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I have started learning homological algebra recently. It looks like the most abstract subject I've seen so far. The most concrete one is without doubts is combinatorics. So I have very specific reference request -

Can you provide a reference to a book/research paper/whatever that illustrates how abstract homological tools allow to compute something concrete/finite/nontrivial or at least prove it exists.

Thanks a lot for your time!

Hedgehog
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    How about Richard Stanley's Combinatorics and Commutative Algebra? – Angina Seng Oct 24 '17 at 20:19
  • Interesting, let me check. – Hedgehog Oct 24 '17 at 20:21
  • You can use sheaf cohomology of algebraic varieties (which I think qualifies as homological algebra) to prove the existence of the Frobenius number for subsemigroups $S$ of $\mathbf{N}$; see one of my answers. By being more careful, you can even bound the Frobenius number in this way; this is a theorem due to L'vovsky. Of course, there are other bounds available, but L'vovsky's holds for arbitrary $S$. – Takumi Murayama Oct 24 '17 at 22:44
  • Takumi Murayama, thanks for the comment. The world is so small, I saw L'vovsky last week. I will ask him about this for sure :) – Hedgehog Oct 24 '17 at 22:46

2 Answers2

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This isn't exactly an application of bare homological algebra, but it does involve cohomology: the hard Lefschetz theorem in algebraic geometry implies that the sequences of even and odd Betti numbers of a smooth projective variety over $\mathbb{C}$ are both unimodal, meaning that they first increase and then decrease. A simple example is a product of $n$ copies of the complex projective line $\mathbb{CP}^1$; here the even Betti numbers are binomial coefficients ${n \choose k}$.

A more interesting example is the Grassmannian $\text{Gr}_d(\mathbb{C}^n)$, whose even Betti numbers count the number of partitions fitting into a $d \times (n-d)$ box. No purely combinatorial proof that this sequence is unimodal is known (edit: it seems my information is out of date! See this survey by Zeilberger of the result, which is due to O'Hara). See this survey by Stanley for more.

Qiaochu Yuan
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In this paper Power series representing Posets the problem of counting in how many ways can you label a Poset (while preserving the order) is solved for a family of Posets called Wixárika Posets (they look like Wixárika collars). The main results are proven using homological algebra ideas. As a consequence several combinatorial identities were discovered (we checked on books of identities and we coulnd't find them).

Ѕᴀᴀᴅ
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Eric
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