# Support varieties for Lie superalgebras in characteristic 2

Christopher M. Drupieski  Department of Mathematical Sciences, DePaul
University, Chicago, IL 60614, USA<EMAIL_ADDRESS>and  Jonathan R.
Kujawa  Department of Mathematics
University of Oklahoma
Norman, OK 73019, USA<EMAIL_ADDRESS>Dedicated to Brian Parshall with
gratitude.

###### Abstract.

This paper investigates cohomology and support varieties for Lie superalgebras
and restricted Lie superalgebras over a field of characteristic $2$. The
existence of an underlying ordinary Lie algebra allows us to obtain results
that are still open in odd characteristic, and also to establish results that
have no non-super analogues in characteristic $2$.

###### 1991 Mathematics Subject Classification:

Primary 18G10. Secondary 18G15, 18G40, 20G10.

The first author was supported in part by Simons Collaboration Grant for
Mathematicians No. 4269055, and by a DePaul University Research Council paid
faculty leave in Spring 2022. The second author was supported in part by
Simons Collaboration Grant for Mathematicians No. 525043.

##  1\. Introduction

###  1.1. Overview

In the mid-1980s, Friedlander and Parshall [19, 18] introduced support
varieties to the representation theory of restricted Lie algebras. Through
their work and the work of those who followed, support varieties for
restricted Lie algebras are comparatively well-understood. Almost four decades
later, support varieties and their generalizations remain a subject of active
investigation in non-semisimple representation theory, and the work of
Friedlander and Parshall continues to serve as a model for those hoping to
bring their methods to other settings. Relevant to this paper, one such
setting is that of Lie superalgebras.

Roughly, a ‘super’ version of a mathematical structure incorporates
$\mathbb{Z}_{2}$-gradings on objects, together with $\pm$ signs that are
inserted into the defining equations whenever elements of odd superdegree move
past one another. Lie superalgebras arise in algebraic topology, spin
representations of the symmetric group, mathematical physics, and elsewhere.
Work by Boe, Kujawa, and Nakano [7, 8], the authors [14, 15, 16], Benson,
Iyengar, Krause, and Pevtsova [4, 5], and others demonstrates that support
varieties generalize in interesting and nontrivial ways to the super setting.

The $\pm$ signs that show up in super formulas are one of the defining
features of the theory. However, when the ground field is of characteristic
$2$ the $\pm$ signs effectively disappear, and by forgetting the
$\mathbb{Z}_{2}$-gradings one seems to obtain just a classical, non-super
version of the object in question. Indeed, it is sometimes assumed that super-
mathematics over fields of characteristic $2$ is uninteresting. Nevertheless,
there is an extensive literature on the study of Lie superalgebras over fields
of characteristic $2$; see, for example, [10] and the references therein. The
current paper hopefully adds additional evidence against that assumption.

The goal of this paper is to study support varieties for Lie superalgebras
(both restricted and un-restricted) over fields of characteristic $2$. As we
will see, the overlap between the super and non-super theories in
characteristic $2$ is a feature, not a bug. One can use this overlap as
leverage to prove results for the super-theory that are merely conjectural in
odd characteristics, as well as results that have no non-super analogues in
characteristic $2$. This paper thus exists at the intersection of topics that
are well-understood (support varieties for restricted Lie algebras) and topics
that are not (support varieties for graded objects).

###  1.2. Main Results

We begin in Section 2 by recalling the definition of a Lie superalgebra over a
field of arbitrary characteristic. Roughly, a Lie superalgebra in
characteristic $2$ is a $\mathbb{Z}_{2}$-graded ordinary Lie algebra
$L=L_{\overline{0}}\oplus L_{\overline{1}}$, together with a distinguished
quadratic operator $q:L_{\overline{1}}\to L_{\overline{0}}$. The quadratic
operator effectively serves as a replacement for the map given in other
characteristics by $x\mapsto\frac{1}{2}[x,x]$; see Remark 2.2. A
representation of a Lie superalgebra in characteristic $2$ is then a
$\mathbb{Z}_{2}$-graded representation of the underlying ordinary Lie algebra
$\overline{L}$ that is ‘partially restricted’ to ensure compatibility with the
quadratic operator.

A restricted Lie superalgebra is a Lie superalgebra $L$ equipped with a
$p$-operation $x\mapsto x^{[p]}$ on $L_{\overline{0}}$ that makes
$L_{\overline{0}}$ into an ordinary restricted Lie algebra, and that makes
$L_{\overline{1}}$ into a restricted $L_{\overline{0}}$-module. In Lemma 2.14
we observe that, given a restricted Lie superalgebra over a field of
characteristic $p=2$, the $2$-operation $(-)^{[2]}:L_{\overline{0}}\to
L_{\overline{0}}$ and the quadratic operator $q:L_{\overline{1}}\to
L_{\overline{0}}$ can be combined to define a $2$-operation $z\mapsto
z^{\\{2\\}}$ on all of $L$,

$z=z_{\overline{0}}+z_{\overline{1}}\quad\mapsto\quad
z^{\\{2\\}}:=(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})+[z_{\overline{1}},z_{\overline{0}}],$

which then makes the underlying ordinary Lie algebra $\overline{L}$ of $L$
into a restricted Lie algebra.111This observation has doubtlessly been made
before and is presumably well-known to characteristic $2$ experts. Moreover,
this restricted structure on $\overline{L}$ then induces a canonical
identification between $V(L)$, the restricted enveloping superalgebra of the
restricted Lie superalgebra $L$, and $V(\overline{L})$, the restricted
enveloping algebra of the ordinary restricted Lie algebra $\overline{L}$.

####  1.2.1. Restricted Lie superalgebras

In Section 3 we explore in detail the implications for support varieties of
the canonical identification $V(L)=V(\overline{L})$, for $L$ a finite-
dimensional restricted Lie superalgebra over a (for simplicity, algebraically
closed) field $k$ of characteristic $2$. The cohomology ring
$\operatorname{H}^{\bullet}(V(L),k)$ is a finitely-generated commutative
$k$-algebra that is naturally graded by $\mathbb{Z}\times\mathbb{Z}_{2}$; the
first component of the grading is the ‘external’ cohomological grading and the
second component is the ‘internal’ grading induced by the superalgebra
structure of $V(L)$. Write $|V(L)|$ for the maximal ideal spectrum of the
$k$-algebra $\operatorname{H}^{\bullet}(V(L),k)$. We show in (3.9) that
$|V(L)|$ identifies with the restricted nullcone of $\overline{L}$:

$|V(L)|\simeq\mathcal{N}_{1}(\overline{L})=\\{z\in\overline{L}:z^{\\{2\\}}=0\\}=\\{z_{\overline{0}}+z_{\overline{1}}\in
L:(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})=0\text{ and
}[z_{\overline{1}},z_{\overline{0}}]=0\\}.$

This extends to characteristic $p=2$ a description previously obtained by the
authors for restricted Lie superalgebras in characteristics $p\geq 3$; cf.
[14, Proposition 5.4.4].

Given a finite-dimensional $V(L)$-supermodule $M$, let $I(M)$ be the
annihilator ideal for the cup product action of
$\operatorname{H}^{\bullet}(V(L),k)$ on
$\operatorname{Ext}_{V(L)}^{\bullet}(M,M)$. The support variety of $M$,
denoted $|V(L)|_{M}$, is the Zariski closed subset of $|V(L)|$ defined by the
ideal $I(M)$. Each nonzero element $z\in\mathcal{N}_{1}(\overline{L})$
generates a $k$-subalgebra of $V(L)=V(\overline{L})$ of the form
$k[z]/(z^{2})$. Appealing to support variety theory for ordinary restricted
Lie algebras, the support variety $|V(L)|_{M}$ then admits the following “rank
variety” description:

(1.1) $|V(L)|_{M}\simeq\\{z\in\mathcal{N}_{1}(\overline{L}):M\text{ is not
free as a $k[z]/(z^{2})$-module}\\}\cup\\{0\\}.$

This immediately implies that support varieties for $V(L)$-supermodules
satisfy the Tensor Product Property (Proposition 3.2). In Proposition 3.4, we
reformulate the condition that $M$ is not free as a $k[z]/(z^{2})$-module in
terms of the projective dimension of a pullback of $M$ to a hypersurface ring.
As discussed in Remark 3.5, this alternate characterization of $|V(L)|_{M}$
can be interpreted as confirming, for characteristic $2$, a conjecture we
previously made for the support varieties of restricted Lie superalgebras (and
more generally, for arbitrary infinitesimal supergroup schemes) over fields of
odd characteristic.

Now let $\mathsf{st\text{-}smod}_{V(L)}$ be the stable category of finite-
dimensional $V(L)$-supermodules. The Hopf algebra structure on $V(L)$ makes
$\mathsf{st\text{-}smod}_{V(L)}$ into a tensor triangulated category.
Following Balmer, one can use the tensor triangulated structure to define the
spectrum $\operatorname{Spc}(\mathsf{st\text{-}smod}_{V(L)})$ of
$\mathsf{st\text{-}smod}_{V(L)}$ and the support function

$\operatorname{supp}^{\prime}:\mathsf{st\text{-}smod}_{V(L)}\to
2^{\operatorname{Spc}(\mathsf{st\text{-}smod}_{V(L)})}.$

Here $2^{\operatorname{Spc}(\mathsf{st\text{-}smod}_{V(L)})}$ denotes the
power set of $\operatorname{Spc}(\mathsf{st\text{-}smod}_{V(L)})$. Given its
abstract definition, it is not surprising that it is desirable to have a more
concrete description of the Balmer spectrum and support function.

Let $X_{s}=\operatorname{Proj}_{s}(A)$ denote the set of all bi-homogeneous
ideals in $A=\operatorname{H}^{\bullet}(V(L),k)$ (i.e., ideals that are
homogeneous with respect to the $\mathbb{Z}\times\mathbb{Z}_{2}$-grading) that
are prime among the bi-homogeneous ideals, and which are properly contained in
the maximal ideal $\operatorname{H}^{>0}(V(L),k)$ of elements of positive
cohomological degree. We view $X_{s}$ as a topological space via the Zariski
topology. We note that $X_{s}$ is a coarser space than the usual projective
spectrum $\operatorname{Proj}(A)$ (which consists of prime ideals in
$\operatorname{H}^{\bullet}(V(L),k)$ which are homogeneous with respect to
just the $\mathbb{Z}$-component of the grading).

In Section 3.4 we show that $X_{s}$ is a Zariski space and that there is a
support function

$\operatorname{supp}_{s}:\mathsf{st\text{-}smod}_{V(L)}\to 2^{X_{s}}.$

Applying work of Dell’Ambrogio [12] and of Benson, Iyengar, Krause, and
Pevtsova [3], we show in Theorem 3.10 that there is a canonical homeomorphism

$\operatorname{Proj}_{s}(A)\simeq\operatorname{Spc}(\mathsf{st\text{-}smod}_{V(L)})$

that identifies the two support functions $\operatorname{supp}^{\prime}$ and
$\operatorname{supp}_{s}$. This gives a non-categorical realization of the
Balmer spectrum and support function. We further deduce that the pair
$(\operatorname{Proj}_{s}(A),\operatorname{supp}_{s})$ provides a
classification of the thick tensor ideal subcategories of
$\mathsf{st\text{-}smod}_{V(L)}$. Concretely, this implies that if $M$ and $N$
are two finite-dimensional $V(L)$-supermodules, then $M$ can be obtained from
$N$ by a finite sequence of the standard operations available in
$\mathsf{st\text{-}smod}_{V(L)}$ (direct sums and summands, tensor products,
extensions, syzygies) if and only if $|V(L)|_{M}\subseteq|V(L)|_{N}$. Thus the
question of whether one supermodule can be built from another can be
reformulated into a statement about their support varieties, which in turn is
reduced to a linear-algebraic problem by (1.1).

####  1.2.2. Lie superalgebras

In Section 4 we turn to the case of an arbitrary finite-dimensional Lie
superalgebra $L$ over $k$. Again, the cohomology ring
$\operatorname{H}^{\bullet}(L,k)=\operatorname{H}^{\bullet}(U(L),k)$ is a
finitely-generated commutative $k$-algebra graded by
$\mathbb{Z}\times\mathbb{Z}_{2}$, and one can define support varieties for
finite-dimensional $L$-supermodules. Writing $|U(L)|$ for the maximal ideal
spectrum of $\operatorname{H}^{\bullet}(L,k)$, we show in Theorem 4.5 that
$|U(L)|$ identifies with the odd nullcone of $L$:

$|U(L)|\simeq\mathcal{N}_{\textup{odd}}(L):=\\{x\in
L_{\overline{1}}:q(x)=0\\}.$

Likewise, Theorem 4.6 shows that the support variety of a finite-dimensional
$L$-supermodule $M$ admits a rank variety description:

$|U(L)|_{M}\simeq\\{x\in\mathcal{N}_{\textup{odd}}(L):M\text{ is not free as a
$k[x]/(x^{2})$-supermodule}\\}\cup\\{0\\}.$

This description implies that the support varieties of finite-dimensional
$L$-supermodules satisfy the Tensor Product Property (Corollary 4.7). Then
Theorem 4.8 shows that $|U(L)|_{M}=\\{0\\}$ if and only if $M$ has finite
projective dimension as a $U(L)$-supermodule. In particular,
$\mathcal{N}_{\textup{odd}}(L)=\\{0\\}$ if and only if $U(L)$ is of finite
global dimension. This extends to $p=2$ a result due of Bøgvad [9] in
characteristic zero (see also [25, Chapter 17]), and due to the authors in
characteristic $p\geq 3$ [16].

###  1.3. Future directions

Since Premet’s proof of the Kac–Weisfieler conjecture [26], $p$-divisibility
of a module’s dimension has been known to be closely related to the dimension
of the module’s support variety. Some divisibility results along these lines
are known in super-representation theory (see, e.g., [6, 30]), but much
remains unknown, however, and it would be interesting to improve upon these
results.

It would also be interesting to compute, in all positive characteristics, the
Balmer spectrum for the stable module category of finite-dimensional
$V(L)$-supermodules, and for the singularity category of finite-dimensional
$U(L)$-supermodules. More generally, there are conjectures and questions for
Lie superalgebras and infinitesimal supergroup schemes that the present paper
addresses, but which remain open in odd characteristics. For $V(L)$ we were
able to take advantage of coincidences that happen only in characteristic $2$.
In odd characteristic new ideas will be required for both superalgebras.

Finally, interesting alternate notions of super-mathematics in characteristic
$2$ were proposed in [29, 23]. It would be interesting to consider the
questions covered in this paper within that framework.

###  1.4. Conventions

Throughout, $k$ will be a field of positive characteristic $p$ (usually
$p=2$). All vector spaces will be $k$-vector spaces, and all unadorned tensor
products will denote tensor products over $k$. Let
$\mathbb{N}=\\{0,1,2,3,\ldots\\}$ be the set of non-negative integers.

