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• The Yoneda Lemma (used everywhere!)
• In algebraic geometry, the vanishing locus of a map of vector bundles on a scheme $X$ is a closed subscheme of $X$.

I'll give an example where we use both. Say we want to show that there is such a thing as a scheme of "flags" on a vector space $V$, and that it is a closed subscheme of projective space. Such a flag scheme $\text{Fl}_{i_1, ..., i_k}$ represents the functor sending a scheme $S$ to the set of $M_1\hookrightarrow ...\hookrightarrow M_k\hookrightarrow V\otimes \mathcal{O}_S$ where $M_j$ is a vector bundle on $S$ of rank $i_j$ and where all the maps are injective bundle maps. We could do this by working with coordinates, but that would be a huge pain, so let's use our lemmas.

It suffices to do this for $k=2$, as larger flag schemes can be written as fiber products of $\text{Fl}_{i_1, i_2}$. We embed $\text{Fl}_{i_1, i_2}$ as a closed subscheme of the scheme $T:=\text{Gr}(i_1, V)\times \text{Gr}(i_2, V)$. We know that this is a closed subscheme of $\mathbb{P}(\Lambda^{i_1}(V)\otimes \Lambda^{i_2}(V))$ so this suffices.

But indeed, letting $M_j$ be the canonical rank $i_j$ bundle over $\text{Gr}(i_j, V)$ (induced by the identity map through the universal property of the Grassmannian and Yoneda), we have that the flag bundle $\text{Fl}_{i_1, i_2}$ is the vanishing locus of the map $M_1\to p_1^*M_1\to \text{coker}(M_2\to text{coker}(p_2^*M_2\to V\otimes \mathcal{O}_T)$. It's easy to check that this represents the desired functor.

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• The Yoneda Lemma (used everywhere!)
• In algebraic geometry, the vanishing locus of a map of vector bundles on a scheme $X$ is a closed subscheme of $X$.

I'll give an example where we use both. Say we want to show that there is such a thing as a scheme of "flags" on a vector space $V$, and that it is a closed , subscheme of projective schemespace. Such a flag scheme $\text{Fl}_{i_1, ..., i_k}$ represents the functor sending a scheme $S$ to the set of $M_1\hookrightarrow ...\hookrightarrow M_k\hookrightarrow V\otimes \mathcal{O}_S$ where $M_j$ is a vector bundle on $S$ of rank $i_j$ and where all the maps are injective bundle maps. We could do this by working with coordinates, but that would be a huge pain, so let's use our lemmas.

It suffices to do this for $k=2$, as larger flag schemes can be written as fiber products of $\text{Fl}_{i_1, i_2}$. We embed $\text{Fl}_{i_1, i_2}$ as a closed subscheme of the scheme $T:=\text{Gr}(i_1, V)\times \text{Gr}(i_2, V)$. We know that this is a closed subscheme of $\mathbb{P}(\Lambda^{i_1}(V)\otimes \Lambda^{i_2}(V))$ so this suffices.

But indeed, letting $M_j$ be the canonical rank $i_j$ bundle over $\text{Gr}(i_j, V)$ (induced by the identity map through the universal property of the Grassmannian and Yoneda), we have that the flag bundle $\text{Fl}_{i_1, i_2}$ is the vanishing locus of the map $M_1\to \text{coker}(M_2\to V\otimes \mathcal{O}_T)$. It's easy to check that this represents the desired functor.

• In algebraic geometry, the vanishing locus of a map of vector bundles on a scheme $X$ is a closed subscheme of $X$.
I'll give an example where we use both. Say we want to show that there is such a thing as a scheme of "flags" on a vector space $V$, and that it is a closed, projective scheme. Such a flag scheme $\text{Fl}_{i_1, ..., i_k}$ represents the functor sending a scheme $S$ to the set of $M_1\hookrightarrow ...\hookrightarrow M_k\hookrightarrow V\otimes \mathcal{O}_S$ where $M_j$ is a vector bundle on $S$ of rank $i_j$ and where all the maps are injective bundle maps. We could do this by working with coordinates, but that would be a huge pain, so let's use our lemmas.
It suffices to do this for $k=2$, as larger flag schemes can be written as fiber products of $\text{Fl}_{i_1, i_2}$. We embed $\text{Fl}_{i_1, i_2}$ as a closed subscheme of the scheme $T:=\text{Gr}(i_1, V)\times \text{Gr}(i_2, V)$. We know that this is a closed subscheme of $\mathbb{P}(\Lambda^{i_1}(V)\otimes \Lambda^{i_2}(V))$ so this suffices.
But indeed, letting $M_j$ be the canonical rank $i_j$ bundle over $\text{Gr}(i_j, V)$ (induced by the identity map through the universal property of the Grassmannian and Yoneda), we have that the flag bundle $\text{Fl}_{i_1, i_2}$ is the vanishing locus of the map $M_1\to \text{coker}(M_2\to V\otimes \mathcal{O}_T)$. It's easy to check that this represents the desired functor.