ScalingStacks

Denote by $\mbox{Kom}(\mathcal{B})$ the category of complexes of an abelian category $\mathcal{B}.$ An object $N$ of $\mbox{Kom}(\mathcal{B})$ is a collection of objects $N^{i}\in\mathcal{B},i\in\mathbb{Z}$ together with morphisms $d^{i}:N^{i}\longrightarrow N^{i+1},i\in\mathbb{Z}$ such that $d^{i+1}d^{i}=0.$ A morphism $f:M\to N$ of complexes is a collection of morphisms $f^{i}:M^{i}\to N^{i}$ such that $f^{i+1}d^{i}=d^{i}f^{i},i\in\mathbb{Z}.$ A morphism $f:M\to N$ is called a quasi-isomorphism if the induced map of the cohomology groups $H^{i}(f):H^{i}(M)\to H^{i}(N)$ is an isomorphism for all $i\in\mathbb{Z}.$

For $n\in\mathbb{Z}$ denote by $[n]$ the automorphism of $\mbox{Kom}(\mathcal{B})$ that is defined on objects by $N[n]^{i}=N^{i+n},d[n]^{i}=(-1)^{n}d^{i+n}$ and continued to morphisms in the obvious way.

The cone of a morphism $f:M\to N$ of complexes is a complex $C(f)$ with

The automorphism $\{n\}$ of shifting the grading down by $n,$ introduced in Section 2.1, can be naturally extended to an automorphism of the category $\mbox{Kom}(R{\mbox{-mod}_{0}})$ of complexes of graded $R$-modules. This automorphism of $\mbox{Kom}(R{\mbox{-mod}_{0}})$ will also be denoted $\{n\}.$

To a complex $M$ of graded $R$-modules we associate a graded $R$-module ${\mathop{\oplus}\limits_{i\in\mathbb{Z}}}M^{i}.$ Each $M^{i}$ is a graded $R$-module, $M^{i}={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}M^{i}_{j}$ and thus ${\mathop{\oplus}\limits_{i\in\mathbb{Z}}}M^{i}$ is a bigraded $R$-module when we extend our usual grading of $R$ to a bigrading with $c\in R$ having degree $(0,2).$

From this viewpoint the differential $d_{M}$ of a complex $M$ is a homogeneous map of degree $(1,0)$ of bigraded $R$-modules.

Let $\mathcal{I}$ be a finite set. Denote by $|\mathcal{I}|$ the cardinality of $\mathcal{I}$ and by $r(\mathcal{I})$ the set of all pairs $(\mathcal{L},a)$ where $\mathcal{L}$ is a subset of $\mathcal{I}$ and $a$ an element of $\mathcal{I}$ that does not belong to $\mathcal{L}.$ To simplify notation we will often

(a) denote a one-element set $\{a\}$ by $a,$

(b) denote a finite set $\{a,b,\dots,d\}$ by $ab\dots d,$

(c) denote the disjoint union $\mathcal{L}_{1}\sqcup\mathcal{L}_{2}$ of two sets $\mathcal{L}_{1},\mathcal{L}_{2}$ by $\mathcal{L}_{1}\mathcal{L}_{2},$ in particular, we denote by $\mathcal{L}a$ the disjoint union of a set $\mathcal{L}$ and a one-element set $\{a\},$ similarly, $\mathcal{L}ab$ means $\mathcal{L}\sqcup\{a\}\sqcup\{b\},$ etc.

We will call a commutative $\mathcal{I}$-cube an $\mathcal{I}$-cube or, sometimes, a cube when it is clear what $\mathcal{I}$ is. Maps $\xi_{a}^{V}$ are called the structure maps of $V.$

Example If $\mathcal{I}$ is the empty set, an $\mathcal{I}$-cube is an object in $\mathcal{B}.$ If $\mathcal{I}$ consists of one element, an $\mathcal{I}$-cube is a morphism in $\mathcal{B}.$ If $\mathcal{I}$ consists of two elements, $\mathcal{I}=\{a,b\},$ an $\mathcal{I}$-cube is a commutative square of objects and morphisms in $\mathcal{B}$:

In general, an $\mathcal{I}$-cube can be visualized in the following manner. Let $n$ be the cardinality of $\mathcal{I}.$ We take an $n$-dimensional cube in standard position in the Euclidean $n$-dimensional space, i.e. each vertex has coordinates $(a_{1},\dots,a_{n})$ where $a_{i}\in\{0,1\}$ and orient each edge in the direction of the vertex with the bigger sum of the coordinates. Then the edges of any 2-dimensional facet of this cube are oriented as shown on the diagram below.