Set $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}=\\{\overline{0},\overline{1}\\}$.
Following the literature, we use the prefix ‘super’ to indicate that an object
is $\mathbb{Z}_{2}$-graded. We denote the decomposition of a vector superspace
into its $\mathbb{Z}_{2}$-homogeneous components by $V=V_{\overline{0}}\oplus
V_{\overline{1}}$, calling $V_{\overline{0}}$ and $V_{\overline{1}}$ the even
and odd subspaces of $V$, respectively, and writing
$\overline{v}\in\mathbb{Z}_{2}$ to denote the superdegree of a homogeneous
element $v\in V_{\overline{0}}\cup V_{\overline{1}}$. When written without
additional adornment, we consider the field $k$ to be a superspace
concentrated in even superdegree. Whenever we state a formula in which
homogeneous degrees of elements are specified, we mean that the formula is
true as written for homogeneous elements, and that it extends linearly to non-
homogeneous elements.

###  1.5. Acknowledgments

The authors thank Luchezar Avramov for prodding us to address what happens in
characteristic two.

##  2\. Definitions

###  2.1. Lie superalgebras

######  Definition 2.1.

Let $k$ be a field (of any characteristic). A Lie superalgebra over $k$ is a
$\mathbb{Z}_{2}$-graded $k$-vector space $L=L_{\overline{0}}\oplus
L_{\overline{1}}$, together with an even (i.e., $\mathbb{Z}_{2}$-degree
preserving) bilinear map $[\,,\,]:L\otimes L\to L$ and a quadratic operator
$q:L_{\overline{1}}\to L_{\overline{0}}$, such that the following axioms hold:

  1. (L1)

$[x,y]=-(-1)^{\overline{x}\cdot\overline{y}}[y,x]$ for all homogeneous $x,y\in
L$.

  2. (L2)

$[x,[y,z]]=[[x,y],z]+(-1)^{\overline{x}\cdot\overline{y}}[y,[x,z]]$ for all
homogeneous $x,y,z\in L$ (Jacobi identity).

  3. (L3)

$[x,x]=0$ for all $x\in L_{\overline{0}}$.

  4. (L4)

$[y,[y,y]]=0$ for all $y\in L_{\overline{1}}$.

  5. (L5)

$[x,y]=q(x+y)-q(x)-q(y)$ for all $x,y\in L_{\overline{1}}$.

  6. (L6)

$[y,[y,z]]=[q(y),z]$ for all $y\in L_{\overline{1}}$ and $z\in L$.

######  Remark 2.2.

Let $L$ be a Lie superalgebra, and let $y\in L_{\overline{1}}$. Then (L5)
asserts that

(2.1) $[y,y]=q(y+y)-q(y)-q(y)=q(2\cdot y)-2\cdot q(y)=4\cdot q(y)-2\cdot
q(y)=2\cdot q(y).$

For $p\neq 2$, this shows that $q(y)=\frac{1}{2}[y,y]$, while for $p=2$, it
shows that $[y,y]=0$ for all $y\in L_{\overline{1}}$. Thus for $p=2$, if we
forget the quadratic operator $q:L_{\overline{1}}\to L_{\overline{0}}$, the
Lie bracket $[\,,\,]:L\otimes L\to L$ makes $L$ into an ordinary Lie algebra,
which we may denote $\overline{L}$.

######  Remark 2.3.

Depending on the characteristic $p$ of the field $k$, some of the axioms of a
Lie superalgebra are redundant:

  1. (1)

If $p\neq 2$, then (L3) is a consequence of (L1).

  2. (2)

If $p\neq 3$, then (L4) is a consequence of (L1) and (L2).

  3. (3)

Suppose $p\neq 2$. Then (L5) and (L6) follow from (L1), (L2), and the formula
$q(y)=\frac{1}{2}[y,y]$. Conversely, if $L$ is a $k$-superspace equipped with
an even bilinear map $[\,,\,]:L\otimes L\to L$ satisfying (L1)–(L4), then
$q:L_{\overline{1}}\to L_{\overline{0}}$ defined by $q(y)=\frac{1}{2}[y,y]$
satisfies (L5) and (L6).

It is necessary to assume (L3) when $p=2$, and to assume (L4) when $p=3$, to
ensure that $L$ can be embedded into its universal enveloping algebra $U(L)$;
see [2, 1.1.10 and 3.2.3].

######  Remark 2.4.

For those who (like the authors of this paper) have spent most of their lives
only considering Lie superalgebras over fields of characteristic $p\neq 2$,
the inclusion of the quadratic operator $q:L_{\overline{1}}\to
L_{\overline{0}}$ in Definition 2.1 may require explanation. Remarks 2.2 and
2.3 provide some motivation for why it is natural to include the quadratic
operator in the definition of a Lie superalgebra. Another source of motivation
is the fact that it permits a uniform, characteristic-free description of the
Koszul resolution (Theorem 4.1). Here is yet another motivating observation:

Let $k$ be a field (of any characteristic), let $A$ be a $k$-superalgebra, and
let $D\in\operatorname{Der}_{k}(A,A)_{\overline{1}}$ be an odd superderivation
of $A$. Thus $D$ is an odd linear map such that for all homogeneous elements
$a,b\in A$, one has $D(ab)=D(a)\cdot b+(-1)^{\overline{a}}a\cdot D(b)$. Then

$\displaystyle D^{2}(ab)$ $\displaystyle=D\left(D(a)\cdot
b+(-1)^{\overline{a}}a\cdot D(b)\right)$ $\displaystyle=D^{2}(a)\cdot
b+(-1)^{\overline{a}+\overline{1}}D(a)\cdot D(b)+(-1)^{\overline{a}}D(a)\cdot
D(b)+a\cdot D^{2}(b)$ $\displaystyle=D^{2}(a)\cdot b+a\cdot D^{2}(b),$

so $D^{2}\in\operatorname{Der}_{k}(A,A)_{\overline{0}}$. Now, one of the ways
to describe the Lie superalgebra $\operatorname{Lie}(G)$ of an affine
algebraic supergroup $G$ is as the space of left-invariant superderivations of
its coordinate superalgebra $k[G]$.333Recall that, writing $\Delta:k[G]\to
k[G]\otimes k[G]$ for the coproduct on $k[G]$, a superderivation $D:k[G]\to
k[G]$ is left-invariant if $\Delta\circ D=(\operatorname{id}\otimes
D)\circ\Delta$. The preceding calculation then implies that, in arbitrary
characteristic, the squaring map $D\mapsto D^{2}$ defines a quadratic operator
$q:\operatorname{Lie}(G)_{\overline{1}}\to\operatorname{Lie}(G)_{\overline{0}}$.

######  Example 2.5.

Let $k$ be a field (of any characteristic), and let $A$ be a $k$-superalgebra.
Define an even bilinear map $[\,,\,]:A\otimes A\to A$ and a quadratic operator
$q:A_{\overline{1}}\to A_{\overline{0}}$ by

  * •

$[a,b]=ab-(-1)^{\overline{a}\cdot\overline{b}}ba$ for $a,b\in A$ homogeneous,
and

  * •

$q(a)=a^{2}$ for $a\in A_{\overline{1}}$.

These operations define a Lie superalgebra structure on the underlying
$k$-superspace of $A$, which we may denote by $\operatorname{Lie}(A)$.

######  Definition 2.6.

Let $k$ be a field (of any characteristic), let $L$ be a Lie superalgebra over
$k$, and let $T(L)=\bigoplus_{i\geq 0}L^{\otimes i}$ be the tensor
(super)algebra on $L$. The universal enveloping algebra of $L$ is the
$k$-superalgebra $U(L):=T(L)/I$, where $I$ is the ideal in $T(L)$ generated by
the following elements:

  * •

$x\otimes y-(-1)^{\overline{x}\cdot\overline{y}}y\otimes x-[x,y]$ for $x,y\in
L$ homogeneous, and

  * •

$y\otimes y-q(y)$ for $y\in L_{\overline{1}}$.

The proof of the following theorem is standard.

######  Theorem 2.7 (Poincaré–Birkhoff–Witt Theorem for $U(L)$).

Let $L$ be a Lie superalgebra over a field $k$ (of any characteristic), and
let $X=\\{x_{\lambda}:\lambda\in\Omega\\}$ be an ordered basis for $L$. Then
$U(L)$ admits a $k$-basis consisting of all monomials of the form
$x_{1}x_{2}\cdots x_{n}$, where $n\in\mathbb{N}$, $x_{i}\in X$, $x_{i}\leq
x_{i+1}$, and $x_{i}\neq x_{i+1}$ if $x_{i}\in L_{\overline{1}}$. In
particular, the inclusion $L=L^{\otimes 1}\hookrightarrow T(L)$ induces an
injective Lie superalgebra homomorphism
$\iota:L\hookrightarrow\operatorname{Lie}(U(L))$.

######  Remark 2.8.

Let $L$ be a Lie superalgebra over a field $k$ of characteristic $p$.

  1. (1)

The algebra $U(L)$ is a cocommutative Hopf superalgebra, with the subspace $L$
consisting of primitive elements. If $p=2$, then $U(L)$ is a Hopf algebra in
the ordinary sense.

  2. (2)

Suppose $p=2$, and let $\overline{L}$ be the ordinary
($\mathbb{Z}_{2}$-graded) Lie algebra obtained by forgetting the quadratic
operator $q:L_{\overline{1}}\to L_{\overline{0}}$. Then

$U(L)=U(\overline{L})/\langle y^{2}-q(y):y\in L_{\overline{1}}\rangle,$

and the canonical quotient map $U(\overline{L})\twoheadrightarrow U(L)$ is a
Hopf (super)algebra homomorphism.

  3. (3)

Suppose $p=2$. More-or-less by definition, a (left) $L$-supermodule is a
(left) supermodule over the universal enveloping superalgebra $U(L)$. Because
$y^{2}-q(y)=0$ in $U(L)$ for each $y\in L_{\overline{1}}$, but $y^{2}-q(y)\neq
0$ in $U(\overline{L})$, this shows that the category of $L$-supermodules is
not simply the category of ($\mathbb{Z}_{2}$-graded) $\overline{L}$-modules.

###  2.2. Restricted Lie superalgebras

######  Definition 2.9.

Let $k$ be a field of characteristic $p\geq 2$, and let $L$ be a Lie
superalgebra over $k$. We say that $L$ is a restricted Lie superalgebra if
there exists a map $x\mapsto x^{[p]}$ on $L_{\overline{0}}$ (called a _$p$
-map_ or a _$p$ -operation_) such that the following are satisfied for all
$x,y\in L_{\overline{0}}$:

  1. (R1)

$(\alpha\cdot x)^{[p]}=\alpha^{p}\cdot x^{[p]}$ for all $\alpha\in k$,

  2. (R2)

$\operatorname{ad}(x^{[p]})=\operatorname{ad}(x)^{p}$

  3. (R3)

$(x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}s_{i}(x,y)$, where $i\cdot
s_{i}(x,y)$ is the coefficient of $t^{i-1}$ in the formal expression
$\operatorname{ad}(t\cdot x+y)^{p-1}(x)$.

In other words, $L$ is a restricted Lie superalgebra if $L_{\overline{0}}$ is
a restricted Lie algebra in the usual sense, and $L_{\overline{1}}$ is a
restricted $L_{\overline{0}}$-module.

######  Remark 2.10.

For $p=2$, the condition (R3) becomes $(x+y)^{[2]}=x^{[2]}+y^{[2]}+[y,x]$.

######  Example 2.11.

Let $k$ be a field of characteristic $p\geq 2$, and let $A$ be a
$k$-superalgebra. Then the map $a\mapsto a^{p}$ defines a $p$-operation on
$A_{\overline{0}}$ that makes $\operatorname{Lie}(A)$ into a restricted Lie
superalgebra.

######  Definition 2.12.

Let $k$ be a field of characteristic $p\geq 2$, and let $L$ be a restricted
Lie superalgebra over $k$. The restricted enveloping algebra of $L$ is the
$k$-superalgebra

$V(L):=U(L)/\langle x^{p}-x^{[p]}:x\in L_{\overline{0}}\rangle.$

The proof of the following theorem is also standard.

######  Theorem 2.13 (Poincaré–Birkhoff–Witt Theorem for $V(L)$).

Let $L$ be a restricted Lie superalgebra over a field $k$ of characteristic
$p\geq 2$, and let $X=\\{x_{\lambda}:\lambda\in\Omega\\}$ be an ordered basis
for $L$. Then $V(L)$ admits a $k$-basis consisting of all monomials of the
form $x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{n}^{\lambda_{n}}$, where
$n\in\mathbb{N}$, $x_{i}\in X$, $x_{1}<x_{2}<\cdots<x_{n}$,
$0\leq\lambda_{i}<p$ if $x_{i}\in L_{\overline{0}}$, and $0\leq\lambda_{i}<2$
if $x_{i}\in L_{\overline{1}}$. The natural inclusion $L\hookrightarrow T(L)$
induces an injective homomorphism of restricted Lie superalgebras
$\iota:L\hookrightarrow\operatorname{Lie}(V(L))$.

Let $G$ be an abelian group, written additively. Recall that a _$G$ -graded
Lie algebra_ is a (ordinary) Lie algebra $L$ together with a family of
subspaces $(L_{g})_{g\in G}$ of $L$ such that $L=\bigoplus_{g\in G}L_{g}$ and
$[L_{g},L_{h}]\subseteq L_{g+h}$ for all $g,h\in G$. If $L$ is a (ordinary)
restricted Lie algebra with $p$-map $x\mapsto x^{[p]}$, then $L$ is a
_restricted $G$-graded Lie algebra_ if the $p$-map satisfies
$(L_{g})^{[p]}\subseteq L_{pg}$ for all $g\in G$.

######  Lemma 2.14.

Let $L$ be a restricted Lie superalgebra over a field $k$ of characteristic
$p=2$, with $p$-operation $x\mapsto x^{[2]}$ on $L_{\overline{0}}$, and let
$\overline{L}$ be the ordinary Lie algebra over $k$ obtained by forgetting the
quadratic operator $q:L_{\overline{1}}\to L_{\overline{0}}$. Given $z\in L$,
let $z=z_{\overline{0}}+z_{\overline{1}}$ be the decomposition of $z$ into its
even and odd parts. Then the map

(2.2) $z\mapsto
z^{\\{2\\}}:=(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})+[z_{\overline{1}},z_{\overline{0}}]$

defines a $p$-operation on $\overline{L}$ that makes $\overline{L}$ into a
restricted ($\mathbb{Z}_{2}$-graded) Lie algebra over $k$. The canonical
quotient map $U(\overline{L})\twoheadrightarrow U(L)$ induces a Hopf
(super)algebra isomorphism $V(\overline{L})\cong V(L)$.

