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An extension group is a value of an Ext functor. In degree one it classifies short exact sequences $0\to N\to E\to M\to 0$ up to isomorphisms fixing the endpoints. The zero class is the split extension; its addition is the Baer sum. Higher degrees can be represented by longer exact extensions, or computed from a projective resolution.

Applying the contravariant Hom functor ${\mathrm{Hom}}_R(-,W)$ to $0\to I\to X\to C\to 0$ givesThe connecting map sends $I\to W$ to its pushout of a module extension. Its vanishing means that map extends to $X\to W$. For a hereditary ring, ${\mathrm{Ext}}^2(C,W)=0$, making the indicated restriction on first extension groups surjective. The covariant variable has its corresponding long exact sequence.

The identification with ${\mathbf{P}}^1$ means that $C$ is a smooth plane conic. An arbitrary singular or nonreduced conic over an algebraically closed field cannot be identified with the projective line, so this is a necessary qualification implicit in the requested description.

Choose an equation $\ell =0$ for $L\subset {\mathbf{P}}^3$ and a degree-two equation $q=0$ whose restriction to $L$ defines $C$. Its homogeneous ideal in ${\mathbf{P}}^3$ is $(\ell ,q)$. These equations form a regular sequence: $\ell$ is a non-zero-divisor in the polynomial ring, and modulo $\ell$ the ring is a polynomial ring in three variables, in which the nonzero quadratic $q$ is again a non-zero-divisor. The generator classes therefore form a basis of the conormal sheaf, with their grading shifts:One can verify the absence of relations from the two-generator Koszul complex: a relation between $\ell ,q$ is a multiple of $(q,-\ell )$, whose coefficients become zero modulo ${\mathcal{I}}_C$. This is the normal sheaf of a projective complete intersection calculation. Dualizing the conormal sheaf gives

These twists are restrictions from ${\mathbf{P}}^3$, not intrinsic degree-one twists on ${\mathbf{P}}^1$. A line in the plane cuts the degree-two curve in two points counted with multiplicity, so the degree of a line bundle ${\mathcal{O}}_C(1)$ is two. The Picard group of the projective line then givesEquivalently, the degree-two Veronese embedding $[s:t]\mapsto [s^2:st:t^2]$ pulls each linear coordinate back to a quadratic. Thus the normal bundle of a smooth plane conic isThe same splitting follows from the normal sequence for the two embeddings:Its outer terms are ${\mathcal{O}}_{{\mathbf{P}}^1}(4)$ and ${\mathcal{O}}_{{\mathbf{P}}^1}(2)$. The extension group class lies in ${\mathrm{Ext}}^1(\mathcal{O}(2),\mathcal{O}(4))=H^1({\mathbf{P}}^1,\mathcal{O}(2))=0$, by the preceding Čech cohomology calculation, so the sequence splits. Both methods distinguish the ambient twist from the intrinsic degree on the curve.

The extension group ${\mathrm{Ext}}_Q^1(V,W)$ consists of equivalence classes of short exact sequences $0\to W\to E\to V\to 0$, with the zero class represented by a split sequence and addition given by the Baer sum. The extension complex of quiver representations givesThe printed map has $\gamma (u{)}_{\rho }=u_j{f}_{\rho }-g_{\rho }{u}_i$, so its kernel is ${\mathrm{Hom}}_Q(V,W)$ and its cokernel is ${\mathrm{Ext}}_Q^1(V,W)$. Reversing the overall differential sign changes neither identification.

For dimension vectors $\mathbf{v},\mathbf{w}$, the Ringel form isThe first expression makes its dependence only on the dimension vectors explicit.

For the one-loop representation $V=k$ with loop scalar $\lambda$, $\gamma (u)=u\lambda -\lambda u=0$ on $k$. Both cochain spaces have dimension one, so $\dim {\mathrm{Ext}}_Q^1(V,V)=1$, for every $\lambda$. Concretely, a self-extension has loop matrix $\left(\begin{array}{cc}\lambda & c\\ 0& \lambda \end{array}\right)$, with $c$ the extension parameter.

The standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus every path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to $0\to I\to K\to C\to 0$ gives a surjection ${\mathrm{Ext}}^1(K,W)\twoheadrightarrow {\mathrm{Ext}}^1(I,W)$, since the next term ${\mathrm{Ext}}^2(C,W)$ is zero. This is the hereditary step in the Ringel lemma on bricks.