# Tachikawa conjecture for finite dimensional commutative monomial algebras

Let $$A=K[x_1,...,x_n]/I$$ be a finite dimensional local algebra with a monomial ideal $$I$$.

The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such monomial algebras.

The first Tachikawa conjecture states that $$Ext_A^i(D(A),A) \neq 0$$ in case $$A$$ is not Gorenstein for some $$i>0$$.

The second Tachikawa conjecture states that $$Ext_A^i(M,M) \neq 0$$ for any non-projective $$A$$-module $$M$$ in case $$A$$ is Gorenstein for some $$i>0$$.

Question : Are the Tachikawa conjectures known for such $$A$$? Is it even known whether the conjecture are true with $$i=1$$?

Since $$I$$ is a monomial ideal, your algebra $$A$$ is graded, by putting the generators $$x_1, \dots ,x_n$$ in degree $$1$$. Your algebra $$A$$ is furthermore connected, that is, it vanishes in negative degrees and in degree zero it is simply a copy of the ground field $$k$$ generated as a $$k$$-vector space by the unit element $$1$$. A 1983 article by George Wilson, https://www.sciencedirect.com/science/article/pii/0021869383901035 , evidently uses an elementary calculation of the Cartan map from algebraic $$K_0$$ to $$G_0$$ to show that that the generalized Nakayama conjecture holds for connected graded algebras. (Caution: Wilson writes "graded algebras" in that paper to mean what I have here been careful to call "connected graded algebras".)
• Unfortunately the proof (or Tachikawa's original proof, at least) that the Nakayama Conjecture implies the two Tachikawa Conjectures for an algebra $A$ applies the statement of the Nakayama Conjecture to different algebras, $\operatorname{End}_A(A\oplus D(A))$ and $\operatorname{End}_A( A\oplus M)$. Nov 4, 2020 at 9:56