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Consider a monoidal category which is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a -algebra which is -quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This -algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted -coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined -algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of -algebras.

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $\mathbb{G}$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$., Comment: 52 pages