# Algebras derived equivalent to quasi-hereditary algebras

Let an algebra always be finite dimensional over a field and connected. It is well known that a quasi-hereditary algebra with $$n$$ simple modules has global dimension at most $$2n-2$$.

Questions: 1. Do algebras derived equivalent to quasi-hereditary algebras with $$n$$ simple modules also have global dimension at most $$2n-2$$?

1. Is there a good discrete test (for example with a computer) for checking whether an algebra of finite global dimension is derived equivalent to a quasi-hereditary algebra?

2. Is every Nakayama algebra of finite global dimension derived equivalent to a quasi-hereditary algebra?

For example the Nakayama algebras with $$n$$ simples and Kupisch series $$[n,n+1,...,n+1]$$ are not quasi-hereditary for $$n \geq 3$$, but they are derived equivalent to the representation-finite block of a Schur algebra, which is quasi-hereditary. Here are some Kupisch series of Nakayama algebras of finite global dimension that are not quasi-hereditary, but Im not sure whether they are derived equivalent to a quasi-hereditary algebra: [ [ 3, 3, 3, 4 ], [ 3, 5, 5, 4 ], [ 4, 5, 6, 5 ], [ 4, 6, 5, 5 ], [ 4, 6, 6, 5 ] ]

It would be interesting to see how to deal with this problem even in small examples.

The answer to question 1 is "no": there are algebras with two simple modules, derived equivalent to quasi-hereditary algebras, but with global dimension three.

This can probably be extracted from one or both of the papers

Dubnov, Dmitry, On derived categories of modules over 2-vertex basic algebras, Commun. Algebra 28, No. 9, 4355-4374 (2000). ZBL0983.16009.

Volkov, Y. On the derived category of quasi-hereditary algebras with two simple modules, arXiv:1812.00351,

but I'll give an explicit example.

Let $$A$$ be the algebra given by a quiver with two vertices $$1$$ and $$2$$, one arrow $$\alpha$$ from vertex $$1$$ to vertex $$2$$, two arrows $$\beta_1$$, $$\beta_2$$ from vertex $$2$$ to vertex $$1$$, and relations $$\beta_1\alpha=0=\beta_2\alpha$$.

Denoting by $$S(i)$$ and $$P(i)$$ the simple and indecomposable projective modules corresponding to vertex $$i$$, $$P(2)$$ is $$3$$-dimensional with head $$S(2)$$ and radical $$S(1)\oplus S(1)$$, and $$P(1)$$ is $$4$$-dimensional with head $$S(1)$$ and radical isomorphic to $$P(2)$$. So $$A$$ is quasihereditary with standard modules $$\Delta(1)=S(1)$$ and $$\Delta(2)=P(2)$$.

There is a tilting complex $$T:=\dots\to0\to P(1)^3\to P(2)\to0\to\dots,$$ with $$P(2)$$ in degree zero, where one copy of $$P(1)$$ is in the kernel of the differential, and the other two copies map onto $$\text{rad }P(2)$$.

Let $$B=\text{End}_{D^b(A)}(T)$$, so there is an equivalence of triangulated categories $$F:D^b(A)\to D^b(B)$$ sending $$T$$ to $$B$$.

Since $$\text{Hom}_{D^b(A)}(T,S(2)[i])=0$$ for $$i\neq0$$, $$FS(2)$$ has homology only in degree zero, and so is a $$B$$-module $$M$$.

If $$X$$ is the quotient of $$P(2)$$ by a $$1$$-dimensional submodule of its socle, then $$\text{Hom}_{D^b(A)}(T,X[i])=0$$ for $$i\neq1$$, and so $$FX[1]$$ is a $$B$$-module $$N$$.

Now $$\text{Ext}^3_B(N,M)\cong\text{Hom}_{D^b(B)}(N,M[3]) \cong\text{Hom}_{D^b(A)}(X,S(2)[2])\cong\text{Ext}^2_A(X,S(2)),$$ which is easily calculated to be non-zero.

Hence $$B$$ has global dimension at least three (in fact, it's equal to three).

• Thanks, Ill take a look at the articles. The next natural question might be whether there is a bound for the global dimension for algebras derived equivalent to quasi-hereditary algebras with $n$ simple modules. – Mare Jan 14 '19 at 13:03