###### Proof.

Let $x,y\in\overline{L}$, and let $x=x_{\overline{0}}+x_{\overline{1}}$ and
$y=y_{\overline{0}}+y_{\overline{1}}$ be the decompositions of $x$ and $y$
into their $\mathbb{Z}_{2}$-homogeneous components. First, for $\alpha\in k$,
one has

$\displaystyle(\alpha\cdot x)^{\\{2\\}}$ $\displaystyle=(\alpha\cdot
x_{\overline{0}}+\alpha\cdot x_{\overline{1}})^{\\{2\\}}$
$\displaystyle=(\alpha\cdot x_{\overline{0}})^{[2]}+q(\alpha\cdot
x_{\overline{1}})+[\alpha\cdot x_{\overline{1}},\alpha\cdot x_{\overline{0}}]$
$\displaystyle=\alpha^{2}\cdot(x_{\overline{0}})^{[2]}+\alpha^{2}\cdot
q(x_{\overline{1}})+\alpha^{2}\cdot[x_{\overline{1}},x_{\overline{0}}]$
$\displaystyle=\alpha^{2}\cdot x^{\\{2\\}},$

so (R1) is satisfied by the map $z\mapsto z^{\\{2\\}}$. Second,

$\displaystyle\operatorname{ad}(x^{\\{2\\}})(y)$
$\displaystyle=[x^{\\{2\\}},y]$
$\displaystyle=[(x_{\overline{0}})^{[2]}+q(x_{\overline{1}})+[x_{\overline{1}},x_{\overline{0}}],y]$
$\displaystyle=[(x_{\overline{0}})^{[2]},y]+[q(x_{\overline{1}}),y]+[[x_{\overline{1}},x_{\overline{0}}],y]$
$\displaystyle=[x_{\overline{0}},[x_{\overline{0}},y]]+[x_{\overline{1}},[x_{\overline{1}},y]]+[x_{\overline{1}},[x_{\overline{0}},y]]+[x_{\overline{0}},[x_{\overline{1}},y]]$
$\displaystyle=[x_{\overline{0}}+x_{\overline{1}},[x_{\overline{0}}+x_{\overline{1}},y]]$
$\displaystyle=\operatorname{ad}(x_{\overline{0}}+x_{\overline{1}})^{2}(y),$

so (R2) is satisfied by the map $z\mapsto z^{\\{2\\}}$. Third,

$\displaystyle(x+y)^{\\{2\\}}$
$\displaystyle=((x_{\overline{0}}+y_{\overline{0}})+(x_{\overline{1}}+y_{\overline{1}}))^{\\{2\\}}$
$\displaystyle=(x_{\overline{0}}+y_{\overline{0}})^{[2]}+q(x_{\overline{1}}+y_{\overline{1}})+[x_{\overline{1}}+y_{\overline{1}},x_{\overline{0}}+y_{\overline{0}}]$
$\displaystyle=(x_{\overline{0}})^{[2]}+(y_{\overline{0}})^{[2]}+[y_{\overline{0}},x_{\overline{0}}]+[x_{\overline{1}},y_{\overline{1}}]+q(x_{\overline{1}})+q(y_{\overline{1}})+[x_{\overline{1}},x_{\overline{0}}]+[x_{\overline{1}},y_{\overline{0}}]+[y_{\overline{1}},x_{\overline{0}}]+[y_{\overline{1}},y_{\overline{0}}]$
$\displaystyle=x^{\\{2\\}}+y^{\\{2\\}}+[y_{\overline{0}},x_{\overline{0}}]+[y_{\overline{1}},x_{\overline{1}}]+[y_{\overline{0}},x_{\overline{1}}]+[y_{\overline{1}},x_{\overline{0}}]$
$\displaystyle=x^{\\{2\\}}+y^{\\{2\\}}+[y_{\overline{0}}+y_{\overline{1}},x_{\overline{0}}+x_{\overline{1}}]$
$\displaystyle=x^{\\{2\\}}+y^{\\{2\\}}+[y,x]$

so (R3) is satisfied by the map $z\mapsto z^{\\{2\\}}$. Finally, let
$j:U(\overline{L})\twoheadrightarrow U(L)$ and $\pi:U(L)\twoheadrightarrow
V(L)$ be the canonical quotient maps. For
$z=z_{\overline{0}}+z_{\overline{1}}\in L$, in the algebra $U(\overline{L})$
one has

$\displaystyle z^{2}-z^{\\{2\\}}$
$\displaystyle=(z_{\overline{0}}+z_{\overline{1}})^{2}-\big{(}(z_{\overline{0}})^{[2]}+q(z_{1})+[z_{\overline{1}},z_{\overline{0}}]\big{)}$
$\displaystyle=\big{(}(z_{\overline{0}})^{2}+z_{\overline{0}}z_{\overline{1}}+z_{\overline{1}}z_{\overline{0}}+(z_{\overline{1}})^{2}\big{)}-\big{(}(z_{\overline{0}})^{[2]}+q(z_{1})+[z_{\overline{1}},z_{\overline{0}}]\big{)}$
$\displaystyle=\big{(}(z_{\overline{0}})^{2}-(z_{\overline{0}})^{[2]}\big{)}+\big{(}(z_{\overline{1}})^{2}-q(z_{1})\big{)}.$

This is a sum of a defining relation in $U(L)$ and a defining relation in
$V(L)$, so the composite algebra homomorphism $\pi\circ
j:U(\overline{L})\twoheadrightarrow V(L)$ factors through a surjective
homomorphism $V(\overline{L})\twoheadrightarrow V(L)$, which can then be seen
to be an injection (and hence an isomorphism) by Theorem 2.13. ∎

######  Remark 2.15.

The converse of Lemma 2.14 is true. That is, let $\overline{L}$ be a
restricted $\mathbb{Z}_{2}$-graded Lie algebra over a field of characteristic
$p=2$, with $2$-map $z\mapsto z^{\\{2\\}}$. For
$z\in\overline{L}_{\overline{1}}$, set
$q(z)=z^{\\{2\\}}\in\overline{L}_{2\cdot\overline{1}}=\overline{L}_{\overline{0}}$,
and for $z\in\overline{L}_{\overline{0}}$ set
$z^{[2]}=z^{\\{2\\}}\in\overline{L}_{2\cdot\overline{0}}=\overline{L}_{\overline{0}}$.
Then $q:\overline{L}_{\overline{1}}\to\overline{L}_{\overline{0}}$ defines a
quadratic operator that makes the underlying $\mathbb{Z}_{2}$-graded Lie
algebra of $\overline{L}$ into a Lie superalgebra, which we may denote $L$,
and the map $z\mapsto z^{[2]}$ defines a $p$-operation on $L_{\overline{0}}$
that makes $L$ into a restricted Lie superalgebra.

##  3\. Support for restricted Lie superalgebras in characteristic 2

###  3.1. Cohomology

Let $L$ be a finite-dimensional restricted Lie superalgebra over a field $k$
of characteristic $p=2$, and let $V(L)$ be its restricted enveloping algebra.
Let $\mathsf{sMod}=\mathsf{sMod}_{V(L)}$ be the category of left
$V(L)$-supermodules, and let $\mathsf{smod}=\mathsf{smod}_{V(L)}$ be the
subcategory of finite-dimensional $V(L)$-supermodules. An object in
$\mathsf{sMod}_{V(L)}$ is a $k$-superspace $M$ equipped with an even bilinear
map $V(L)\otimes M\to M$ that makes $M$ into a left $V(L)$-module. Given
$V(L)$-supermodules $M$ and $N$, one has

$\operatorname{Hom}_{\mathsf{sMod}}(M,N)=\operatorname{Hom}_{\mathsf{sMod}}(M,N)_{\overline{0}}\oplus\operatorname{Hom}_{\mathsf{sMod}}(M,N)_{\overline{1}}$

where $\operatorname{Hom}_{\mathsf{sMod}}(M,N)_{\overline{0}}$ (resp.
$\operatorname{Hom}_{\mathsf{sMod}}(M,N)_{\overline{1}}$) is the subspace of
all even (resp. odd) $k$-linear maps $f:M\to N$ such that $f(u.m)=u.f(m)$ for
all $u\in V(L)$ and $m\in M$.444If $p\neq 2$, then
$\operatorname{Hom}_{\mathsf{sMod}}(M,N)_{\overline{1}}$ consists of all odd
$k$-linear maps $f:M\to N$ such that $f(u.m)=(-1)^{\overline{u}}u.f(m)$ for
all homogeneous $u\in V(L)$ and all $m\in M$. The category
$\mathsf{sMod}_{V(L)}$ is not an abelian category (because the kernel or image
of a non-homogeneous homomorphism need not be the direct sum of its even and
odd subspaces), but the underlying even subcategory
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ of $\mathsf{sMod}_{V(L)}$, having the
same objects as $\mathsf{sMod}_{V(L)}$ but only the even homomorphisms as
morphisms, is an abelian category.

######  Remark 3.1.

For $p\neq 2$, the category $(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ identifies
with the category of left modules for the smash product algebra
$V(L)\\#k\mathbb{Z}_{2}$. Here $V(L)$ is considered as a
$k\mathbb{Z}_{2}$-module algebra, with the nontrivial element of
$\mathbb{Z}_{2}$ acting on $V(L)$ via the sign automorphism
$u\mapsto(-1)^{\overline{u}}u$. This identification of categories no longer
holds for $p=2$, because then $k\mathbb{Z}_{2}$-modules no longer decompose
into $\pm 1$ eigenspaces, and hence you can no longer use the action of
$k\mathbb{Z}_{2}$ to recover the superspace structure of modules, and you
cannot use $k\mathbb{Z}_{2}$-equivariance of maps to ensure that you are only
dealing with even $V(L)$-supermodule homomorphisms.

Each $V(L)$-supermodule is the image, via an even homomorphism, of a direct
sum of copies of $V(L)$ and its parity shift $\Pi(V(L))$.555For $p=2$, the
parity shift $\Pi(M)$ of a $V(L)$-supermodule $M$ is the $V(L)$-supermodule
that has the same underlying $k$-vector space as $M$, and the same action of
$V(L)$ on that $k$-vector space, but with
$\Pi(M)_{\overline{0}}=M_{\overline{1}}$ and
$\Pi(M)_{\overline{1}}=M_{\overline{0}}$. For $p\neq 2$, there is more than
once choice of parity change functor; see [15, §2.3]. Then the category
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ has enough projectives, so given
$V(L)$-supermodules $M$ and $N$, we can define
$\operatorname{Ext}_{\mathsf{sMod}}^{n}(M,N)$ to be the $n$-th right derived
functor of the contravariant functor
$\operatorname{Hom}_{\mathsf{sMod}}(-,N):(\mathsf{sMod}_{V(L)})_{\textup{ev}}\to\mathsf{svec}_{k}$,
i.e.,

$\operatorname{Ext}_{\mathsf{sMod}_{V(L)}}^{n}(M,N)=\big{(}R^{n}\operatorname{Hom}_{\mathsf{sMod}_{V(L)}}(-,N)\big{)}(M).$

The category $(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ also has enough
injectives: Given a $V(L)$-supermodule $N$, define
$\varphi:N\to\operatorname{Hom}_{k}(V(L),N)$ by $\varphi(n)=\varphi_{n}$,
where $\varphi_{n}(u)=(-1)^{\overline{u}\cdot\overline{n}}u.n$. Then $\varphi$
is an even $V(L)$-supermodule homomorphism, where the left $V(L)$-module
structure on $\operatorname{Hom}_{k}(V(L),N)$ comes from the right action of
$V(L)$ on itself, and the Hom-tensor adjunction

$\operatorname{Hom}_{V(L)}(M,\operatorname{Hom}_{k}(V(L),N))\cong\operatorname{Hom}_{k}(V(L)\otimes_{V(L)}M,N)\cong\operatorname{Hom}_{k}(M,N)$

implies that $\operatorname{Hom}_{k}(V(L),N)$ is injective in
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$. In fact, $V(L)$ is itself an injective
object in $(\mathsf{sMod}_{V(L)})_{\textup{ev}}$. This can be seen from (3.1),
using the fact that $V(L)=V(\overline{L})$ is self-injective in the category
of arbitrary $V(\overline{L})$-modules because $V(\overline{L})$ is a finite-
dimensional Hopf algebra; cf. [14, Lemma 2.3.2] for $p\neq 2$. From this it
follows that $(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ is a Frobenius category.

Let $\mathsf{Mod}=\mathsf{Mod}_{V(L)}$ be the category of arbitrary (i.e., not
necessarily $\mathbb{Z}_{2}$-graded) $V(L)$-modules. The forgetful functor,
$\operatorname{Forget}:\mathsf{sMod}_{V(L)}\to\mathsf{Mod}_{V(L)}$, which
simply ignores the $\mathbb{Z}_{2}$-gradings on objects, defines a canonical
identification

(3.1)
$\operatorname{Hom}_{\mathsf{sMod}}(M,N)=\operatorname{Hom}_{\mathsf{Mod}}(M,N)\quad\text{for
all $M,N\in\mathsf{sMod}_{V(L)}$.}$

More generally, since each object in $\mathsf{sMod}_{V(L)}$ can be resolved in
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ by free $V(L)$-modules, it follows that
the forgetful functor induces for each $n\in\mathbb{N}$ a canonical
identification

(3.2)
$\operatorname{Ext}_{\mathsf{sMod}}^{n}(M,N)=\operatorname{Ext}_{\mathsf{Mod}}^{n}(M,N)\quad\text{for
all $M,N\in\mathsf{sMod}_{V(L)}$.}$

From now on, we will simply write $\operatorname{Hom}_{V(L)}(M,N)$ and
$\operatorname{Ext}_{V(L)}^{n}(M,N)$ to denote the common values indicated by
(3.1) and (3.2).

The identification in (3.2) is also compatible with cup products. This can be
seen using the description of cup products given, for example, in [31, §9.3].
The key point to check to ensure that this description makes sense for
$\mathsf{sMod}_{V(L)}$ is that if $P$ and $Q$ are projective modules in
$\mathsf{sMod}_{V(L)}$, then the tensor product $P\otimes Q$ with the diagonal
$V(L)$-action is also projective in $\mathsf{sMod}_{V(L)}$. It suffices to
assume that $P$ is a free $V(L)$-supermodule, and then the proof of [31, Lemma
9.2.9] produces an isomorphism in $(\mathsf{sMod}_{V(L)})_{\textup{ev}}$
between $P\otimes Q$ and the free $V(L)$-supermodule $P\otimes Q_{triv}$. Here
$Q_{triv}$ denotes the $V(L)$-supermodule having the same underlying
superspace as $Q$, but with the action of $V(L)$ defined by the augmentation
map $\varepsilon:V(L)\to k$.

