p), \mathcal{O}_C)$ where $k(p) = \mathcal{O}_C/\mathfrak{m}_p$ is the skyscraper sheaf at $p$.

The local-to-global spectral sequence gives: $E_2^{p,q} = H^p(C, \mathcal{E}xt^q_{\mathcal{O}_C}(k(p), \mathcal{O}_C))$

Locally at $p$, we have $\mathcal{E}xt^1_{\mathcal{O}_C}(k(p), \mathcal{O}_C)_p = k(p)$ and the sheaf $\mathcal{E}xt^1$ is supported at $p$: $\mathcal{E}xt^1_{\mathcal{O}_C}(k(p), \mathcal{O}_C) = k(p)$

Therefore:

• $\text{Ext}^0_{\mathcal{O}_C}(k(p), \mathcal{O}_C) = 0$ (no global sections)
• $\text{Ext}^1_{\mathcal{O}_C}(k(p), \mathcal{O}_C) = H^0(C, k(p)) \oplus H^1(C, 0) = k$
• $\text{Ext}^2_{\mathcal{O}_C}(k(p), \mathcal{O}_C) = H^1(C, k(p)) = 0$

Let $C$ be a smooth projective curve and $\mathcal{I}_p$ the ideal sheaf of a point $p$. We have the exact sequence: $0 \to \mathcal{I}_p \to \mathcal{O}_C \to k(p) \to 0$

For any coherent sheaf $\mathcal{F}$, applying $\text{Hom}(-, \mathcal{F})$: $0 \to \text{Hom}(k(p), \mathcal{F}) \to \text{Hom}(\mathcal{O}_C, \mathcal{F}) \to \text{Hom}(\mathcal{I}_p, \mathcal{F})$ $\to \text{Ext}^1(k(p), \mathcal{F}) \to \text{Ext}^1(\mathcal{O}_C, \mathcal{F}) \to \text{Ext}^1(\mathcal{I}_p, \mathcal{F}) \to \cdots$

Since $\text{Hom}(k(p), \mathcal{F}) = \mathcal{F}_p/\mathfrak{m}_p \mathcal{F}_p$ and $\text{Hom}(\mathcal{O}_C, \mathcal{F}) = H^0(C, \mathcal{F})$, this sequence computes Ext groups with ideal sheaves.

Let $C$ be a smooth projective curve and $\mathcal{L}, \mathcal{M}$ line bundles on $C$. Then: $\text{Ext}^i_{\mathcal{O}_C}(\mathcal{L}, \mathcal{M}) = H^i(C, \mathcal{L}^\vee \otimes \mathcal{M})$

By Serre duality: $\text{Ext}^1_{\mathcal{O}_C}(\mathcal{L}, \mathcal{M}) = H^1(C, \mathcal{L}^\vee \otimes \mathcal{M}) \cong H^0(C, \mathcal{L} \otimes \mathcal{M}^\vee \otimes \omega_C)^\vee$

In particular, for the structure sheaf: $\text{Ext}^1_{\mathcal{O}_C}(\mathcal{O}_C, \mathcal{O}_C) = H^1(C, \mathcal{O}_C) = k^g$ where $g$ is the genus of $C$.

Let $X = \mathbb{P}^n$ over a field $k$. For integers $a, b$: $\text{Ext}^i_{\mathcal{O}_X}(\mathcal{O}(a), \mathcal{O}(b)) = H^i(\mathbb{P}^n, \mathcal{O}(b-a))$

Using the cohomology of $\mathcal{O}(m)$ on $\mathbb{P}^n$:

• $H^0(\mathbb{P}^n, \mathcal{O}(m)) = k^{\binom{m+n}{n}}$ for $m \geq 0$
• $H^n(\mathbb{P}^n, \mathcal{O}(m)) = k^{\binom{-m-1}{n}}$ for $m \leq -n-1$
• $H^i(\mathbb{P}^n, \mathcal{O}(m)) = 0$ otherwise

Therefore:

• $\text{Ext}^0(\mathcal{O}(a), \mathcal{O}(b)) = k^{\binom{b-a+n}{n}}$ for $b \geq a$
• $\text{Ext}^n(\mathcal{O}(a), \mathcal{O}(b)) = k^{\binom{a-b-1}{n}}$ for $b \leq a - n - 1$
• $\text{Ext}^i(\mathcal{O}(a), \mathcal{O}(b)) = 0$ otherwise

Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^n$ with a locally free resolution: $0 \to \mathcal{E}_r \to \cdots \to \mathcal{E}_1 \to \mathcal{E}_0 \to \mathcal{F} \to 0$

For any coherent sheaf $\mathcal{G}$: $\text{Ext}^i(\mathcal{F}, \mathcal{G}) = H^i(\text{Hom}^\bullet(\mathcal{E}_\bullet, \mathcal{G}))$

Since each $\mathcal{E}_j$ is locally free: $\text{Hom}(\mathcal{E}_j, \mathcal{G}) = \mathcal{E}_j^\vee \otimes \mathcal{G}$

This reduces Ext computations to cohomology of vector bundles.