Choose a bijection between elements of $\mathcal{I}$ and coordinates of $\mathbb{R}^{n}.$ This bijection defines a bijection between vertices of the $n$-cube and subsets of $\mathcal{I},$ with the $(0,\dots,0)$ vertex associated to the empty set. Oriented edges of the $n$-cube correspond to pairs $(\mathcal{L},a)\in r(\mathcal{I}).$

Given an $\mathcal{I}$-cube $V,$ put object $V(\mathcal{L})$ into the vertex associated to the set $\mathcal{L}$ and assign morphism $\xi^{V}_{a}(\mathcal{L})$ to the arrow going from the vertex associated to $\mathcal{L}$ to the vertex associated to $\mathcal{L}a.$ Equation (28) is equivalent to the commutativity of diagrams in all 2-dimensional faces of the $n$-cube.

Given two $\mathcal{I}$-cubes $V,W$ over a category $\mathcal{B},$ an $\mathcal{I}$-cube map $\psi:V\longrightarrow W$ is a collection of maps

that make diagrams

commutative for all $(\mathcal{L},a)\in r(\mathcal{I}).$ The map $\psi$ is called an isomorphism if $\psi(\mathcal{L})$ is an isomorphism for all $\mathcal{L}\subset\mathcal{I}.$ The map $\psi$ of $\mathcal{I}$-cubes over an abelian category $\mathcal{B}$ is called injective/surjective if $\psi(\mathcal{L})$ is injective/surjective for all $\mathcal{L}\subset\mathcal{I}.$

The class of $\mathcal{I}$-cubes over an abelian category $\mathcal{B}$ and maps between $\mathcal{I}$-cubes constitute an abelian category in the obvious way. In particular, direct sums of $\mathcal{I}$-cubes are defined.

For a finite set $\mathcal{I}$ and $a\in\mathcal{I},$ let $\mathcal{J}$ be the complement, $\mathcal{I}=\mathcal{J}\sqcup\{a\}.$ Given an $\mathcal{I}$-cube $V,$ let $V_{a}(\ast 0),V_{a}(\ast 1)$ be $\mathcal{J}$-cubes defined as follows:

and the structure maps of $V_{a}(\ast 0),V_{a}(\ast 1)$ are determined by the structure maps $\xi_{b}^{V},b\in\mathcal{I}_{1}$ of $V$ in the obvious fashion. Sometimes we will write $V(\ast 0)$ for $V_{a}(\ast 0),$ etc. The structure map $\xi_{a}^{V}$ of $V$ defines an $\mathcal{I}_{1}$-cube map $\xi_{a}^{V}:V_{a}(\ast 0)\longrightarrow V_{a}(\ast 1).$ This provides a one-to-one correspondence between $\mathcal{I}$-cubes and maps of $\mathcal{I}_{1}$-cubes.

We say that a map $\psi:V\to W$ of $\mathcal{I}$-cubes over the category $R$-mod of graded $R$-modules and graded maps is graded of degree $i$ if the map $\psi(\mathcal{L}):V(\mathcal{L})\to W(\mathcal{L})$ has degree $i$ for all $\mathcal{L}\subset\mathcal{I}.$

For a cube $V$ over $R{\mbox{-mod}_{0}}$ denote by $V\{i\}$ the cube $V$ with the grading shifted by $i$:

and the structure maps being appropriate shifts of the structure maps of $V.$ A degree $i$ map $\psi:V\to W$ of $\mathcal{I}$-cubes over $R{\mbox{-mod}_{0}}$ induces a grading-preserving map $V\to W\{i\},$ also denoted $\psi.$

We next define skew-commutative $\mathcal{I}$-cubes over an additive category $\mathcal{B}.$ A skew-commutative $\mathcal{I}$-cube is almost the same as a commutative $\mathcal{I}$-cube but now we require that for every square facet of the cube the associated diagram of objects and morphisms of $\mathcal{B}$ anticommutes.

We will call a skew-commutative $\mathcal{I}$-cube over $\mathcal{B}$ a skew $\mathcal{I}$-cube or, without specifying $\mathcal{I}$, a skew cube.