By Lemma 2.14, $V(L)$ identifies with $V(\overline{L})$, the restricted
enveloping algebra of the ordinary restricted Lie algebra $\overline{L}$,
whose underlying Lie algebra is obtained by forgetting the quadratic operator
$q:L_{\overline{1}}\to L_{\overline{0}}$, and whose $p$-operation $z\mapsto
z^{\\{2\\}}$ is defined by (2.2). Then the module categories
$\mathsf{Mod}_{V(L)}$ and $\mathsf{Mod}_{V(\overline{L})}$ are the same, so by
(3.2) one gets a canonical identification

(3.3)
$\operatorname{Ext}_{V(L)}^{\bullet}(M,N)=\operatorname{Ext}_{V(\overline{L})}^{\bullet}(M,N)\quad\text{for
all $M,N\in\mathsf{sMod}_{V(L)}$,}$

which is again evidently compatible with cup products. To emphasize, the left-
hand side of (3.3) represents the value calculated in the category of
$V(L)$-supermodules, while the right-hand side represents the value calculated
without regard for the $\mathbb{Z}_{2}$-gradings.

###  3.2. Support

Given a Hopf superalgebra $A$ over $k$, the cohomology ring
$\operatorname{H}^{\bullet}(A,k)=\operatorname{Ext}_{A}^{\bullet}(k,k)$ is a
graded-commutative superalgebra [14, Corollary 2.3.6]. For $p=2$, this means
that $\operatorname{H}^{\bullet}(A,k)$ is an ordinary commutative ring. We
define the cohomological spectrum of $A$, denoted $|A|$, to be

(3.4) $|A|=\operatorname{Spec}\operatorname{H}^{\bullet}(A,k),$

the prime ideal spectrum of $\operatorname{H}^{\bullet}(A,k)$, considered as a
topological space via the Zariski topology. If the field $k$ is algebraically
closed, one can take the maximal ideal spectrum instead, and then $|A|$ will
be an affine algebraic variety (assuming that
$\operatorname{H}^{\bullet}(A,k)$ is a finitely-generated $k$-algebra, which
it is for the examples we consider) rather than an affine scheme. We will
adopt this latter, more classical perspective later in Section 4 when
discussing support varieties for $U(L)$.

Given $A$-supermodules $M$ and $N$, let $I_{A}(M,N)$ be the annihilator ideal
for the cup product action of $\operatorname{H}^{\bullet}(A,k)$ on
$\operatorname{Ext}_{A}^{\bullet}(M,N)$. The ideal $I_{A}(M,N)$ is bi-
homogeneous (i.e., homogeneous with respect to both the ‘external’
cohomological degree and the ‘internal’ super degree) because the cup product
action is compatible with both the cohomological and super degrees. The
relative support variety of the pair $(M,N)$ is then defined to be

(3.5)
$|A|_{(M,N)}=\operatorname{Spec}\left(\operatorname{H}^{\bullet}(A,k)/I_{A}(M,N)\right),$

a Zariski closed subset of $|A|$, and the support variety of $M$ is defined to
be

(3.6)
$|A|_{M}=|A|_{(M,M)}=\operatorname{Spec}\left(\operatorname{H}^{\bullet}(A,k)/I_{A}(M,M)\right).$

For more details on basic notions concerning support varieties for Hopf
(super)algebras, see for example [14, §2.3] or [31, §8.3].

Taking $M=N=k$ in (3.3), the identification of cohomology rings
$\operatorname{H}^{\bullet}(V(L),k)=\operatorname{H}^{\bullet}(V(\overline{L}),k)$
implies the equality of cohomological spectra:

$|V(L)|=|V(\overline{L})|.$

By [27, Lemma 1.6] and [28, Theorem 5.2], there is a finite universal
homeomorphism between $|V(\overline{L})|$ and the restricted nullcone of
$\overline{L}$:

(3.7)
$|V(\overline{L})|\simeq\mathcal{N}_{1}(\overline{L}):=\\{z\in\overline{L}:z^{\\{2\\}}=0\\}.$

Writing $z=z_{\overline{0}}+z_{\overline{1}}$ for the decomposition of $z$
into its even and odd parts, one has $z^{\\{2\\}}=0$ if and only if
$(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})+[z_{\overline{1}},z_{\overline{0}}]=0$.
Since $(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})$ is of even superdegree
and $[z_{\overline{1}},z_{\overline{0}}]$ is of odd superdegree, this implies
that

(3.8) $z^{\\{2\\}}=0\quad\text{if and only
if}\quad(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})=0\text{ and
}[z_{\overline{1}},z_{\overline{0}}]=0.$

Thus, for the spectrum of $V(L)$ one gets

(3.9) $|V(L)|\simeq\\{z=z_{\overline{0}}+z_{\overline{1}}\in
L:(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})=0\text{ and
}[z_{\overline{1}},z_{\overline{0}}]=0\\}.$

This extends to $p=2$ the description of $|V(L)|$ we previously obtained for
characteristic $p\geq 3$ in [14, Proposition 5.4.4]. In fact, the result here
for $p=2$ is stronger than the result in [14], since the identification there
is only proved up to a finite morphism of varieties.

Next, the compatibility of (3.3) with cup products implies that

(3.10) $|V(L)|_{M}=|V(\overline{L})|_{M}\quad\text{for all
$M\in\mathsf{sMod}_{V(L)}$.}$

By [28, Corollary 6.3.1], the homeomorphism (3.7) restricts for $M$ finite-
dimensional to

(3.11) $|V(\overline{L})|_{M}\simeq\\{z\in\overline{L}:z^{\\{2\\}}=0\text{ and
}M|_{\langle z\rangle}\text{ is not free}\\}\cup\\{0\\}.$

Here $M|_{\langle z\rangle}$ denotes the restriction of $M$ to the
$k$-subalgebra of $V(\overline{L})$ generated by $z$. For $z\neq 0$ this
subalgebra has the form $k[z]/(z^{2})$, while for $z=0$ the subalgebra is just
the field $k$. (This subalgebra is not a sub-_superalgebra_ of
$V(\overline{L})$ unless $z$ is homogeneous.) An immediate consequence of
(3.10), (3.11), and the representation theory of $k[z]/(z^{2})$ is the
following (cf. [28, Theorem 7.2]):

######  Proposition 3.2 (Tensor Product Property).

Let $L$ be a finite-dimensional restricted Lie superalgebra over an
algebraically closed field of characteristic $p=2$, and let $M$ and $N$ be
finite-dimensional $V(L)$-supermodules. Then

$|V(L)|_{M\otimes N}=|V(L)|_{M}\cap|V(L)|_{N}.$

Next we give an alternative characterization of the condition that
$M|_{\langle z\rangle}$ is not free.

######  Lemma 3.3.

Let $M$ be a finite-dimensional $V(\overline{L})$-module, and let
$z=z_{\overline{0}}+z_{\overline{1}}\in\overline{L}$ with $z^{\\{2\\}}=0$. Set
$P=k[u,v]/(u^{2}+v^{2})$, and let $\sigma_{z}:P\to V(\overline{L})$ be the
homomorphism defined by $\sigma_{z}(u)=z_{\overline{0}}$ and
$\sigma_{z}(v)=z_{\overline{1}}$. Write $M\\!\\!\downarrow_{\sigma_{z}}$ for
the $P$-module obtained by pulling back $M$ along $\sigma_{z}$. Then for
$z\neq 0$,

$M|_{\langle z\rangle}$ is not free if and only if
$\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\infty$.

For $z=0$, one has that $M|_{\langle z\rangle}$ is free (i.e., $M$ is free
over $k$) and
$\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\infty$.

###### Proof.

Suppose $z\neq 0$, and let $Q$ be the $k$-subalgebra of $P$ generated by
$u+v$. Then $Q\cong k[z]/(z^{2})$, so $M|_{\langle z\rangle}$ is not free if
and only if
$\operatorname{projdim}_{Q}(M\\!\\!\downarrow_{\sigma_{z}})=\infty$. Next,
$P=Q[u]$ is a polynomial extension of the ring $Q$, so
$\operatorname{flatdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\infty$ if and
only if $\operatorname{flatdim}_{Q}(M\\!\\!\downarrow_{\sigma_{z}})=\infty$ by
[1, §5.3]. Finally,
$\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\operatorname{flatdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})$
and
$\operatorname{projdim}_{Q}(M\\!\\!\downarrow_{\sigma_{z}})=\operatorname{flatdim}_{Q}(M\\!\\!\downarrow_{\sigma_{z}})$
because $P$ and $Q$ are commutative Noetherian rings over which $M$ is
finitely generated. If $z=0$, then $\sigma_{z}:P\to k$ is the augmentation
map, and
$\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\operatorname{projdim}_{P}(k)=\infty$.
∎

######  Proposition 3.4.

Let $L$ be a finite-dimensional restricted Lie superalgebra over an
algebraically closed field of characteristic $p=2$, and let $M$ be a finite-
dimensional $V(L)$-supermodule. Let $P$ and $\sigma_{z}:P\to V(\overline{L})$
be as defined in Lemma 3.3. Then there exists a homeomorphism

(3.12) $|V(L)|_{M}\simeq\\{z\in
L:(z_{\overline{0}})^{[2]}+q(z_{\overline{1}})=0\text{ and
}[z_{\overline{1}},z_{\overline{0}}]=0\text{ and
}\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})=\infty\\}.$

A $V(L)$-supermodule $M$ is projective if and only if $|V(L)|_{M}=\\{0\\}$, or
equivalently, if and only if
$\operatorname{projdim}_{P}(M\\!\\!\downarrow_{\sigma_{z}})<\infty$ for all
nonzero $z\in L$ such that $z^{\\{2\\}}=0$.

###### Proof.

The asserted homeomorphism follows immediately from (3.8), (3.10), (3.11), and
the lemma. Next, the fact that the forgetful functor
$\operatorname{Forget}:\mathsf{sMod}_{V(L)}\to\mathsf{Mod}_{V(L)}=\mathsf{Mod}_{V(\overline{L})}$
is surjective on morphisms implies that if a $V(L)$-supermodule is projective
as an object in $\mathsf{Mod}_{V(\overline{L})}$, then it is also projective
as an object in (the underlying even subcategory of) $\mathsf{sMod}_{V(L)}$.
Then the last sentence of of the proposition is true by the well-known fact
that a finite-dimensional $V(\overline{L})$-module is projective if and and
only if $|V(\overline{L})|_{M}=\\{0\\}$; cf. [19, Proposition 1.5]. ∎

######  Remark 3.5.

In [16, Lemma 3.1.3], we interpreted our previous work from [15] to show that
an identification analogous to (3.12) holds for the support varieties of
arbitrary finite-dimensional _$p$ -nilpotent_ restricted Lie superalgebras
over (algebraically closed) fields of characteristic $p\geq 3$. In [15, §1.2],
we conjectured that a similar identification should hold for all finite-
dimensional restricted Lie superalgebras (and more generally, for all
infinitesimal supergroup schemes) in characteristic $p\geq 3$. The primary
obstruction to proving such a generalization in characteristic $p\geq 3$ is
the lack of a characterization of projectivity similar to that stated in the
last sentence of Proposition 3.4.

Recall that a closed set $W\subseteq|A|$ is _conical_ if it is defined by a
homogeneous (respect to the cohomological degree) ideal in
$\operatorname{H}^{\bullet}(A,k)$. Given a Hopf superalgebra $A$, let us say
that a closed set $W\subseteq|A|$ is _super-conical_ if it is defined by a bi-
homogeneous ideal in $\operatorname{H}^{\bullet}(A,k)$, i.e., an ideal that is
homogeneous with respect to both the ‘external’ cohomological degree and the
‘internal’ super degree. Given a $V(L)$-supermodule $M$, the support variety
$|V(L)|_{M}$ is super-conical because its defining ideal $I_{V(L)}(M,M)$ is
automatically bi-homogeneous. The next proposition states that every super-
conical subset of $|V(L)|$ is of this form.

######  Proposition 3.6 (Realizability).

Let $L$ be a finite-dimensional restricted Lie superalgebra over an
algebraically closed field of characteristic $2$. Then $W\subseteq|V(L)|$ is
the support variety $|V(L)|_{M}$ of a finite-dimensional $V(L)$-supermodule if
and only if $W$ is a closed, super-conical subset of $|V(L)|$.

###### Proof.

We have already explained the ‘only if’ portion of the statement, so we just
need to prove the ‘if’ portion. Let $W\subseteq|V(L)|$ be a closed, super-
conical subset. Then $W$ is defined by a bi-homogeneous ideal in
$\operatorname{H}^{\bullet}(V(L),k)$, which in turn is generated by a finite
set of bi-homogeneous elements. By the Tensor Product Property (Proposition
3.2), it suffices to show that if $W=Z(\zeta)$ is the closed set defined by a
single bi-homogeneous element $\zeta\in\operatorname{H}^{n}(V(L),k)$, then
$W=|V(L)|_{M}$ for some finite-dimensional $V(L)$-supermodule $M$.

Choose a projective resolution $(P_{\bullet},d)$ in
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ of the trivial $V(L)$-supermodule $k$,
and a homogeneous (with respect to the super degree) cocycle representative
$g:P_{n}\to k$ for $\zeta$. Let $M$ be the kernel of the induced map
$\widehat{g}:P_{n}/d(P_{n+1})\to k$. (As an ungraded module in
$\mathsf{Mod}_{V(\overline{L})}$, and perhaps modulo a projective summand,
$P_{n}/d(P_{n+1})$ is the $n$-th syzygy module $\Omega^{n}(k)$ for the trivial
$V(\overline{L})$-module $k$.) Then $M$ is a $V(L)$-supermodule because
$d:P_{n+1}\to P_{n}$ and $g:P_{n}\to k$ are both homogeneous
$V(L)$-supermodule homomorphisms. Now the proof of [28, Theorem 7.5] (cf. also
[19, §4]) shows that $|V(\overline{L})|_{M}=Z(\zeta)$, hence
$|V(L)|_{M}=Z(\zeta)$ by (3.10). ∎

###  3.3. $\operatorname{Proj}(A)$ and $\operatorname{Proj}_{s}(A)$

In anticipation of Section 3.4, we now discuss a modification of the
projective spectrum that is useful for graded(-commutative) superalgebras in
characteristic $2$. If we were in characteristic other than $2$, then
$\mathbb{Z}_{2}$-gradings could be encoded by an action of the cyclic group
$C_{2}$ and the following would essentially be the theory of
$C_{2}\text{-}\operatorname{Proj}(A)$ as described in the literature; see for
example [24]. Since we are in characteristic $2$, and for the sake of
completeness, we provide the relevant details.

