Finite poset diagrams have bounded finite bar models

Kernels, images and cokernels of morphisms of diagrams of $k$-vector spaces are formed objectwise, so a morphism of diagrams is an epimorphism exactly when it is surjective at every object, and a complex of diagrams is exact exactly when it is exact at every object. In particular exactness questions about ${\mathrm{Diag}}_k(P)$ reduce to the vector spaces $N(p)$, and no diagram-level machinery beyond the objectwise one is used below.

For $p\in P$ define the diagram $Q_p$ by $Q_p(q)=k$ for $p\le q$ and $Q_p(q)=0$ otherwise, with the identity as structure map whenever both values are nonzero. A natural transformation $\phi :Q_p\to N$ is determined by ${\phi }_p(1)\in N(p)$, since for $q\ge p$ the component ${\phi }_q$ must equal $N(p\to q)\circ {\phi }_p$ and for $q\ngeqq p$ the source $Q_p(q)$ is zero; conversely every vector $v\in N(p)$ extends by these formulas. Hence $\mathrm{Hom}(Q_p,N)\cong N(p)$ by evaluation at $p$. Pulling back along the identification, a surjection $\pi :N\to N^{\mathrm{\prime }}$ induces the surjection $N(p)\to N^{\mathrm{\prime }}(p)$ by step 1.1, so every map $Q_p\to N^{\mathrm{\prime }}$ lifts to $N$ and $Q_p$ is projective. For a finite-dimensional $V$, a natural transformation $Q_p{\otimes }_{k}V\to N$ is likewise determined by its value at $p$, a $k$-linear map $V\to N(p)$, and conversely every such map extends uniquely by the structure maps of $N$, so $\mathrm{Hom}(Q_p{\otimes }_{k}V,N)\cong {\mathrm{Hom}}_k(V,N(p))$; since ${\mathrm{Hom}}_k(V,-)$ carries surjections to surjections — lift the images of a finite basis and extend linearly — the diagram $Q_p{\otimes }_{k}V$ is projective as well. A finite direct sum of projectives is projective, because $\mathrm{Hom}({⨁}_t{X}_t,N)\cong {\prod }_t\mathrm{Hom}(X_t,N)$ and a finite product of surjections of vector spaces is surjective by induction on the number of factors. Only finitely many basis vectors and summands occur at each step, so no choice principle is used.

For a diagram $N$ put ${\mathcal{B}}_n(N)={⨁}_{p_0<\cdots <p_n}{Q}_{p_n}{\otimes }_{k}N(p_0)$ for $n\ge 0$, with the face maps ${\mathrm{\partial }}_i:{\mathcal{B}}_n(N)\to {\mathcal{B}}_{n-1}(N)$ ($n\ge 1$, $0\le i\le n$) given on the summand of the chain $p_0<\cdots <p_n$ by: for $i=0$ apply the structure map $N(p_0)\to N(p_1)$ and delete $p_0$; for $0<i<n$ delete the inner vertex $p_i$, keeping the coefficient in $N(p_0)$ and the representable index $p_n$; for $i=n$ delete $p_n$ and use the natural map $Q_{p_n}\to Q_{p_{n-1}}$ (the identity where the source is $k$, and zero otherwise) while keeping the coefficient. Let $\mathrm{\partial }={\sum }_i(-1{)}^i{\mathrm{\partial }}_i$ for $n\ge 1$ and set $\mathrm{\partial }=0$ in bar degree zero, before adjoining the augmentation. The simplicial identities ${\mathrm{\partial }}_i{\mathrm{\partial }}_j={\mathrm{\partial }}_{j-1}{\mathrm{\partial }}_i$ for $i<j$ hold because diagram transport $p\mapsto N(p)$ is functorial and the maps between representables compose, so ${\mathrm{\partial }}^2=0$; this is a sign-alternating sum of a simplicial identity. Since chains of $P$ have at most $h$ strict inequalities, ${\mathcal{B}}_n(N)=0$ for $n>h$. Augment by the structure maps $ϵ:{\mathcal{B}}_0(N)\to N$, $Q_p{\otimes }_{k}N(p)\to N$, which are natural.

