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The Grothendieck group of an \(n\)-angulated category. (English) Zbl 1291.18015

An \(n\)-angulated category is a “higher-dimensional” analogue of a triangulated category. Where the latter is an additive category with a class of so-called triangles which are in some sense the equivalent of short exact sequences in an abelian category, in the former we are given sequences of length \(n\) which have a similar purpose. In fact, a \(3\)-angulated category is precisely a triangulated category. In the paper under review the authors define and study the Grothendieck group of an \(n\)-angulated category. Roughly speaking, one takes, similar to the triangulated case, the free abelian group on the set of isomorphism clases of objects modulo the Euler relations corresponding to the \(n\)-angles.
In Section 2 the authors recall the definition of an \(n\)-angulated category, define its Grothendieck group and establish some of its basic properties. In the following section the following example is studied: If \(\mathcal{T}\) is a triangulated category and \(\mathcal{C}\) a so-called \((n-2)\)-cluster tilting subcategory satisfying an additional technical condition, then \(\mathcal{C}\) can be made into an \(n\)-angulated category. It is then shown that its Grothendieck group \(K_0(\mathcal{C})\) surjects onto the Grothendieck group of \(\mathcal{T}\).
In Section 4 the main result of the paper is proved. Namely, if \(\mathcal{C}\) is an \(n\)-angulated category and \(n\) is odd, there is a one-to-one correspondence between subgroups of \(K_0(\mathcal{C})\) and complete and dense \(n\)-angulated subcategories of \(\mathcal{C}\). Here, a subcategory \(\mathcal{A}\) is dense if every object in \(\mathcal{C}\) is a direct summand of an object of \(\mathcal{C}\) and it is complete if the following holds: in an \(n\)-angle in \(\mathcal{C}\) in which \(n-1\) vertices are in \(\mathcal{A}\), the last vertex is also in \(\mathcal{A}\). This result generalizes the corresponding statement for a triangulated category proved by Thomason.
In the final section the authors define tensor \(n\)-angulated categories, which are \(n\)-angulated categories admitting a symmetric monoidal product compatible with the \(n\)-angulated structure. They show that in this case \(K_0(\mathcal{C})\) becomes a commutative ring and there is a one-to-one correspondence between (prime) ideals of \(K_0(\mathcal{C})\) and complete dense \(n\)-angulated (prime) tensor ideals of \(\mathcal{C}\).

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
18F30 Grothendieck groups (category-theoretic aspects)
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References:

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