Let $A$ be a commutative $k$-algebra that is graded by
$\mathbb{N}\times\mathbb{Z}_{2}$. Let $A=\bigoplus_{i\in\mathbb{N}}A^{i}$ be
the decomposition of $A$ into its $\mathbb{N}$-graded components, and assume
that $A^{0}=k$. Let $\operatorname{Proj}(A)$ be the subset of
$\operatorname{Spec}(A)$ consisting of the prime ideals that are homogeneous
with respect to the $\mathbb{N}$-grading on $A$, and which are properly
contained in the maximal ideal $A^{+}=\bigoplus_{i\geq 1}A^{i}$. A subset $W$
of $\operatorname{Proj}(A)$ is closed in the Zariski topology if there exists
an $\mathbb{N}$-homogeneous ideal $I$ of $A$ such that

$W=Z(I):=\\{\mathfrak{p}\in\operatorname{Proj}(A):I\subseteq\mathfrak{p}\\}.$

We say that an ideal $I$ of $A$ is _bi-homogeneous_ if it is homogeneous with
respect to the full $\mathbb{N}\times\mathbb{Z}_{2}$-grading on $A$. For
example, the ideal $A^{+}$ is bi-homogeneous. Given an ideal $I$ of $A$, let
$I_{s}\subseteq I$ be the largest bi-homogeneous subideal of $I$.
Equivalently, $I_{s}$ is the ideal in $A$ generated by the bi-homogeneous
elements of $I$. Of course, $I=I_{s}$ if $I$ is bi-homogeneous. If $J$ is a
bi-homogeneous ideal and $I$ is an arbitrary ideal, then $J\subseteq I$ if and
only if $J\subseteq I_{s}$. If $\\{I_{\lambda}:\lambda\in\Omega\\}$ is any
family of ideals in $A$, then
$(\bigcap_{\lambda\in\Omega}I_{\lambda})_{s}=\bigcap_{\lambda\in\Omega}(I_{\lambda})_{s}$.

Given an ideal $I$ of $A$, let $\sqrt{I}$ be the usual radical of $I$, and set
$\sqrt[s]{I}=(\sqrt{I})_{s}$. Then $I_{s}\subseteq\sqrt[s]{I}$, and
$\sqrt[s]{I}$ has the property that if $a\in A$ is a bi-homogeneous element
with $a^{n}\in I_{s}$ for some $n\geq 1$, then $a\in\sqrt[s]{I}$. We say that
a bi-homogeneous ideal $J$ is an _$s$ -radical ideal_ if $J=\sqrt[s]{J}$. We
say that a bi-homogeneous ideal $P$ is _$s$ -prime_ if whenever $I$ and $J$
are bi-homogeneous ideals with $IJ\subseteq P$, then either $I\subseteq P$ or
$J\subseteq P$. Equivalently, $P$ is $s$-prime if whenever $a$ and $b$ are bi-
homogeneous elements with $ab\in P$, either $a\in P$ or $b\in P$. If $P$ is a
prime ideal in $A$, then $P_{s}$ is an $s$-prime ideal. Every $s$-prime ideal
in $A$ is an $s$-radical ideal.

######  Lemma 3.7.

Let $P$ be an $s$-prime ideal of $A$. Then $\sqrt{P}$ is an
$\mathbb{N}$-homogeneous prime ideal of $A$.

###### Proof.

Ignoring the $\mathbb{Z}_{2}$-gradings, $P$ is an $\mathbb{N}$-homogeneous
ideal of $A$, so its radical is also an $\mathbb{N}$-homogeneous ideal. To
show that $\sqrt{P}$ is prime, let $x,y\in A$, and suppose $xy\in\sqrt{P}$,
say with $x^{n}y^{n}=(xy)^{n}\in P$. Replacing $x$ and $y$ with $x^{n}$ and
$y^{n}$, we may assume that $xy\in P$. Then also $x^{2}y^{2}=(xy)^{2}\in P$.
Write $x=\sum x_{i}$ with $x_{i}\in A^{i}$, and write
$x_{i}=x_{i,\overline{0}}+x_{i,\overline{1}}$ with
$x_{i,\overline{0}}\in(A^{i})_{\overline{0}}$ and
$x_{i,\overline{1}}\in(A^{i})_{\overline{1}}$. Since $A$ is commutative, the
binomial theorem modulo $2$ implies that
$x^{2}=\sum(x_{i})^{2}=\sum[(x_{i,\overline{0}})^{2}+(x_{i,\overline{1}})^{2}]$.
Then $x^{2}$ is a sum of terms of even superdegree, and we can rewrite $x^{2}$
as $x^{2}=\sum x_{2i}^{\prime}$, with
$x_{2i}^{\prime}\in(A^{2i})_{\overline{0}}$. Similarly, $y^{2}=\sum
y_{2i}^{\prime}$ with $y_{2i}^{\prime}\in(A^{2i})_{\overline{0}}$.

Now suppose that $x^{2}\notin P$ and $y^{2}\notin P$. Then there exist minimal
indices $s$ and $t$ such that $x_{2s}^{\prime}\notin P$ and
$y_{2t}^{\prime}\notin P$. Since $x^{2}$ and $y^{2}$ are each of even
superdegree, the $\mathbb{N}$-graded components of $x^{2}y^{2}$ are each bi-
homogeneous. Since $P$ is bi-homogeneous, and since $x^{2}y^{2}\in P$, this
means in particular that the component of $\mathbb{N}$-degree $s+t$ in
$x^{2}y^{2}$ is an element of $P$. This component is equal to
$\sum_{\ell\in\mathbb{Z}}x_{2s-\ell}^{\prime}y_{2t+\ell}^{\prime}$. By the
minimality of $s$ and $t$, each summand
$x_{2s-\ell}^{\prime}y_{2t+\ell}^{\prime}$ with $\ell\neq 0$ is an element of
$P$, because at least one of the factors is an element of $P$ by assumption.
Then $x_{2s}^{\prime}y_{2t}^{\prime}$ must also be an element of $P$. The
factors $x_{2s}^{\prime}$ and $y_{2t}^{\prime}$ are each bi-homogeneous, so
the fact that $P$ is an $s$-prime ideal implies that either
$x_{2s}^{\prime}\in P$ or $y_{2t}^{\prime}\in P$, in contradiction to the
assumptions on $s$ and $t$. Thus, it must be the case that either $x^{2}\in P$
or $y^{2}\in P$, and hence $x\in\sqrt{P}$ or $y\in\sqrt{P}$. ∎

######  Definition 3.8.

Define $\operatorname{Proj}_{s}(A)$ to be the set of all (bi-homogeneous)
$s$-prime ideals in $A$ that are properly contained in the (bi-homogeneous,
$s$-prime) maximal ideal $A^{+}$. We say that a subset $W$ of
$\operatorname{Proj}_{s}(A)$ is _closed_ if there exists a bi-homogeneous
ideal $I$ of $A$ such that

$W=Z_{s}(I):=\\{P\in\operatorname{Proj}_{s}(A):I\subseteq P\\}.$

The properties listed below are now straightforward to verify. Unless stated
otherwise, $I$ and $J$ denote bi-homogeneous ideals in $A$.

  1. (P1)

$Z_{s}(I)\cup Z_{s}(J)=Z_{s}(IJ)$.

  2. (P2)

$Z_{s}(I)\cap Z_{s}(J)=Z_{s}(I+J)$.

  3. (P3)

If $\mathfrak{p}\in\operatorname{Proj}(A)$, then
$\mathfrak{p}_{s}\in\operatorname{Proj}_{s}(A)$.

  4. (P4)

If $\varphi:\operatorname{Proj}(A)\to\operatorname{Proj}_{s}(A)$ is the
function $\varphi(\mathfrak{p})=\mathfrak{p}_{s}$, then
$\varphi^{-1}(Z_{s}(I))=Z(I)$.

  5. (P5)

If $P\in\operatorname{Proj}_{s}(A)$, then $\sqrt{P}\in\operatorname{Proj}(A)$.

  6. (P6)

If $\psi:\operatorname{Proj}_{s}(A)\to\operatorname{Proj}(A)$ is the function
$\psi(P)=\sqrt{P}$, then $(\varphi\circ\psi)(P)=\sqrt[s]{P}=P$.

The first two properties imply that finite unions and arbitrary intersections
of closed sets are again closed. Thus the closed subsets of
$\operatorname{Proj}_{s}(A)$ define a topology, which we call the Zariski
topology on $\operatorname{Proj}_{s}(A)$. Property (P4) then implies that
$\varphi:\operatorname{Proj}(A)\to\operatorname{Proj}_{s}(A)$ is a continuous
function.

If $K$ is an arbitrary $\mathbb{N}$-homogeneous ideal in $A$, one has
$\varphi(Z(K))\subseteq Z_{s}(K_{s})$. On the other hand, if $I$ is a bi-
homogeneous ideal, and if $P\in Z_{s}(I)$, then $I\subseteq
P\subseteq\sqrt{P}$. Then property (P6) implies:

  1. (P7)

If $I$ is bi-homogeneous, then $\varphi(Z(I))=Z_{s}(I)$.

Next observe that if $Z_{s}(I)\subseteq Z_{s}(J)$, then

$Z(I)=\varphi^{-1}(Z_{s}(I))\subseteq\varphi^{-1}(Z_{s}(J))=Z(J).$

But $Z(I)\subseteq Z(J)$ implies that $\sqrt{I}\supseteq\sqrt{J}$, and hence
$\sqrt[s]{I}\supseteq\sqrt[s]{J}$. Now one can check that:

  1. (P8)

$Z_{s}(I)=Z_{s}(\sqrt[s]{I})$.

  2. (P9)

If $I\subseteq J$, then $Z_{s}(I)\supseteq Z_{s}(J)$.

  3. (P10)

If $Z_{s}(I)\subseteq Z_{s}(J)$, then $\sqrt[s]{I}\supseteq\sqrt[s]{J}$.

  4. (P11)

If $A$ is a Noetherian ring, then $\operatorname{Proj}_{s}(A)$ is a Noetherian
topological space, i.e., if $Z_{s}(I_{1})\supseteq Z_{s}(I_{2})\supseteq
Z_{s}(I_{3})\supseteq\cdots$ is a descending chain of closed sets, then there
exists an integer $m$ such that $Z_{s}(I_{j})=Z_{s}(I_{m})$ for all $j\geq m$.

######  Lemma 3.9.

If $W=Z_{s}(I)$ is an irreducible closed set in $\operatorname{Proj}_{s}(A)$,
then $\sqrt[s]{I}$ is an $s$-prime ideal. In particular, if $W$ is nonempty,
then $W$ is the closure of the point
$\sqrt[s]{I}\in\operatorname{Proj}_{s}(A)$, and $\sqrt[s]{I}$ is the unique
point in $\operatorname{Proj}_{s}(A)$ with this property.

###### Proof.

To show that $\sqrt[s]{I}$ is an $s$-prime ideal, one can apply the usual
argument for $\operatorname{Spec}(A)$; see for example the proof of [11,
Proposition 4.5.3]. Then $W=Z_{s}(I)=Z_{s}(\sqrt[s]{I})$ by property (P8)
above, while property (P10) implies that if $W=\overline{\\{P\\}}=Z_{s}(P)$
for some $P\in\operatorname{Proj}_{s}(A)$, then $P=\sqrt[s]{I}$. ∎

Combining property (P11) and Lemma 3.9, we get:

  1. (P12)

If $A$ is a Noetherian ring, then $\operatorname{Proj}_{s}(A)$ is a Zariski
space, i.e., it is a Noetherian topological space, and every nonempty closed
irreducible subset is the closure of a unique point.

###  3.4. Tensor triangular geometry

Now take $A=\operatorname{H}^{\bullet}(V(L),k)$. In this section we discuss
how the geometry of
$\operatorname{Proj}_{s}(A)=\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(L),k))$
classifies certain categories of $V(L)$-supermodules. First we recall the
corresponding results for the category of ungraded $V(\overline{L})$-modules.

Let $\mathbf{K}$ be a symmetric tensor triangulated category. Recall that a
triangulated subcategory of $\mathbf{K}$ is _thick_ if it is closed under
taking direct summands, and is a _tensor ideal_ (or _$\otimes$ -ideal_) if it
is closed under $M\otimes-$ for any object $M$ in $\mathbf{K}$. A
$\otimes$-ideal $\mathcal{J}$ is _radical_ if whenever $M^{\otimes
n}\in\mathcal{J}$ for some $n\geq 1$, then also $M\in\mathcal{J}$. A thick
$\otimes$-ideal $\mathcal{P}$ is _prime_ if whenever $M,N\in\mathbf{K}$ and
$M\otimes N\in\mathcal{P}$, then either $M\in\mathcal{P}$ or
$N\in\mathcal{P}$. The _Balmer spectrum_ of $\mathbf{K}$, denoted
$\operatorname{Spc}(\mathbf{K})$, is the set of all proper prime thick
$\otimes$-ideals of $\mathbf{K}$, considered as a topological space via the
Zariski topology. Write $2^{\operatorname{Spc}(\mathbf{K})}$ for the power set
of $\operatorname{Spc}(\mathbf{K})$. Then the support function

$\operatorname{supp}^{\prime}:\mathbf{K}\to
2^{\operatorname{Spc}(\mathbf{K})}$

is defined by
$\operatorname{supp}^{\prime}(M)=\\{\mathcal{P}\in\operatorname{Spc}(\mathbf{K}):M\notin\mathcal{P}\\}$
for each object $M\in\mathbf{K}$.

Let $\mathsf{stmod}_{V(\overline{L})}$ (resp.,
$\mathsf{StMod}_{V(\overline{L})}$) be the stable module category of finite-
dimensional (resp., all) $V(\overline{L})$-modules. These are the categories
obtained from $\mathsf{mod}_{V(\overline{L})}$ and
$\mathsf{Mod}_{V(\overline{L})}$, respectively, by quotienting out the
morphisms that factor through a projective. The objects of the stable category
are the same as in the corresponding ordinary module category, but with
projective objects now isomorphic to zero. The categories
$\mathsf{stmod}_{V(\overline{L})}$ and $\mathsf{StMod}_{V(\overline{L})}$ are
tensor triangulated categories, with suspension $\Sigma$ equal to
$\Omega^{-1}(-)$, the inverse of the syzygy functor. The category
$\mathsf{StMod}_{V(\overline{L})}$ is compactly generated, with subcategory of
compact objects
$(\mathsf{StMod}_{V(\overline{L})})^{c}=\mathsf{stmod}_{V(\overline{L})}$.

Now set
$X=\operatorname{Proj}(\operatorname{H}^{\bullet}(V(\overline{L}),k))$. In
[3], Benson, Iyengar, Krause, and Pevtsova use localization functors to define
a support function

(3.13) $\operatorname{supp}:\mathsf{StMod}_{V(\overline{L})}\to 2^{X}$

that satisfies the properties listed below.999Again, the results in [3] are
stated in the language of finite group schemes, so the reader must translate
to the category of rational $G$-modules, where $k[G]=V(\mathfrak{g})^{*}$.
Except when indicated, $M$, $M_{i}$, and $N$ denote arbitrary objects in
$\mathsf{StMod}_{V(\overline{L})}$.

