I need a reference for the description of Ext groups in mixed categories (i.e. abelian categories with a weight filtration and semisimple graded quotients) by using the bar complex, as mentioned in "Notes on motivic cohomology, Beilinson, et al". This result is referred in this paper to a paper of Manin: "Correspondences, Motifs and monoidal transformations". But I can't find anything related in the paper of Manin.


Edit. To clarify the result I restate the definitions and the theorem here:

Mixed Category: An abelian $\mathbb{Q}$-category $\mathcal{M}$ in which every object is supplied by an increasing (weight) filtration $W_\bullet$ such that 1) $W_\bullet$ is finite for every object, 2)Morphisms in $\mathcal{M}$ are strictly compatible with $W_\bullet$, 3)The graded quotients of the weight filtration for every object are semisimple, 4) $\mathrm{Hom}$'s are finite dimensional $\mathbb{Q}$-vector spaces.

Suppose $\mathcal{M}$ is a mixed category with the further property that endomorphisms of simple objects are one dimensional.

For each $i$ we denote pure objects of weight $i$ in $\mathcal{M}$ by $\mathcal{M}_i$. Let $i<j$, $x\in \mathcal{M}_i, y\in \mathcal{M}_j$ denote by $\mathcal{M}(x,y)$ the category whose objects are $z\in \mathcal{M}$ with isomorphisms $gr_i^W z\to x$ and $y\to gr_j^W z$ and morphisms which respect this isomorphisms. Define $A(x,y)$ to be $\pi_0\mathcal{M}(x,y)$. $A(x,y)$ has a natural $\mathbb{Q}$-vector space structure defined similiar to Yoneda's definition of addition for $\mathrm{Ext}$ groups. And for every triple of simple objects $x,y,z$ one can construct a comultiplication $A(x,z)\to A(x,y)\otimes A(y,z)$.

Fix for each $i$ a set $S_i$ of representatives of isomorphism classes of simple objects in $\mathcal{M}_i$.

Suppose $x\in S_i, y \in S_{i+n},n>0$ and $1\leq k\leq n$. Then put $$A^k(x,y)=\oplus \left(A(x_0,x_1)\otimes A(x_1,x_2)\otimes \cdots \otimes A(x_{k-1},x_k) \right),$$
where the sum is taken over all sequnces $(x_0,\dots,x_k)$ with $x_0=x, x_k=y$ and $x_p\in S_{i_p}$ for $i=i_0<i_1<\cdots<i_k=i+n$. Comultiplication in $A$ induces evident maps $\delta_p:A^k(x,y)\to A^{k+1}(x,y)$ for $p=0,\dots, k-1$. Put $d^k= \sum_p (-1)^p \delta_p$. We get a complex: $$A^\bullet (x,y) : 0\to A^1(x,y)\to A^2(x,y)\to \cdots \to A^n(x,y)\to 0.$$

Then the result mentioned above is the following theorem:

Theorem. For every $i$ there is a natural isomorphism $\mathrm{Ext}^i(y,x)\cong \mathrm{H}^i A^\bullet (x,y)$.


Maybe the following lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991, will be useful: http://www.ams.org/bookstore-getitem/item=pspum-55.s

  • $\begingroup$ Thanks! Is there another reference (other than above mentioned paper of Beilinson, et al) which investigates the algebra of mixed categories? $\endgroup$ – Mostafa Feb 2 '15 at 6:53
  • $\begingroup$ I do not know, I'm not a specialist in the field - just curious about applications of motives in physics. Maybe some references can be found through this website: ncatlab.org/nlab/show/motive $\endgroup$ – Zurab Silagadze Feb 2 '15 at 8:18

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