We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We consider rational representations of a connected linear algebraic group 
$\mathbb {G}$ over a field 
$k$ of positive characteristic 
$p > 0$. We introduce a natural extension 
$M \mapsto \Pi (\mathbb {G})_M$ to 
$\mathbb {G}$-modules of the 
$\pi$-point support theory for modules 
$M$ for a finite group scheme 
$G$ and show that this theory is essentially equivalent to the more ‘intrinsic’ and ‘explicit’ theory 
$M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ of supports for an algebraic group of exponential type, a theory which uses 
$1$-parameter subgroups 
$\mathbb {G}_a \to \mathbb {G}$. We extend our support theory to bounded complexes of 
$\mathbb {G}$-modules, 
$C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$. We introduce the tensor triangulated category 
$\mathit {StMod}(\mathbb {G})$, the Verdier quotient of the bounded derived category 
$D^b(\mathit {Mod}(\mathbb {G}))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the ‘standard properties’ for a theory of supports for 
$\mathit {StMod}(\mathbb {G})$. As an application, we employ 
$C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$ to establish the classification of 
$(r)$-complete, thick tensor ideals of 
$\mathit {stmod}(\mathbb {G})$ in terms of locally 
$\mathit {stmod}(\mathbb {G})$-realizable subsets of 
$\Pi (\mathbb {G})$ and the classification of 
$(r)$-complete, localizing subcategories of 
$\mathit {StMod}(\mathbb {G})$ in terms of locally 
$\mathit {StMod}(\mathbb {G})$-realizable subsets of 
$\Pi (\mathbb {G})$.
Let 
be a classical Lie superalgebra and let ℱ be the category of finite-dimensional 
-supermodules which are completely reducible over the reductive Lie algebra 
. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of 
. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra . In both cases we show that the complexity is related to the atypicality of the block containing the module.