Evaluate the augmented complex at $q\in P$. Its degree-$n$ part consists of the chains $p_0<\cdots <p_n\le q$ with a vector of $N(p_0)$, the faces delete a vertex as in step 3.1, the last face being free to delete $q$, and the augmentation sends a length-zero chain $(p_0)$ to $N(p_0\to q)v$. Define a degree-$(-1)$-to-$0$ map $s_{-1}:N(q)\to {\mathcal{B}}_0(N)(q)$ by $v\mapsto (q,v)$, and for $n\ge 0$ a degree-$n$-to-$n+1$ map $s_n$ on the summand of the chain $p_0<\cdots <p_n\le q$ by $s_n(v)=(-1{)}^{n+1}(p_0<\cdots <p_n<q,v)$ when $p_n<q$, and $s_n=0$ when $p_n=q$. One checks $\mathrm{\partial }{s}_n+s_{n-1}\mathrm{\partial }=\mathrm{i}\mathrm{d}$ on each evaluated component: the appended simplex contributes through its last face with sign $(-1{)}^{n+1}\cdot (-1{)}^{n+1}=1$, every other face is matched by the corresponding appended term in $s\mathrm{\partial }$ with opposite sign, and in degree zero the augmentation section cancels exactly the last-face term; when $p_n=q$ and $n\ge 1$, only the last face contributes to $s_{n-1}\mathrm{\partial }$, with coefficient $(-1{)}^n(-1{)}^n=1$, while for $n=0$ the term $s_{-1}ϵ$ is the identity. Moreover $ϵ$ is surjective on the evaluated complexes because $(q,v)\mapsto v$, so the augmented evaluated complex is exact in every degree. Exactness is objectwise by step 1.1, so $ϵ:\mathcal{B}(N)\to N$ is a quasi-isomorphism; and if every $N(p)$ is finite-dimensional then every ${\mathcal{B}}_n(N)$ is a finite direct sum of projectives by step 2.1, so $\mathcal{B}(N)$ is a bounded projective resolution of $N$.

Let $A$ be a bounded complex of diagrams with finite-dimensional values, vanishing outside internal degrees $[a,b]$, with internal differentials $d_A$. Apply the bar construction of step 4.1 to each internal degree $u$, so ${\mathcal{B}}_n(A^u)$ has bar degree $n\in [0,h]$ and carries the internal differential $d_A:{\mathcal{B}}_n(A^u)\to {\mathcal{B}}_n(A^{u+1})$. The face maps $\mathrm{\partial }$ commute with the maps induced by $d_A$, since they are built from the structure maps of the diagrams and from the fixed maps between representables. Take the total complex $\mathrm{Tot}$ with ${\mathrm{Tot}}^m={⨁}_{u-n=m}{\mathcal{B}}_n(A^u)$ and differential $D=d_A+(-1{)}^u\mathrm{\partial }$ on the summand ${\mathcal{B}}_n(A^u)$; the sign $(-1{)}^u$ makes the horizontal and vertical parts anticommute, so $D^2=0$. Each column $u$ maps quasi-isomorphically to $A^u$ by step 4.1, so the rows of the mapping cone of $\mathrm{Tot}\to A$ are the mapping cones of the augmentations $\mathcal{B}(A^u)\to A^u$ and are acyclic; a bounded double complex with acyclic rows has acyclic total complex, by filtering by the finitely many internal degrees and inducting on the number of rows with the long exact homology sequence of the quotient by the top row. Hence $\mathrm{Tot}\to A$ is a quasi-isomorphism, $\mathrm{Tot}$ is a complex of finite-dimensional projective diagrams by steps 2.1 and 4.1, and its total degrees $u-n$ lie in $[a-h,b]$.

