42.2 Periodic complexes and Herbrand quotients
Of course there is a very general notion of periodic complexes. We can require periodicity of the maps, or periodicity of the objects. We will add these here as needed. For the moment we only need the following cases.
Definition 42.2.1. Let $R$ be a ring.
A $2$-periodic complex over $R$ is given by a quadruple $(M, N, \varphi , \psi )$ consisting of $R$-modules $M$, $N$ and $R$-module maps $\varphi : M \to N$, $\psi : N \to M$ such that
\[ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & N \ar[r]^\psi & M \ar[r]^\varphi & N \ar[r] & \ldots } \]
is a complex. In this setting we define the cohomology modules of the complex to be the $R$-modules
\[ H^0(M, N, \varphi , \psi ) = \mathop{\mathrm{Ker}}(\varphi )/\mathop{\mathrm{Im}}(\psi ) \quad \text{and}\quad H^1(M, N, \varphi , \psi ) = \mathop{\mathrm{Ker}}(\psi )/\mathop{\mathrm{Im}}(\varphi ). \]
We say the $2$-periodic complex is exact if the cohomology groups are zero.
A $(2, 1)$-periodic complex over $R$ is given by a triple $(M, \varphi , \psi )$ consisting of an $R$-module $M$ and $R$-module maps $\varphi : M \to M$, $\psi : M \to M$ such that
\[ \xymatrix{ \ldots \ar[r] & M \ar[r]^\varphi & M \ar[r]^\psi & M \ar[r]^\varphi & M \ar[r] & \ldots } \]
is a complex. Since this is a special case of a $2$-periodic complex we have its cohomology modules $H^0(M, \varphi , \psi )$, $H^1(M, \varphi , \psi )$ and a notion of exactness.
In the following we will use any result proved for $2$-periodic complexes without further mention for $(2, 1)$-periodic complexes. It is clear that the collection of $2$-periodic complexes forms a category with morphisms $(f, g) : (M, N, \varphi , \psi ) \to (M', N', \varphi ', \psi ')$ pairs of morphisms $f : M \to M'$ and $g : N \to N'$ such that $\varphi ' \circ f = g \circ \varphi $ and $\psi ' \circ g = f \circ \psi $. We obtain an abelian category, with kernels and cokernels as in Homology, Lemma 12.13.3.
Definition 42.2.2. Let $(M, N, \varphi , \psi )$ be a $2$-periodic complex over a ring $R$ whose cohomology modules have finite length. In this case we define the multiplicity of $(M, N, \varphi , \psi )$ to be the integer
\[ e_ R(M, N, \varphi , \psi ) = \text{length}_ R(H^0(M, N, \varphi , \psi )) - \text{length}_ R(H^1(M, N, \varphi , \psi )) \]
In the case of a $(2, 1)$-periodic complex $(M, \varphi , \psi )$, we denote this by $e_ R(M, \varphi , \psi )$ and we will sometimes call this the (additive) Herbrand quotient.
If the cohomology groups of $(M, \varphi , \psi )$ are finite abelian groups, then it is customary to call the (multiplicative) Herbrand quotient
\[ q(M, \varphi , \psi ) = \frac{\# H^0(M, \varphi , \psi )}{\# H^1(M, \varphi , \psi )} \]
In words: the multiplicative Herbrand quotient is the number of elements of $H^0$ divided by the number of elements of $H^1$. If $R$ is local and if the residue field of $R$ is finite with $q$ elements, then we see that
\[ q(M, \varphi , \psi ) = q^{e_ R(M, \varphi , \psi )} \]
An example of a $(2, 1)$-periodic complex over a ring $R$ is any triple of the form $(M, 0, \psi )$ where $M$ is an $R$-module and $\psi $ is an $R$-linear map. If the kernel and cokernel of $\psi $ have finite length, then we obtain
42.2.2.1
\begin{equation} \label{chow-equation-multiplicity-coker-ker} e_ R(M, 0, \psi ) = \text{length}_ R(\mathop{\mathrm{Coker}}(\psi )) - \text{length}_ R(\mathop{\mathrm{Ker}}(\psi )) \end{equation}
We state and prove the obligatory lemmas on these notations.
Lemma 42.2.3. Let $R$ be a ring. Suppose that we have a short exact sequence of $2$-periodic complexes
\[ 0 \to (M_1, N_1, \varphi _1, \psi _1) \to (M_2, N_2, \varphi _2, \psi _2) \to (M_3, N_3, \varphi _3, \psi _3) \to 0 \]
If two out of three have cohomology modules of finite length so does the third and we have
\[ e_ R(M_2, N_2, \varphi _2, \psi _2) = e_ R(M_1, N_1, \varphi _1, \psi _1) + e_ R(M_3, N_3, \varphi _3, \psi _3). \]
Proof.
We abbreviate $A = (M_1, N_1, \varphi _1, \psi _1)$, $B = (M_2, N_2, \varphi _2, \psi _2)$ and $C = (M_3, N_3, \varphi _3, \psi _3)$. We have a long exact cohomology sequence
\[ \ldots \to H^1(C) \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C) \to \ldots \]
This gives a finite exact sequence
\[ 0 \to I \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to K \to 0 \]
with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of the length function (Algebra, Lemma 10.52.3) we see the result.
$\square$
Lemma 42.2.4. Let $R$ be a ring. If $(M, N, \varphi , \psi )$ is a $2$-periodic complex such that $M$, $N$ have finite length, then $e_ R(M, N, \varphi , \psi ) = \text{length}_ R(M) - \text{length}_ R(N)$. In particular, if $(M, \varphi , \psi )$ is a $(2, 1)$-periodic complex such that $M$ has finite length, then $e_ R(M, \varphi , \psi ) = 0$.
Proof.
Observe that on the category of $2$-periodic complexes with $M$, $N$ of finite length the quantity “$\text{length}_ R(M) - \text{length}_ R(N)$” is additive in short exact sequences (precise statement left to the reader). Consider the short exact sequence
\[ 0 \to (M, \mathop{\mathrm{Im}}(\varphi ), \varphi , 0) \to (M, N, \varphi , \psi ) \to (0, N/\mathop{\mathrm{Im}}(\varphi ), 0, 0) \to 0 \]
The initial remark combined with the additivity of Lemma 42.2.3 reduces us to the cases (a) $M = 0$ and (b) $\varphi $ is surjective. We leave those cases to the reader.
$\square$
Lemma 42.2.5. Let $R$ be a ring. Let $f : (M, \varphi , \psi ) \to (M', \varphi ', \psi ')$ be a map of $(2, 1)$-periodic complexes whose cohomology modules have finite length. If $\mathop{\mathrm{Ker}}(f)$ and $\mathop{\mathrm{Coker}}(f)$ have finite length, then $e_ R(M, \varphi , \psi ) = e_ R(M', \varphi ', \psi ')$.
Proof.
Apply the additivity of Lemma 42.2.3 and observe that $(\mathop{\mathrm{Ker}}(f), \varphi , \psi )$ and $(\mathop{\mathrm{Coker}}(f), \varphi ', \psi ')$ have vanishing multiplicity by Lemma 42.2.4.
$\square$
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