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Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules. Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...

Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$. By Koszul resolution I mean $$\ldots \to A \...