Elementary problems that could be solved using homological algebra

My favorite example is an application of homological methods in algebraic geometry to a question almost anyone can understand:

Question. Given a sequence $A = (a_1 < a_2 < \cdots < a_n)$ of $n$ positive integers such that $\gcd(a_1,a_2,\ldots,a_n) = 1$, what is the largest natural number $g(a_1,\ldots,a_n)$ not expressible as a non-negative integer combination of $a_1,a_2,\ldots,a_n$?

This question has been popularized as the "coin problem," the "stamp problem," and even the "Chicken McNugget problem."

Definition. Given a sequence $A$ as above, we call the number $g(a_1,\ldots,a_n)$ the Frobenius number for the sequence $A$.

Note that the existence of this number is a consequence of Schur's theorem, although we will also prove this below.

Sylvester proved that if $n = 2$, we have the explicit formula $$g(a_1,a_2) = a_1 a_2 − a_1 − a_2.$$ You can see a proof here on Math.SE. On the other hand, Curtis showed that for $n > 2$ it is (in some sense) impossible to find an explicit formula, so instead, we can ask

Question. Are there bounds for $g(a_1,\ldots,a_n)$?

Most bounds, though, rely on some hypotheses on the numbers in $A$, such as symmetry or conditions on pairwise coprimeness, etc. (see the various results in Ramírez). The following bound has the advantage of working in all settings, and it seems to be one of the best upper bounds that people have shown:

Theorem (L'vovsky). Let $a_1 < a_2 < \cdots < a_n$ be an increasing sequence of positive integers such that $\gcd(a_1,a_2,\ldots,a_n)$. Put $a_0 = 0$. Then, letting $$\delta = \max_{1 \le i < j \le n}\!\left\{(a_i − a_{i−1}) + (a_j − a_{j−1})\right\},$$ we have $$g(a_1,\ldots,a_n) \le (\delta - 2)a_n.$$

What is novel about this bound is that the proof uses sheaf cohomology, and that there is no known combinatorial proof of this fact. The point is to translate this question about natural numbers into a question about vanishing of sheaf cohomology on monomial curves, and then apply general machinery about when these cohomology groups vanish due to Gruson–Lazarsfeld–Peskine (this is where the heavy homological machinery comes into play). More precisely, L'vovsky's bound for the Frobenius number is a consequence of the bound $\operatorname{reg} X \le \delta$ for the Castelnuovo–Mumford regularity of the monomial curve $$\begin{aligned} \mathbf{P}^1 &\overset{p}{\longrightarrow} \mathbf{P}^n\\ [s:t] &\longmapsto [1:t^{a_1}:t^{a_2}:\cdots:t^{a_n}] \end{aligned}$$

I was going to also post a proof of this Theorem, but it ended up being really long. I can elaborate on the proof if anyone is interested! In the mean time, you can read Ch. 4 in Eisenbud's The Geometry of Syzygies.

To illustrate this idea, we give a proof of the existence of the Frobenius number using sheaf cohomology, which should be accessible to anyone who has read Hartshorne (I must admit that this is a bit more than just homological algebra!).

Proof that $g(a_1,\ldots,a_n) < \infty$. Let $k = \overline{k}$. Consider the map $p\colon \mathbf{P}^1 \to \mathbf{P}^n$ defined above, with image $X$. Then, $p\colon \mathbf{P}^1 \to X$ is the normalization of $X$, and so we have the short exact sequence $$0 \longrightarrow \mathcal{O}_X \longrightarrow p_*\mathcal{O}_{\mathbf{P}^1} \longrightarrow \mathscr{Q} \longrightarrow 0$$ where $\mathscr{Q}$ is supported on $\{p(0),p(\infty)\}$ (away from these points, $p\colon \mathbf{P}^1 \to X$ is an isomorphism by using that $\gcd(a_1,\ldots,a_n) = 1$) [Hartshorne, Exc. IV.1.8]. Twisting by $\mathcal{O}_{\mathbf{P}^n}(r)$ gives the short exact sequence $$0 \longrightarrow \mathcal{O}_X(r) \longrightarrow p_*(\mathcal{O}_{\mathbf{P}^1}(ra_n)) \longrightarrow \mathscr{Q} \longrightarrow 0$$ after using the projection formula [Hartshorne, Exc. II.5.1(d)]. We can then compute (part of) the long exact sequence on cohomology [Hartshorne, III.1]: $$\require{AMScd}\begin{CD} H^0(\mathbf{P}^1,\mathcal{O}_{\mathbf{P}^1}(ra_n)) @>>> H^0(X,\mathscr{Q}) @>>> H^1(X,\mathcal{O}_X(r))\\ @| @| @|\\ k\langle t^i \mid i \le ra_n\rangle @>>> k\langle t^i \mid i \in \mathbf{N} \setminus S \rangle \oplus \cdots @>>> H^1(X,\mathcal{O}_X(r)) \end{CD}$$ where the bottom row consists of vector spaces spanned by the listed monomials, and $S$ is the subsemigroup of $\mathbf{N}$ generated by $a_1,\ldots,a_n$. The computation of $H^0(\mathbf{P}^1,\mathcal{O}_{\mathbf{P}^1}(ra_n))$ is [Hartshorne, Thm. III.5.1], and you can compute the global sections of $H^0(X,\mathscr{Q})$ by looking at stalks and computing in local coordinates.

The final ingredient is Serre vanishing [Hartshorne, Thm. III.5.2(b)], which states that there exists an integer $r_0$ such that for all $r \ge r_0$, the cohomology group $H^1(X,\mathcal{O}_X(r))$ vanishes. This implies that $$k\langle t^i \mid i \le ra_n\rangle \longrightarrow k\langle t^i \mid i \in \mathbf{N} \setminus S \rangle$$ is surjective for all $r \ge r_0$, and so the largest number in $\mathbf{N} \setminus S$ is at most $r_0 a_n$, that is, $g(a_1,\ldots,a_n) \le r_0a_n < \infty$. $\blacksquare$

The issue with Serre vanishing is that it gives no effective means of figuring out what this number $r_0$ is. This is why Castelnuovo–Mumford regularity is useful; see this question on MathOverflow, for example. The need for an explicit bound explains why the homological machinery of Gruson–Lazarsfeld–Peskine appears in this context.