  1. (S0)

$\operatorname{supp}(0)=\emptyset$

  2. (S1)

$\operatorname{supp}(k)=X$

  3. (S2)

$\operatorname{supp}(M\oplus
N)=\operatorname{supp}(M)\cup\operatorname{supp}(N)$

  4. (S3)

$\operatorname{supp}(\Sigma M)=\operatorname{supp}(M)$

  5. (S4)

For each distinguished triangle $M_{1}\to M_{2}\to M_{3}\to\Sigma M_{1}$ in
$\mathsf{StMod}_{V(\overline{L})}$,

$\operatorname{supp}(M_{2})\subseteq\operatorname{supp}(M_{1})\cup\operatorname{supp}(M_{3})$

  6. (S5)

Tensor Product Property: For all compact $M$ and arbitrary $N$,

$\operatorname{supp}(M\otimes
N)=\operatorname{supp}(M)\cap\operatorname{supp}(N).$

  7. (S6)

$\operatorname{supp}(\bigoplus_{i\in I}M_{i})=\bigcup_{i\in
I}\operatorname{supp}(M_{i})$ for any set $I$.

  8. (S7)

For all compact $M$, $\operatorname{supp}(M)$ is Zariski closed with quasi-
compact complement.111111As used in algebraic geometry, the term _quasi-
compact_ means that every open cover has a finite subcover.

  9. (S8)

For every Zariski closed subset $W\subseteq X$ with quasi-compact complement,
there exists a compact object $M$ such that $\operatorname{supp}(M)=W$.

  10. (S9)

If $\operatorname{supp}(M)=\emptyset$, then $M\cong 0$.

  11. (S10)

For each finite-dimensional $V(\overline{L})$-module $M$,
$\operatorname{supp}(M^{*})=\operatorname{supp}(M)$.

We note that the ‘quasi-compact complement’ condition in (S7) and (S8) is
automatically satisfied by any Zariski closed subset of $X$, as a consequence
of the fact that $\operatorname{H}^{\bullet}(V(\overline{L}),k)$ is a
finitely-generated commutative $k$-algebra and hence a Noetherian ring.
Elaborating further on (S7), if $M$ is a compact object in
$\mathsf{StMod}_{V(\overline{L})}$ (equivalently, if $M$ is isomorphic, modulo
projective summands, to a finite-dimensional $V(\overline{L})$-module), then
$\operatorname{supp}(M)$ is the Zariski closed subset of $X$ defined by
$I(M)$, the annihilator ideal for the cup product action of
$\operatorname{H}^{\bullet}(V(\overline{L}),k)$ on
$\operatorname{Ext}_{V(\overline{L})}^{\bullet}(M,M)$.

Properties (S0)–(S9) imply by Theorem 1.5 of [12] that the restriction of
(3.13) to $\mathsf{stmod}_{V(\overline{L})}$ is a classifying support datum in
the sense of [12, Theorem 2.19]; cf. also Theorems 3.4.1 and 3.5.1 of [8].
Then there is a canonical homeomorphism from
$X=\operatorname{Proj}(\operatorname{H}^{\bullet}(V(\overline{L}),k))$ to the
Balmer spectrum of the category $\mathsf{stmod}_{V(\overline{L})}$,

(3.14)
$\operatorname{Proj}(\operatorname{H}^{\bullet}(V(\overline{L}),k))\simeq\operatorname{Spc}(\mathsf{stmod}_{V(\overline{L})}),$

which identifies the support functions $\operatorname{supp}$ and
$\operatorname{supp}^{\prime}$. Furthermore, there are inverse bijections

(3.15) $\\{\text{specialization closed subsets $\mathcal{V}$ of
$\operatorname{Proj}(\operatorname{H}^{\bullet}(V(\overline{L}),k))$}\\}\overset{\Gamma}{\underset{\Theta}{\rightleftarrows}}\\\
\\{\text{thick $\otimes$-ideal subcategories $\mathcal{J}$ of
$\mathsf{stmod}_{V(\overline{L})}$}\\},$

defined by
$\Gamma(\mathcal{V})=\\{M\in\mathsf{stmod}_{V(\overline{L})}:\operatorname{supp}(M)\subseteq\mathcal{V}\\}$
and $\Theta(\mathcal{J})=\bigcup_{M\in\mathcal{J}}\operatorname{supp}(M)$.
(Recall that a subset of a topological space is _specialization closed_ if it
is a union of arbitrarily many closed sets.) We note that [12, Theorem 2.19]
is stated in terms of _radical_ thick $\otimes$-ideals, but using the fact
that each finite-dimensional $V(\overline{L})$-module $M$ is a direct summand
of $M\otimes M^{*}\otimes M$, one can see that every thick $\otimes$-ideal of
$\mathsf{stmod}_{V(\overline{L})}$ is automatically radical.

Now we consider the super setting. Recall from Section 3.1 that
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$ is a Frobenius category. Then we can
consider the corresponding stable category (see, e.g., [22, I.2]), which we
denote $\mathsf{St\text{-}sMod}_{V(L)}$. It is a compactly generated tensor-
triangulated category, with subcategory of compact objects
$(\mathsf{St\text{-}sMod}_{V(L)})^{c}=\mathsf{st\text{-}smod}_{V(L)}$, the
stable category of $(\mathsf{smod}_{V(L)})_{\textup{ev}}$. The forgetful
functor $\mathsf{sMod}_{V(L)}\to\mathsf{Mod}_{V(L)}$ induces a (exact)
triangulated functor

$\operatorname{Forget}:\mathsf{St\text{-}sMod}_{V(L)}\to\mathsf{StMod}_{V(L)}=\mathsf{StMod}_{V(\overline{L})}.$

In particular, the forgetful functor maps projective objects in
$\mathsf{sMod}_{V(L)}$ to projective objects in $\mathsf{Mod}_{V(L)}$. Via
this functor, we will consider objects from $\mathsf{St\text{-}sMod}_{V(L)}$
as objects in $\mathsf{StMod}_{V(L)}$.

Our goal now is to establish analogues of (3.14) and (3.15) for
$\mathsf{st\text{-}smod}_{V(L)}$. Set

$X_{s}=\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(L),k)),$

and let

$\varphi:\operatorname{Proj}(\operatorname{H}^{\bullet}(V(L),k))\to\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(L),k))$

be the function $\mathfrak{p}\mapsto\mathfrak{p}_{s}$ as defined in (P4)
above. Using the fact that $\operatorname{H}^{\bullet}(V(L),k)$ is a
Noetherian ring and the properties listed in Section 3.3, one can again deduce
that every Zariski closed set in $X_{s}$ has quasi-compact complement. Define

$\operatorname{supp}_{s}:\mathsf{St\text{-}sMod}_{V(L)}\to 2^{X_{s}}$

by

$\operatorname{supp}_{s}(M)=\varphi\left(\operatorname{supp}(\operatorname{Forget}(M))\right).$

######  Theorem 3.10.

Let $L$ be a finite-dimensional restricted Lie superalgebra over a field $k$
of characteristic $p=2$. There is a canonical homeomorphism from
$X_{s}=\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(L),k))$ to the
Balmer spectrum of the category $\mathsf{st\text{-}smod}_{V(L)}$,

(3.16)
$\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(\overline{L}),k))\simeq\operatorname{Spc}(\mathsf{stmod}_{V(L)}),$

which identifies $\operatorname{supp}_{s}$ with
$\operatorname{supp}^{\prime}$. Furthermore, there are inverse bijections

(3.17) $\\{\text{specializataion closed subsets $\mathcal{V}$ of
$\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(V(L),k))$}\\}\overset{\Gamma}{\underset{\Theta}{\rightleftarrows}}\\\
\\{\text{thick $\otimes$-ideal subcategories $\mathcal{J}$ of
$\mathsf{st\text{-}smod}_{V(L)}$}\\},$

defined by
$\Gamma(\mathcal{V})=\\{M\in\mathsf{st\text{-}smod}_{V(L)}:\operatorname{supp}_{s}(M)\subseteq\mathcal{V}\\}$
and $\Theta(\mathcal{J})=\bigcup_{M\in\mathcal{J}}\operatorname{supp}_{s}(M)$.

###### Proof.

The theorem follows directly from [12, Theorem 1.5], once we verify that the
analogues of (S0)–(S9) hold for $\operatorname{supp}_{s}$. Many of the
properties are immediate consequences of the fact that the forgetful functor
$\operatorname{Forget}:\mathsf{St\text{-}sMod}_{V(L)}\to\mathsf{StMod}_{V(L)}$
is a triangulated functor, and the fact that the function $\varphi:X\to X_{s}$
is a continuous surjection. We will make a few comments about those properties
that are not entirely immediate.

(S7): Let $M$ be a finite-dimensional $V(L)$-supermodule. Then
$\operatorname{supp}(M)=Z(I(M))$, the Zariski closed subset of
$\operatorname{Proj}\operatorname{H}^{\bullet}(V(L),k)$ defined by $I(M)$, the
annihilator ideal for the cup product action of
$\operatorname{H}^{\bullet}(V(L),k)$ on
$\operatorname{Ext}_{V(L)}^{\bullet}(M,M)$. The ideal $I(M)$ is bi-homogeneous
in $\operatorname{H}^{\bullet}(V(L),k)$, so then

$\operatorname{supp}_{s}(M)=\varphi\left(Z(I(M))\right)=Z_{s}(I(M))$

by property (P7). So $\operatorname{supp}_{s}(M)$ is Zariski closed (with
quasi-compact complement) in $X_{s}$.

(S5): Let $M$ be a finite-dimensional $V(L)$-supermodule, and let $N$ be an
arbitrary $V(L)$-supermodule. By property (S5) for $\operatorname{supp}$, one
has $\operatorname{supp}(M\otimes
N)=\operatorname{supp}(M)\cap\operatorname{supp}(N)$. Then

$\operatorname{supp}_{s}(M\otimes
N)=\varphi\left(\operatorname{supp}(M)\cap\operatorname{supp}(N)\right)\subseteq\operatorname{supp}_{s}(M)\cap\operatorname{supp}_{s}(N).$

For the reverse set inclusion, let
$P\in\operatorname{supp}_{s}(M)\cap\operatorname{supp}_{s}(N)$. Then there
exist $\mathfrak{p}\in\operatorname{supp}(M)$ and
$\mathfrak{q}\in\operatorname{supp}(N)$ such that
$\varphi(\mathfrak{p})=P=\varphi(\mathfrak{q})$, i.e.,
$\mathfrak{p}_{s}=P=\mathfrak{q}_{s}$. Since $\operatorname{supp}(M)=Z(I(M))$
and $I(M)$ is bi-homogeneous, this implies that
$I(M)\subseteq\mathfrak{p}_{s}=\mathfrak{q}_{s}\subseteq\mathfrak{q}$. Then
$\mathfrak{q}\in\operatorname{supp}(M)\cap\operatorname{supp}(N)=\operatorname{supp}(M\otimes
N)$, and
$P=\mathfrak{q}_{s}=\varphi(\mathfrak{q})\in\operatorname{supp}_{s}(M\otimes
N)$.

(S8): Let $W\subseteq X_{s}$ be a Zarsiki closed set (with quasi-compact
complement). Then $W=Z_{s}(I)$ for some bi-homogeneous ideal $I$ in
$\operatorname{H}^{\bullet}(V(L),k)$. By Proposition 3.6, there exists a
finite-dimensional $V(L)$-supermodule $M$ such that
$\operatorname{supp}(M)=Z(I)$. Then $\operatorname{supp}_{s}(M)=Z_{s}(I)=W$ by
(P7).

(S9): If $M\cong 0$ in $\mathsf{St\text{-}sMod}_{V(L)}$, then
$\operatorname{supp}_{s}(M)=\emptyset$ by property (S0) for
$\operatorname{supp}$ and the fact that the forgetful functor maps the zero
object of $\mathsf{St\text{-}sMod}_{V(L)}$ to the zero object of
$\mathsf{StMod}_{V(L)}$. Conversely, if
$\operatorname{supp}_{s}(M)=\emptyset$, then
$\operatorname{supp}(M)=\emptyset$, and hence $M\cong 0$ in
$\mathsf{StMod}_{V(L)}$ by property (S9) for $\operatorname{supp}$. Then $M$
is projective as an object in $\mathsf{Mod}_{V(L)}$. Since the forgetful
functor $\mathsf{sMod}_{V(L)}\to\mathsf{Mod}_{V(L)}$ is exact and fully
faithful, this implies that $M$ is projective in
$(\mathsf{sMod}_{V(L)})_{\textup{ev}}$, and hence $M\cong 0$ in
$\mathsf{St\text{-}sMod}_{V(L)}$. Thus, $\operatorname{supp}_{s}(M)=\emptyset$
if and only if $M\cong 0$ in $\mathsf{St\text{-}sMod}_{V(L)}$. ∎

##  4\. Support for non-restricted Lie superalgebras in characteristic 2

###  4.1. Lie superalgebra cohomology

Given a $k$-vector space $V$, let $S(V)$, $\Lambda(V)$, and $\Gamma(V)$ be the
symmetric, exterior121212For $p=2$, one has $\Lambda(V)\cong S(V)/\langle
x^{2}:x\in V\rangle$; i.e., $\Lambda(V)$ is isomorphic to a truncated
polynomial ring., and divided power algebras on $V$, respectively. We denote a
typical monomial in $S(V)$ by $v_{1}^{a_{1}}\cdots v_{t}^{a_{t}}$, and a
typical monomial in $\Lambda(V)$ by $\langle v_{1}\cdots v_{n}\rangle$. The
divided power algebra $\Gamma(V)$ is generated by the symbols $\gamma_{r}(v)$
for $r\in\mathbb{N}$ and $v\in V$. These generators satisfy the relations
$\gamma_{0}(v)=1$, $\gamma_{r}(0)=0$ if $r\geq 1$,
$\gamma_{r}(v)\gamma_{s}(v)=\binom{r+s}{r}\gamma_{r+s}(v)$, and
$\gamma_{r}(v+v^{\prime})=\sum_{i=0}^{r}\gamma_{i}(v)\gamma_{r-i}(v^{\prime})$.
Since $p=\operatorname{char}(k)=2$, the algebras $S(V)$, $\Lambda(V)$, and
$\Gamma(V)$ are each naturally Hopf algebras: the subspace $V$ consists of
primitive elements in each of $S(V)$ and $\Lambda(V)$, and the coproduct on
$\Gamma(V)$ satisfies
$\Delta(\gamma_{r}(v))=\sum_{i+j=r}\gamma_{i}(v)\otimes\gamma_{j}(v)$.131313If
$p\neq 2$, then $\Lambda(V)$ is not an ordinary Hopf algebra but is instead a
Hopf superalgebra. The algebras $S(V)$, $\Lambda(V)$, and $\Gamma(V)$ are each
naturally $\mathbb{N}$-graded, and these $\mathbb{N}$-gradings are compatible
with the Hopf algebra structures.