Resolve the constant diagram $\underset{‾}{k}$ by its bar complex $\mathcal{B}(\underset{‾}{k})$, a bounded complex of finite-dimensional projective diagrams of the form of step 4.1 with $\underset{‾}{k}$ in internal degree $0$. These bounded projective models require no DC: Hom from a single projective into an acyclic complex is acyclic by exactness of its Hom functor; filtering a bounded projective complex by its finitely many degrees and applying the cohomology sequence to the resulting termwise split Hom sequences proves the same for that complex. Hence Hom from the model preserves quasi-isomorphisms and computes [F1] without any infinite sequence of lifts. Then $\mathbf{R}\lim M=\mathbf{R}\mathrm{Hom}(\underset{‾}{k},M)$ is computed by the Hom complex $Q\underset{}{\mathrm{Hom}}(\mathcal{B}(\underset{‾}{k}),M)$ of [F1], whose degree-$r$ term is ${\prod }_i\mathrm{Hom}(\mathcal{B}(\underset{‾}{k}{)}^i,M^{i+r})$. Here $\mathcal{B}(\underset{‾}{k}{)}^{-n}={⨁}_{p_0<\cdots <p_n}{Q}_{p_n}$ is nonzero only for $0\le n\le h$ and $M^j=0$ for $j\mathrm{\notin }[a,b]$, so nonzero factors require $-h\le i\le 0$ and $a\le i+r\le b$. Consequently $a\le i+r\le r$ and $r\le b-i\le b+h$, giving $r\in [a,b+h]$. Concretely the contribution of bar degree $n$ to total degree $r$ is the finite product ${\prod }_{p_0<\cdots <p_n}{M}^{r-n}(p_n)$, so every term is a finite product of finite-dimensional spaces, hence finite-dimensional. Thus $\mathbf{R}\lim M$ has a finite-dimensional cochain model concentrated in $[a,b+h]$.

For the pair $A,B$ concentrated in $[a,b]$ and $[c,d]$, resolve the source by the total bar model $P_A:=\mathrm{Tot}$ of step 5.1, concentrated in $[a-h,b]$, and apply [F1]: $\mathbf{R}\mathrm{Hom}(A,B)$ is computed by $Q\underset{}{\mathrm{Hom}}(P_A,B)$, whose degree-$r$ term is the finite product ${\prod }_i\mathrm{Hom}(P_A^i,B^{i+r})$. A nonzero factor requires $P_A^i\mathrm{\ne }0$ and $B^{i+r}\mathrm{\ne }0$, hence $a-h\le i\le b$ and $c\le i+r\le d$; so for $r<c-b$ every $i$ with $P_A^i\mathrm{\ne }0$ has $i+r\le b+r<c$ and for $r>d-a+h$ every such $i$ has $i+r>d$, and in both cases every factor of the term vanishes. Thus the model is concentrated in $[c-b,d-a+h]$. By the evaluation identity in step 2.1, each summand with $u-n=i$ and chain $p_0<\cdots <p_n$ contributes ${\mathrm{Hom}}_k(A^u(p_0),B^{i+r}(p_n))$. Thus every term is a finite product of Hom spaces between finite-dimensional vector spaces, hence finite-dimensional.

Let $K\mathrm{/}k$ be a field extension, and write the base-changed inputs as $K{\otimes }_{k}M$, $K{\otimes }_{k}A$, $K{\otimes }_{k}B$, diagrams over $K$. The models built in steps 5.2 and 6.1 have terms obtained from the values $M(p)$, ${\mathrm{Hom}}_k(A(p),B(q))$ by finite sums, finite products and the fixed face and structure maps; extending scalars commutes with finite direct sums, with finite products and, between finite-dimensional spaces, with Hom spaces, so $K{\otimes }_k$ of the model for $(M,A,B)$ is the model for the base-changed inputs, with the same signs and differentials. Tensoring with $K$ preserves kernels and images of linear maps between finite-dimensional $k$-vector spaces and hence commutes with the homology of a bounded complex of such spaces, so the cohomology of the base-changed model is the base change of the cohomology of the original model. This proves both base-change assertions. For the empty poset there are no chains, the diagram sections and the Hom complex vanish by the same product formulas, and the asserted model is zero.