On $\mathbb{P}^2$, consider a twisted cubic curve $C \subset \mathbb{P}^3$ embedded by $\mathcal{O}_{\mathbb{P}^1}(3)$. The structure sheaf $\mathcal{O}_C$ has a resolution: $0 \to \mathcal{O}_{\mathbb{P}^3}(-3)^{\oplus 2} \to \mathcal{O}_{\mathbb{P}^3}(-2)^{\oplus 3} \to \mathcal{O}_{\mathbb{P}^3} \to \mathcal{O}_C \to 0$

For $\mathcal{G} = \mathcal{O}_{\mathbb{P}^3}(n)$: $\text{Ext}^1(\mathcal{O}_C, \mathcal{O}(n)) = H^1(\text{Hom}^\bullet(\mathcal{E}_\bullet, \mathcal{O}(n)))$

The complex $\text{Hom}^\bullet(\mathcal{E}_\bullet, \mathcal{O}(n))$ is: $0 \to \mathcal{O}(n) \to \mathcal{O}(n+2)^{\oplus 3} \to \mathcal{O}(n+3)^{\oplus 2} \to 0$

Computing cohomology determines the Ext groups.

Let $C$ be a smooth projective curve of genus $g$ with canonical bundle $\omega_C$. For line bundles $\mathcal{L}, \mathcal{M}$: $\text{Ext}^0(\mathcal{L}, \mathcal{M})^\vee = \text{Hom}(\mathcal{L}, \mathcal{M})^\vee \cong \text{Ext}^1(\mathcal{M}, \mathcal{L} \otimes \omega_C)$

Explicitly: $H^0(C, \mathcal{L}^\vee \otimes \mathcal{M})^\vee \cong H^1(C, \mathcal{M}^\vee \otimes \mathcal{L} \otimes \omega_C)$

This is the classical Serre duality for curves, expressed in Ext language.

On a smooth projective surface $S$ with canonical bundle $\omega_S$, consider a coherent sheaf $\mathcal{F}$. Then: $\text{Ext}^i(\mathcal{F}, \mathcal{F})^\vee \cong \text{Ext}^{2-i}(\mathcal{F}, \mathcal{F} \otimes \omega_S)$

For $i = 1$: $\text{Ext}^1(\mathcal{F}, \mathcal{F})^\vee \cong \text{Ext}^1(\mathcal{F}, \mathcal{F} \otimes \omega_S)$

This duality is important in deformation theory, where $\text{Ext}^1(\mathcal{F}, \mathcal{F})$ parametrizes first-order deformations of $\mathcal{F}$.

Let $C$ be a smooth curve (regular of dimension 1). For any coherent sheaves $\mathcal{F}, \mathcal{G}$: $\mathcal{E}xt^i(\mathcal{F}, \mathcal{G}) = 0 \quad \text{for } i \geq 2$

This implies the local-to-global spectral sequence has only two rows: $E_2^{p,q} = H^p(C, \mathcal{E}xt^q(\mathcal{F}, \mathcal{G})) \quad \text{with } q \in \{0, 1\}$

The spectral sequence gives exact sequences: $0 \to H^1(C, \mathcal{H}om(\mathcal{F}, \mathcal{G})) \to \text{Ext}^1(\mathcal{F}, \mathcal{G}) \to H^0(C, \mathcal{E}xt^1(\mathcal{F}, \mathcal{G})) \to 0$

Let $X$ be a smooth projective variety of dimension $n$. For a locally free sheaf $\mathcal{E}$ and coherent sheaf $\mathcal{F}$: $\text{Ext}^i(\mathcal{E}, \mathcal{F}) = H^i(X, \mathcal{E}^\vee \otimes \mathcal{F})$

By standard vanishing (e.g., for ample line bundles $\mathcal{L}$ and large $m$): $H^i(X, \mathcal{E}^\vee \otimes \mathcal{F} \otimes \mathcal{L}^m) = 0 \quad \text{for } i > 0, m \gg 0$

This gives: $\text{Ext}^i(\mathcal{E} \otimes \mathcal{L}^{-m}, \mathcal{F}) = 0 \quad \text{for } i > 0, m \gg 0$

Given a short exact sequence of coherent sheaves: $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$

For any coherent sheaf $\mathcal{G}$, we obtain long exact sequences: $0 \to \text{Hom}(\mathcal{F}'', \mathcal{G}) \to \text{Hom}(\mathcal{F}, \mathcal{G}) \to \text{Hom}(\mathcal{F}', \mathcal{G})$ $\to \text{Ext}^1(\mathcal{F}'', \mathcal{G}) \to \text{Ext}^1(\mathcal{F}, \mathcal{G}) \to \text{Ext}^1(\mathcal{F}', \mathcal{G}) \to \cdots$

And in the other variable: $0 \to \text{Hom}(\mathcal{G}, \mathcal{F}') \to \text{Hom}(\mathcal{G}, \mathcal{F}) \to \text{Hom}(\mathcal{G}, \mathcal{F}'')$ $\to \text{Ext}^1(\mathcal{G}, \mathcal{F}') \to \text{Ext}^1(\mathcal{G}, \mathcal{F}) \to \text{Ext}^1(\mathcal{G}, \mathcal{F}'') \to \cdots$

These sequences are essential for inductive computations.