Preprojective partitions and the determinant of the Hom matrix

Igusa, K.; Todorov, G.

doi:10.1090/S0002-9939-1985-0784160-0

Preprojective partitions and the determinant of the Hom matrix
HTML articles powered by AMS MathViewer

by K. Igusa and G. Todorov

Proc. Amer. Math. Soc. 94 (1985), 189-197

DOI: https://doi.org/10.1090/S0002-9939-1985-0784160-0

PDF | Request permission

Abstract:

If $\Lambda$ is an artin algebra and $\Lambda$ is the set of isomorphism classes of indecomposable finitely generated $\Lambda$-modules, then there is a partition $\operatorname {ind}\Lambda = { \cup _{i \geqslant 0}}{\underline {\underline P} _i}$, called the preprojective partition. We give an algorithm for computing this partition, which is given only in terms of numerical properties of the Auslander-Reiten quiver of $\Lambda$. If $\Lambda$ is of finite representation type, then there are two essentially different proofs that the matrix $\operatorname {Hom} = ({\text {lengt}}{{\text {h}}_{{\text {End(}}N)/{\text {rEnd(}}N{\text {)}}}}\operatorname {Hom}_{\Lambda }(M,N))$, where $M,n \in \operatorname {ind}\Lambda$ has determinant $+ 1[{\mathbf {IT1}},{\mathbf {W1}},{\mathbf {Z1}}]$. We show that the paths between the Hom matrix and the identity matrix in $\operatorname {GL}_{n}(\mathbf {R})$ given by these two proofs are homotopic.

References

Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. VI. A functorial approach to almost split sequences, Comm. Algebra 6 (1978), no. 3, 257–300. MR 472919, DOI 10.1080/00927877808822246
M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
P. Donovan and M. R. Freislich, Representable functions on the category of modular representations of a finite group with cyclic Sylow subgroup, J. Algebra 32 (1974), 356–364. MR 360786, DOI 10.1016/0021-8693(74)90144-6
Kiyoshi Igusa and Gordana Todorov, Radical layers of representable functors, J. Algebra 89 (1984), no. 1, 105–147. MR 748231, DOI 10.1016/0021-8693(84)90238-2
Kiyoshi Igusa and Gordana Todorov, Radical layers of representable functors, J. Algebra 89 (1984), no. 1, 105–147. MR 748231, DOI 10.1016/0021-8693(84)90238-2
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
Christine Riedtmann, Preprojective partitions for self-injective algebras, J. Algebra 76 (1982), no. 2, 532–539. MR 661871, DOI 10.1016/0021-8693(82)90230-7
Brit Rohnes, Preprojective partitions for trivial extensions of hereditary algebras, Comm. Algebra 11 (1983), no. 9, 949–972. MR 696480, DOI 10.1080/00927878308822888
Gordana Todorov, A note on preprojective partitions over hereditary Artin algebras, Proc. Amer. Math. Soc. 85 (1982), no. 4, 523–528. MR 660596, DOI 10.1090/S0002-9939-1982-0660596-0
George V. Wilson, The Cartan map on categories of graded modules, J. Algebra 85 (1983), no. 2, 390–398. MR 725091, DOI 10.1016/0021-8693(83)90103-5

The preprojective partition and Poincare-Betti series for finite dimensional algebras

G. Wilson and D. Zacharia, Morphisms between preprojective modules over hereditary Artin algebras, Comm. Algebra 11 (1983), no. 1, 95–107. MR 687407, DOI 10.1080/00927878308822838
Dan Zacharia, On the Cartan matrix of an Artin algebra of global dimension two, J. Algebra 82 (1983), no. 2, 353–357. MR 704760, DOI 10.1016/0021-8693(83)90156-4
D. Zacharia, The preprojective partition for hereditary Artin algebras, Trans. Amer. Math. Soc. 274 (1982), no. 1, 327–343. MR 670936, DOI 10.1090/S0002-9947-1982-0670936-9