- ALGEBRAIC COMBINATORICS
- Math-Conf - Abstracts
- Internally Calabi–Yau algebras and cluster-tilting objects

First Online: 03 January The structure of the paper is as follows. This section introduces our main definitions. Definition 2. Remark 2. Lemma 2. Theorem 2. Proof Pick a triangle Open image in new window. Corollary 2. Example 2. Definition 3. Theorem 3. Remark 3.

## ALGEBRAIC COMBINATORICS

Proposition 3. Proof As already noted, any perfect A -module is finitely presented. For the converse, let M be a finitely presented A -module with presentation Open image in new window. By [ 25 , Prop. Combining these to form the exact sequence Open image in new window. It follows from the above calculations that the sequence Open image in new window. In this case, the left and right injective dimensions coincide, and are called the Gorenstein dimension of B. For brevity, an Iwanaga—Gorenstein algebra with Gorenstein dimension d will be called d - Iwanaga—Gorenstein.

Corollary 3. Example 3. One description of B is as follows. Theorem 4. Remark 4. Proposition 4.

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Proof Both A and Ae are finitely generated projective A -modules, and so are in particular perfect. Open image in new window. From 4. Lemma 4. Proof Pick an exact sequence Open image in new window. By [ 30 , Prop. Definition 5. A potential on Q is a linear combination W of cycles of Q. A vertex or arrow of Q is called frozen if it is a vertex or arrow of F , and mutable or unfrozen otherwise.

Remark 5.

## Math-Conf - Abstracts

Example 5. Lemma 5. If the map Open image in new window.

Theorem 5. Now consider the commutative diagram Open image in new window. The diagram 5. However, we also have many more examples, some of which are infinite dimensional. For example, let Open image in new window.

Proposition 5. We have Open image in new window. Acknowledgements Open access funding provided by the Max Planck Society. Amiot, C. Fourier Grenoble 59 6 , — Auslander, M. If the map Open image in new window. Theorem 5. Now consider the commutative diagram Open image in new window. The diagram 5. However, we also have many more examples, some of which are infinite dimensional. For example, let Open image in new window. Proposition 5. We have Open image in new window. Acknowledgements Open access funding provided by the Max Planck Society.

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Amiot, C. Fourier Grenoble 59 6 , — Auslander, M. Baur, K. Bocklandt, R. Pure Appl.

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Butler, M. Caldero, P. Demonet, L. Algebra Number Theory 10 7 , — Dwyer, W. Non-commutative localization in algebra and topology, pp. Franco, S. High Energy Phys. Fourier Grenoble 58 3 , — Ginzburg, V. Goncharov, A. Guo, L. Algebra 9 , — Happel, D. Herzog, J. Iyama, O. Jensen, B. Kalck, M.

Keller, B. In: Trends in representation theory of algebras and related topics, pp. In: Triangulated categories. Leuschke, G. Palu, Y. Fourier Grenoble 58 6 , — Postnikov, A.

## Internally Calabi–Yau algebras and cluster-tilting objects

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