Now let $L=L_{\overline{0}}\oplus L_{\overline{1}}$ be Lie superalgebra over
$k$, and let
$\overline{Y}(L)=\Lambda(L_{\overline{0}})\otimes\Gamma(L_{\overline{1}})$ be
the tensor product of algebras.141414If $p\neq 2$, take the graded tensor
product of superalgebras, in which $(a\otimes b)(c\otimes
d)=(-1)^{\overline{b}\cdot\overline{c}+\deg(b)\cdot\deg(c)}ac\otimes bd$. Then
$\overline{Y}(L)$ inherits the structure of $\mathbb{N}$-graded Hopf algebra,
with
$\overline{Y}_{\ell}(L)=\bigoplus_{i+j=\ell}\Lambda^{i}(L_{\overline{0}})\otimes\Gamma^{j}(L_{\overline{1}})$.
In particular, $\overline{Y}_{1}(L)\cong L$, and we write
$s:L\rightarrow\overline{Y}_{1}(L)$ for the natural identification. The right
adjoint action of $L$ on itself extends to a right action of $L$ on
$\overline{Y}(L)$ such that
$(z_{1}z_{2}).u=z_{1}(z_{2}.u)+(-1)^{\overline{u}\cdot\overline{z}_{2}}(z_{1}.u)z_{2}$
for all $z_{1},z_{2}\in\overline{Y}(L)$ and $u\in L$ (i.e., $u$ acts by right
superderivations), and such that $\langle x\rangle.u=s([x,u])$ and
$\gamma_{r}(y).u=\gamma_{r-1}(y)s([y,u])$ for all $x\in L_{\overline{0}}$,
$y\in L_{\overline{1}}$, and $r\in\mathbb{N}$. The right action of $L$ on
$\overline{Y}(L)$ then extends to a right action of the universal enveloping
superalgebra $U(L)$ on $\overline{Y}(L)$. Now define $Y(L)$ to be the
corresponding smash product algebra $U(L)\\#\overline{Y}(L)$. The coproducts
on $U(L)$ and $\overline{Y}(L)$ induce a bialgebra structure on $Y(L)$. Then
$Y(L)$ is an $\mathbb{N}$-graded bialgebra with $U(L)$ concentrated in
$\mathbb{N}$-degree $0$. As a vector space and as a left $U(L)$-module,
$Y(L)=U(L)\otimes\overline{Y}(L)$. From now on we denote the product in $Y(L)$
by juxtaposition. Then $Y(L)$ is spanned by the monomials

(4.1) $u\langle x_{1}\cdots
x_{s}\rangle\gamma_{a_{1}}(y_{1})\cdots\gamma_{a_{t}}(y_{t}),$

with $u\in U(L)$, $x_{i}\in L_{\overline{0}}$, $y_{j}\in L_{\overline{1}}$,
and $a_{j}\in\mathbb{N}$.

######  Theorem 4.1.

Let $L$ be a finite-dimensional Lie superalgebra over the field $k$. There
exists a differential $d:Y(L)\rightarrow Y(L)$ (which we call the Koszul
differential) making $Y(L)$ into a differential graded bialgebra and into a
left $U(L)$-free resolution of the trivial module $k$ (which we call the
Koszul resolution of $k$). Given $u\in U(L)$, $x\in L_{\overline{0}}$, and
$y\in L_{\overline{1}}$, one has

$\displaystyle d(u)$ $\displaystyle=0,$ $\displaystyle d(\langle x\rangle)$
$\displaystyle=x,\quad\text{and}$ $\displaystyle d(\gamma_{r}(y))$
$\displaystyle=y\gamma_{r-1}(y)-\langle q(y)\rangle\gamma_{r-2}(y).$

###### Proof.

Apply the same line of reasoning as in the proof of [13, Theorem 3.1.1]. ∎

Let $M$ and $N$ be left $U(L)$-supermodules. Given $n\in\mathbb{N}$, set

$C^{n}(L,M,N)=\operatorname{Hom}_{U(L)}(Y_{n}(L)\otimes M,N).$

Here we consider $Y(L)\otimes M$ as a $U(L)$-module via the coproduct, and as
a chain complex with differential induced by the Koszul differential $d$ on
$Y(L)$. In particular, set $C^{\bullet}(L,k)=C^{\bullet}(L,k,k)$. The Koszul
differential induces a differential $\partial$ on $C^{\bullet}(L,M,N)$, making
$C^{\bullet}(L,M,N)$ into a cochain complex. Then the Lie superalgebra
cohomology group
$\operatorname{Ext}_{L}^{\bullet}(M,N)=\operatorname{Ext}_{U(L)}^{\bullet}(M,N)$
can be computed as the $n$-th cohomology group of the complex
$C^{\bullet}(L,M,N)$. The coalgebra structure of $Y(L)$ induces the structure
of a differential graded algebra on $C^{\bullet}(L,k)$, and the structure of a
differential graded $C^{\bullet}(L,k)$-bimodule on $C^{\bullet}(L,M,N)$.
Specifically,

(4.2)
$C^{\bullet}(L,k)\cong(\overline{Y}_{\bullet}(L))^{*}\cong\Lambda(L_{\overline{0}}^{*})\otimes
S(L_{\overline{1}}^{*})$

as an $\mathbb{N}$-graded algebra.151515If $p\neq 2$, this is the graded
tensor product of algebras. Passing to cohomology, one gets the cup product
structure on

$\operatorname{H}^{\bullet}(L,k)=\operatorname{Ext}_{L}^{\bullet}(k,k)=\operatorname{Ext}_{U(L)}^{\bullet}(k,k),$

and the $\operatorname{H}^{\bullet}(L,k)$-bimodule structure on

$\operatorname{Ext}_{L}^{\bullet}(M,N)=\operatorname{Ext}_{U(L)}^{\bullet}(M,N).$

As in Section 3, let $\mathsf{sMod}_{U(L)}$ be the category of left
$U(L)$-supermodules, and let $\mathsf{Mod}_{U(L)}$ be the category of
arbitrary (not necessarily $\mathbb{Z}_{2}$-graded) left $U(L)$-modules. For
each $M\in\mathsf{sMod}_{U(L)}$, the tensor product $Y(L)\otimes M$ provides a
resolution of $M$ in $(\mathsf{sMod}_{U(L)})_{\textup{ev}}$ by free
$U(L)$-supermodules, so by the same reasoning as for (3.2), one gets

(4.3)
$\operatorname{Ext}_{\mathsf{sMod}_{U(L)}}^{\bullet}(M,N)=\operatorname{Ext}_{\mathsf{Mod}_{U(L)}}^{\bullet}(M,N)\quad\text{for
all $M,N\in\mathsf{sMod}_{U(L)}$.}$

We will simply write $\operatorname{Ext}_{U(L)}^{\bullet}(M,N)$ for the common
value in (4.3).

Let $\varphi:k\to k$ be the Frobenius map, $\lambda\mapsto\lambda^{2}$, and
given a $k$-vector space $V$, let $V^{(1)}=k\otimes_{\varphi}V$ denotes its
Frobenius twist. Let $L_{\overline{1}}^{*}[2]$ be the superspace
$L_{\overline{1}}^{*}$ concentrated in $\mathbb{N}$-degree $2$. The squaring
operation on $S(L_{\overline{1}}^{*})$ induces a homomorphism of graded
algebras $S(L_{\overline{1}}^{*}[2])^{(1)}\to S(L_{\overline{1}}^{*})$,
$\lambda\otimes_{\varphi}z\mapsto\lambda\cdot z^{2}$. This is not a
homomorphism of graded superalgebras, because the odd superdegree generators
in $S(L_{\overline{1}}^{*}[2])^{(1)}$ are mapped to even superdegree elements
in $S(L_{\overline{1}}^{*})$. Composing with the inclusion
$S(L_{\overline{1}}^{*})\hookrightarrow\Lambda(L_{\overline{0}}^{*})\otimes
S(L_{\overline{1}}^{*})\cong C^{\bullet}(L,k)$, one gets a homomorphism of
graded algebras $\widehat{\varphi}:S(L_{\overline{1}}^{*}[2])^{(1)}\to
C^{\bullet}(L,k)$. Since $\partial$ acts by derivations on the ring
$C^{\bullet}(L,k)$, it follows that the image of $\widehat{\varphi}$ consists
of cocycles in $C^{\bullet}(L,k)$, and hence $\widehat{\varphi}$ induces a
graded algebra map

(4.4)
$\varphi:S(L_{\overline{1}}^{*}[2])^{(1)}\to\operatorname{H}^{\bullet}(L,k),$

which by abuse of notation we also denote $\varphi$. The map $\varphi$ is
natural with respect to $L$, i.e., with respect to even homomorphisms $L\to
L^{\prime}$ of Lie superalgebras.

######  Theorem 4.2.

Let $L$ be a finite-dimensional Lie superalgebra over $k$, and let $M$ and $N$
be finite-dimensional $U(L)$-supermodules. Then
$\operatorname{H}^{\bullet}(L,k)$ is finite over the image of $\varphi$, and
$\operatorname{Ext}_{L}^{\bullet}(M,N)\cong\operatorname{Ext}_{L}^{\bullet}(k,\operatorname{Hom}_{k}(M,N))$
is finite as either a left or right $\operatorname{H}^{\bullet}(L,k)$-module.

###### Proof.

Same as for $p\geq 3$; see [13]. ∎

In fact, a more general version of Theorem 4.2 is true:

######  Theorem 4.3.

Let $L$ be a finite-dimensional Lie superalgebra over $k$. Let $M$ and $N$ be
arbitrary (i.e., not necessarily $\mathbb{Z}_{2}$-graded) $U(L)$-modules, with
$M$ finite-dimensional and $N$ finitely-generated over $U(L)$. Then
$\operatorname{Ext}_{U(L)}^{\bullet}(M,N)$ is finite under the cup product
action of $\operatorname{H}^{\bullet}(L,k)$.

###### Proof.

This can be shown in precisely the same manner as in [16, §§2.2–2.3]. For the
arguments presented there, it is immaterial whether or not the modules are
$\mathbb{Z}_{2}$-graded. The exterior algebra
$\Lambda(\widetilde{\mathfrak{g}}_{1})$ is a genuine cocommutative Hopf
algebra in characteristic $2$, so in the argument at the end of the proof of
[16, Lemma 2.3.1], one can appeal to the results of [20] rather than [13,
Theorem 3.2.4] to argue that the $E_{2}$-page of [16, (2.3.5)] is a finite
module over the cohomology ring
$\operatorname{H}^{\bullet}(\Lambda(\widetilde{\mathfrak{g}}_{1}),k)$. ∎

###  4.2. Support varieties

_For the rest of this section, assume that the field $k$ is algebraically
closed._ Taking $A=U(L)$ in (3.4), (3.5), and (3.6), and using the maximal
ideal spectrum rather than the prime ideal spectrum, we get the definition of
the cohomological spectrum $|U(L)|$, and the definition of (relative) support
varieties for $U(L)$-supermodules. Given $M,N\in\mathsf{sMod}_{U(L)}$, let
$J_{U(L)}(M,N)=\varphi^{-1}(I_{U(L)}(M,N))$, the inverse image of
$I_{U(L)}(M,N)$ under the map (4.4). Then $J_{U(L)}(M,N)$ is homogeneous with
respect to the $\mathbb{N}$-grading on $S(L_{\overline{1}}^{*}[2])^{(1)}$. Set

$\mathcal{X}_{L}(M,N)=\operatorname{MaxSpec}\left(S(L_{\overline{1}}^{*}[2])^{(1)}/J_{L}(M,N)\right).$

and set $\mathcal{X}_{L}(M)=\mathcal{X}_{L}(M,M)$.

######  Lemma 4.4.

The graded algebra homomorphism
$\varphi:S(L_{\overline{1}}^{*}[2])^{(1)}\to\operatorname{H}^{\bullet}(L,k)$
induces for each pair of finite-dimensional $U(L)$-supermodules $M$ and $N$ a
homeomorphism

$\varphi_{M,N}^{*}:|U(L)|_{(M,N)}\simeq\mathcal{X}_{L}(M,N).$

###### Proof.

Same as for $p\geq 3$; see [14, §4.2]. ∎

If $X$ is an affine variety with coordinate algebra $k[X]$, we write $X^{(1)}$
for the affine variety with coordinate algebra $k[X^{(1)}]=k[X]^{(1)}$. If
$Y=X^{(1)}$, then we may write $X=Y^{(-1)}$.

######  Theorem 4.5.

Let $L$ be a finite-dimensional Lie superalgebra over an algebraically closed
field $k$ of characteristic $p=2$. Then

$\mathcal{X}_{L}(k)^{(-1)}=\mathcal{N}_{\textup{odd}}(L):=\\{x\in
L_{\overline{1}}:q(x)=0\\}.$

###### Proof.

Essentially the same as for $p\geq 3$; see the proof of [14, Theorem 4.2.4]. ∎

We call $\mathcal{N}_{\textup{odd}}(L)$ the _odd nullcone_ of $L$. For $p\neq
2$, one has $\mathcal{N}_{odd}(L)=\\{x\in L_{\overline{1}}:[x,x]=0\\}$.

######  Theorem 4.6.

Let $L$ be a finite-dimensional Lie superalgebra over an algebraically closed
field $k$ of characteristic $p=2$, and let $M$ be a finite-dimensional
$L$-supermodule. Then

(4.5)
$\mathcal{X}_{L}(M)^{(-1)}=\\{x\in\mathcal{N}_{\textup{odd}}(L):M|_{\langle
x\rangle}\text{ is not free}\\}\cup\\{0\\}.$

In (4.5), $M|_{\langle x\rangle}$ denotes the restriction of $M$ to the
$k$-subalgebra of $U(L)$ generated by $x$. If $x\neq 0$, this subalgebra has
the form $k[x]/(x^{2})$, while for $x=0$ the subalgebra is just the field $k$.
The set on the right-hand side of (4.5) corresponds to the _associated
variety_ originally considered in characteristic zero by Duflo and Serganova;
see [17, 21].