Given $\mathcal{I}$-cubes or skew $\mathcal{I}$-cubes $V$ and $W$ over $R{\mbox{-mod}_{0}}$, their tensor product is defined to be an $\mathcal{I}$-cube (if $V$ and $W$ are both cubes or both skew cubes) or a skew $\mathcal{I}$-cube (if one of $V,W$ is a cube and the other is a skew cube), denoted $V\otimes W,$ given by

where, recall, the tensor products are taken over $R.$

For a finite set $\mathcal{L}$ denote by $o(\mathcal{L})$ the set of complete orderings or elements of $\mathcal{L}.$ For $x,y\in o(\mathcal{L})$ let $p(x,y)$ be the parity function, $p(x,y)=0$ if $y$ can be obtained by from $x$ via an even number of transpositions of two neighboring elements in the ordering, otherwise, $p(x,y)=1.$ To a finite set $\mathcal{L}$ associate a graded $R$-module $E(\mathcal{L})$ defined as the quotient of the graded $R$-module, freely generated by elements $x$ for all $x\in o(\mathcal{L}),$ by relations $x=(-1)^{p(x,y)}y$ for all pairs $x,y\in o(\mathcal{L}).$ Module $E(\mathcal{L})$ is a free graded $R$-module of rank $1.$ For $a\not\in\mathcal{L}$ there is a canonical isomorphism of graded $R$-modules $E(\mathcal{L})\longrightarrow E(\mathcal{L}a)$ induced by the map $o(L)\to o(La)$ that takes $x\in o(L)$ to $xa\in o(La).$ Moreover, for $a,b,a\not=b,$ the diagram below anticommutes

Denote by $E_{\mathcal{I}}$ the skew $\mathcal{I}$-cube with $E_{\mathcal{I}}(\mathcal{L})=E(\mathcal{L})$ for $\mathcal{L}\subset\mathcal{I}$ and the structure map $E_{\mathcal{I}}(\mathcal{L})\to E_{\mathcal{I}}(\mathcal{L}a)$ being canonical isomorphism $E(\mathcal{L})\to E(\mathcal{L}a).$

We will use $E_{\mathcal{I}}$ to pass from $\mathcal{I}$-cubes over $R{\mbox{-mod}_{0}}$ to skew $\mathcal{I}$-cubes over $R{\mbox{-mod}_{0}}$ by tensoring an $\mathcal{I}$-cube with $E_{\mathcal{I}}.$

Let $V$ be a skew $\mathcal{I}$-cube over an abelian category $\mathcal{B}.$ To $V$ we associate a complex $\overline{C}(V)=(\overline{C}^{i}(V),d^{i}),i\in\mathbb{Z}$ of objects of $\mathcal{B}$ by

The differential $d^{i}:\overline{C}^{i}(V)\to\overline{C}^{i+1}(V)$ is given on an element $x\in V(\mathcal{L}),|\mathcal{L}|=i$ by

Examples:

1. 1.

   If $|\mathcal{I}|=1,\mathcal{I}=\{a\},$

   $$\overline{C}^{i}(V)=\left\{\begin{array}[]{ll}V(\emptyset)&\mbox{ if $i=0$ }\\ V(a)&\mbox{ if $i=1$ }\\ 0&\mbox{ otherwise }\end{array}\right.$$

   The differential $d^{0}=\xi_{a}^{V}(\emptyset)$ and $d^{i}=0$ if $i\not=0,$ so $\overline{C}(V)$ is the complex

   $$\cdots\longrightarrow 0\longrightarrow V(\emptyset)\stackrel{{\scriptstyle\xi^{V}_{a}(\emptyset)}}{{\longrightarrow}}V(\mathcal{I})\longrightarrow 0\longrightarrow\cdots$$

   (35)

2. 2.

   If $\mathcal{I}$ contains two elements, say, $\mathcal{I}=\{a,b\},$ then

   $$\overline{C}^{i}(V)=\left\{\begin{array}[]{ll}V(\emptyset)&\mbox{ if $i=0$ }\\ V(a)\oplus V(b)&\mbox{ if $i=1$ }\\ V(ab)&\mbox{ if $i=2$ }\\ 0&\mbox{ otherwise }\end{array}\right.$$

   and differentials

   		$\displaystyle d^{0}:$	$\displaystyle V(\emptyset)\longrightarrow V(a)\oplus V(b)$
   		$\displaystyle d^{0}=$	$\displaystyle\xi_{b}^{V}(\emptyset)+\xi_{a}^{V}(\emptyset)$
   		$\displaystyle d^{1}:$	$\displaystyle V(b)\oplus V(a)\longrightarrow V(ab)$
   		$\displaystyle d^{1}=$	$\displaystyle(\xi_{a}^{V}(b),\xi_{b}^{V}(a))$

Proof: The complex $\overline{C}(V)$ is isomorphic to the cone of the identity map of the complex $\overline{C}(V_{a}(\ast 1))[-1]$ and, therefore, acyclic. $\square$

Every map of $\mathcal{I}$-cubes $\phi:V\longrightarrow W$ over $R{\mbox{-mod}_{0}}$ induces a map of complexes

If $\phi$ is an isomorphism of commutative cubes, $\overline{C}(\phi)$ is an isomorphism of complexes.

Proof: Immediate from Proposition 4. $\square$

The following proposition and its corollary are obvious.