###### Proof of Theorem 4.6.

Let
$\mathcal{X}_{L}^{\prime}(M)^{(-1)}=\\{x\in\mathcal{N}_{\textup{odd}}(L):M|_{\langle
x\rangle}\text{ is not free}\\}\cup\\{0\\}$ be the ‘rank variety’ defined by
the right-hand side of (4.5). For convenience, we will henceforth omit the
$(-1)$ superscript. First, using naturality with respect to $L$, one can argue
as in the proof of [14, Proposition 4.3.1] to show that
$\mathcal{X}_{L}^{\prime}(M)\subseteq\mathcal{X}_{L}(M)$. To establish the
reverse inclusion, we relate support varieties for $U(L)$ to support varieties
for $V(L)$ by way of the quotient map $\pi:U(L)\twoheadrightarrow V(L)$. The
following argument replaces, for $p=2$, the arguments given in [16, §§3.2–3.3]
for $p\geq 3$.

Consider the increasing monomial-length filtrations on $U(L)$ and $V(L)$, and
the corresponding associated graded algebras

$\displaystyle\operatorname{gr}U(L)\cong
S(L_{\overline{0}})\otimes\Lambda(L_{\overline{1}})\quad\text{and}$
$\displaystyle\operatorname{gr}V(L)\cong S(L_{\overline{0}})/(x^{2}:x\in
L_{\overline{0}})\otimes\Lambda(L_{\overline{1}})\cong\Lambda(L_{\overline{0}})\otimes\Lambda(L_{\overline{1}})\cong\Lambda(L).$

The algebras $\operatorname{gr}U(L)$ and $\operatorname{gr}V(L)$ are
$\mathbb{N}$-graded, with the degree-$0$ component equal to the field $k$, and
the degree-$1$ component identifying with the superspace $L$. Now in the same
manner described in [13, §3.4] (cf. also [16, §2.3]), the monomial-length
filtrations on $U(L)$ and $V(L)$ induce decreasing filtrations on the
corresponding bar complexes for each algebra, and the decreasing filtrations
on the bar complexes then give rise to spectral sequences of algebras

(4.6) $\displaystyle{}^{V}E_{1}^{i,j}$
$\displaystyle=\operatorname{H}^{i+j}(\operatorname{gr}V(L),k)_{-i}\Rightarrow\operatorname{H}^{\bullet}(V(L),k),\quad\text{and}$
(4.7) $\displaystyle{}^{U}E_{1}^{i,j}$
$\displaystyle=\operatorname{H}^{i+j}(\operatorname{gr}U(L),k)_{-i}\Rightarrow\operatorname{H}^{\bullet}(U(L),k).$

The non-negative gradings on $\operatorname{gr}U(L)$ and
$\operatorname{gr}V(L)$ induce non-positive gradings on their cohomology
rings, which are indicated by the subscript $-i$ in (4.7) and (4.6).

By the descriptions of $\operatorname{gr}U(L)$ and $\operatorname{gr}V(L)$
given above, one has

$\displaystyle\operatorname{H}^{\bullet}(\operatorname{gr}V(L),k)$
$\displaystyle\cong S(L^{*}),\quad\text{and}$
$\displaystyle\operatorname{H}^{\bullet}(\operatorname{gr}U(L),k)$
$\displaystyle\cong\Lambda(L_{\overline{0}}^{*})\otimes
S(L_{\overline{1}}^{*})\cong S(L^{*})/(x^{2}:x\in L_{\overline{0}}^{*}).$

In both cases, the superspace $L^{*}=L_{\overline{0}}^{*}\oplus
L_{\overline{1}}^{*}$ is concentrated in the $E_{1}^{1,0}$ term of the
spectral sequence. By multiplicativity, this implies that both spectral
sequences are concentrated in the row $j=0$, and consequently the abutment is
equal to the cohomology of the $E_{1}$-page. Since the differential of each
spectral sequence acts by algebra derivations, it follows that the square of
any element in $S(L^{*})$ is a cocycle in ${}^{V}E_{1}^{\bullet,0}$. As a
complex, ${}^{U}E_{1}^{\bullet,0}$ is simply the Koszul cochain complex
described above in (4.2). The quotient map $\pi:U(L)\twoheadrightarrow V(L)$
induces a morphism of spectral sequences ${}^{V}E\to{}^{U}E$. On the abutment
this is just the inflation map induced by $\pi$, while on the $E_{1}$-page
this is the quotient homomorphism $S(L^{*})\twoheadrightarrow
S(L^{*})/(x^{2}:x\in L_{\overline{0}}^{*})$.

From the observations of the preceding paragraph, we get the commutative
diagram

(4.8)
$\textstyle{S(L^{*}[2])^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\operatorname{res}}$$\textstyle{\operatorname{H}^{\bullet}(V(L),k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{S(L_{\overline{1}}^{*}[2])^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\operatorname{H}^{\bullet}(U(L),k).}$

Here $\varphi$ is the homomorphism of (4.4), and $\phi$ is the analogous map
for $\operatorname{H}^{\bullet}(V(L),k)$ that arises from the graded
$k$-algebra homomorphism $S(L^{*}[2])^{(1)}\to
S(L^{*})\cong{}^{V}E_{1}^{\bullet,0}$ induced by the squaring operation on
$S(L^{*})$. Commutativity of (4.8) then implies commutativity of the diagram

(4.9)
$\textstyle{L^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}}$$\scriptstyle{\operatorname{res}^{*}}$$\textstyle{|V(L)|\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{|V(L)|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{L_{\overline{1}}^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}_{L}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{|U(L)|\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|U(\widetilde{\mathfrak{g}})|_{M}.}$

By the classical theory (cf. [18]),

$\phi^{*}(|V(L)|_{M})=\\{z\in L:z^{\\{2\\}}=0\text{ and }M|_{\langle
z\rangle}\text{ is not free}\\}\cup\\{0\\}.$

Then commutativity of the diagram implies that

$\mathcal{X}_{L}(M)\subseteq\\{z\in\mathcal{N}_{\textup{odd}}(L):M|_{\langle
z\rangle}\text{ is not free}\\}\cup\\{0\\}=\mathcal{X}_{L}^{\prime}(M).$

Then $\mathcal{X}_{L}^{\prime}(M)\subseteq\mathcal{X}_{L}(M)$ and
$\mathcal{X}_{L}(M)\subseteq\mathcal{X}_{L}^{\prime}(M)$, so
$\mathcal{X}_{L}^{\prime}(M)=\mathcal{X}_{L}(M)$. ∎

######  Corollary 4.7 (Tensor Product Property).

Let $L$ be a finite-dimensional Lie superalgebra over an algebraically closed
field $k$ of characteristic $p=2$, and let $M$ and $N$ be finite-dimensional
$L$-supermodules. Then

$|U(L)|_{M\otimes N}=|U(L)|_{M}\cap|U(L)|_{N}.$

###### Proof.

Follows immediately from the theorem, in the same manner as Proposition 3.2. ∎

######  Theorem 4.8.

Let $L$ be a finite-dimensional Lie superalgebra over an algebraically closed
field $k$ of characteristic $p=2$. Let $M$ be a finite-dimensional
$L$-supermodule. Then

$\\{x\in\mathcal{N}_{\textup{odd}}(L):M|_{\langle x\rangle}\text{ is not
free}\\}=\emptyset\quad\text{if and only
if}\quad\operatorname{projdim}_{U(L)}(M)<\infty.$

###### Proof.

The ‘if’ portion of the statement is an immediate consequence of the
identification (4.5). For the ‘only if’ portion of statement, the line of
reasoning is essentially the same as that for the proof of [16, Theorem
3.4.2]. First, one argues as in the proof of [16, Lemma 2.4.1] that $U(L)$ is
free of finite rank over a central (purely even) polynomial subalgebra
$\mathcal{O}$, and hence $U(L)$ is a Noether $\mathcal{O}$-algebra in the
terminology of [16, Definition A.1.1]. Next, one uses Theorems 4.6 and 4.3 to
deduce for each finitely-generated $U(L)$-module $N$ that
$\operatorname{Ext}_{U(L)}^{i}(M,N)=0$ for all $i\gg 0$. By [16, Theorem
A.1.2], $M$ has finite projective dimension in $\mathsf{Mod}_{U(L)}$. By
(4.3), this implies that $M$ has finite projective dimension in
$\mathsf{sMod}_{U(L)}$. ∎

######  Remark 4.9.

An immediate consequence of Theorem 4.8 is that

$\mathcal{N}_{\textup{odd}}(L)=\\{0\\}$ if and only if $U(L)$ is of finite
global dimension.

On the one hand, if the global dimension of $U(L)$ is finite, then
$\operatorname{H}^{\bullet}(U(L),k)$ is a finite-dimensional $k$-algebra, and
hence $\mathcal{N}_{\textup{odd}}(L)\simeq|U(L)|=\\{0\\}$. On the other hand,
if $\mathcal{N}_{\textup{odd}}(L)=\\{0\\}$, then
$\operatorname{projdim}_{U(L)}(k)<\infty$. Since
$\operatorname{projdim}_{U(L)}(M)\leq\operatorname{projdim}_{U(L)}(k)$ for any
$U(L)$-supermodule $M$, this implies that $U(L)$ has finite global dimension.
This extends to characteristic $2$ a result originally due to Bøgvad [9] in
characteristic zero (see [25, Theorem 17.1.2] for a general statement in
characteristic zero) and due to the authors in odd characteristic [16,
Corollary 3.4.3].

In any positive characteristic it would be interesting to find a tensor
triangulated category whose thick tensor ideals are governed by
$\operatorname{Proj}_{s}(\operatorname{H}^{\bullet}(U(L),k))$ in a manner
analogous to (3.16) and (3.17). Instead of quotienting out the projective
supermodules, as one does for the stable category, one would evidently need to
factor out the supermodules of finite projective dimension, since these are
the modules on which the support theory for $U(L)$ vanishes. This suggests a
version of the singularity category will be the appropriate setting to
consider these questions.

## References

  * 1. L. L. Avramov and S. B. Iyengar, _Restricting homology to hypersurfaces_ , Geometric and Topological Aspects of the Representation Theory of Finite Groups, PIMS Summer School and Workshop, July 27–August 5, 2016, Springer Proceedings in Mathematics & Statistics, vol. 242, Springer, 2018, pp. 1–23.
  * 2. Y. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, and M. V. Zaicev, _Infinite-dimensional Lie superalgebras_, De Gruyter Expositions in Mathematics, vol. 7, Walter de Gruyter & Co., Berlin, 1992.
  * 3. D. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, _Stratification for module categories of finite group schemes_ , J. Amer. Math. Soc. 31 (2018), no. 1, 265–302.
  * 4. by same author, _Rank varieties and $\pi$-points for elementary supergroup schemes_, Trans. Amer. Math. Soc. Ser. B 8 (2021), 971–998.
  * 5. by same author, _Stratification and duality for unipotent finite supergroup schemes_ , Equivariant topology and derived algebra, London Math. Soc. Lecture Note Ser., vol. 474, Cambridge Univ. Press, Cambridge, 2022, pp. 241–275.
  * 6. B. D. Boe, J. R. Kujawa, and D. K. Nakano, _Cohomology and support varieties for Lie superalgebras. II_, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 19–44.
  * 7. B. D. Boe, J. R. Kujawa, and D. K. Nakano, _Cohomology and support varieties for Lie superalgebras_, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6551–6590.
  * 8. B. D. Boe, J. R. Kujawa, and D. K. Nakano, _Tensor triangular geometry for classical Lie superalgebras_, Adv. Math. 314 (2017), 228–277.
  * 9. R. Bøgvad, _Some elementary results on the cohomology of graded Lie algebras_ , Algebraic homotopy and local algebra (Luminy, 1982), Astérisque, vol. 113, Soc. Math. France, Paris, 1984, pp. 156–166.
  * 10. S. Bouarroudj, P. Grozman, A. Lebedev, D. Leites, and I. Shchepochkina, _Simple vectorial Lie algebras in characteristic 2 and their superizations_, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 089, 101.
  * 11. D. A. Cox, J. Little, and D. O’Shea, _Ideals, varieties, and algorithms_, fourth ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015, An introduction to computational algebraic geometry and commutative algebra.
  * 12. I. Dell’Ambrogio, _Tensor triangular geometry and $KK$-theory_, J. Homotopy Relat. Struct. 5 (2010), no. 1, 319–358.
  * 13. C. M. Drupieski, _Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes_, Represent. Theory 17 (2013), 469–507.
  * 14. C. M. Drupieski and J. R. Kujawa, _On support varieties for Lie superalgebras and finite supergroup schemes_, J. Algebra 525 (2019), 64–110.
  * 15. by same author, _Support schemes for infinitesimal unipotent supergroups_, Adv. Math. 384 (2021), 107754.
  * 16. by same author, _Support varieties and modules of finite projective dimension for modular Lie superalgebras_, Algebra Number Theory 15 (2021), no. 5, 1157–1180, With an appendix by Luchezar L. Avramov and Srikanth B. Iyengar.
  * 17. M. Duflo and V. Serganova, _On associated variety for Lie superalgebras_ , unpublished, 2005, arXiv:0507198.
  * 18. E. M. Friedlander and B. J. Parshall, _Support varieties for restricted Lie algebras_, Invent. Math. 86 (1986), no. 3, 553–562.
  * 19. by same author, _Geometry of $p$-unipotent Lie algebras_, J. Algebra 109 (1987), no. 1, 25–45.
  * 20. E. M. Friedlander and A. Suslin, _Cohomology of finite group schemes over a field_, Invent. Math. 127 (1997), no. 2, 209–270.
  * 21. M. Gorelik, C. Hoyt, V. Serganova, and A. Sherman, _The Duflo-Serganova functor, vingt ans après_ , preprint, 2022, arXiv:2203.00529.
  * 22. D. Happel, _Triangulated categories in the representation theory of finite-dimensional algebras_, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988.
  * 23. A. Kaufer, _Superalgebra in characteristic 2_ , preprint, 2018, arXiv:1804.00824.
  * 24. M. Lorenz, _Algebraic group actions on noncommutative spectra_, Transform. Groups 14 (2009), no. 3, 649–675.
  * 25. I. M. Musson, _Lie superalgebras and enveloping algebras_, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012.
  * 26. A. Premet, _Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture_, Invent. Math. 121 (1995), no. 1, 79–117.
  * 27. A. Suslin, E. M. Friedlander, and C. P. Bendel, _Infinitesimal $1$-parameter subgroups and cohomology_, J. Amer. Math. Soc. 10 (1997), no. 3, 693–728.
  * 28. by same author, _Support varieties for infinitesimal group schemes_, J. Amer. Math. Soc. 10 (1997), no. 3, 729–759.
  * 29. S. Venkatesh, _Hilbert basis theorem and finite generation of invariants in symmetric tensor categories in positive characteristic_, Int. Math. Res. Not. IMRN (2016), no. 16, 5106–5133.
  * 30. W. Wang and L. Zhao, _Representations of Lie superalgebras in prime characteristic. I_, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 145–167.
  * 31. S. J. Witherspoon, _Hochschild cohomology for algebras_, Graduate Studies in Mathematics, vol. 204, American Mathematical Society, Providence, RI, 2019.