论文标题
代数超过广义环
Algebra over generalized rings
论文作者
论文摘要
对于交换戒指$ a $，我们拥有$ a $ a $ modules $ ch _ {+}（a \ mymod）$的（有限孔）链综合体的类别，这是一个封闭的对称单体类别，具有兼容的稳定Quillen模型结构。相关的同型类别是派生的类别$ \ mathbbm {d}（a \ mymod）$，其中一个倒置了所有的准同态，它具有很好的描述，因为链接综合体是由投影型$ $ a $ a-module组成的链条中的链条，并在每个尺寸中构成了链映射，而链映射则构成了链型同型。我们在这里给出了\ cite {MR3605614}的（交换性）广义环的类似理论。我们将新概念称为``$ \ aset $''。对于普通的交换戒指$ a $，$ a $ set只是通常的含义$ a $ - 模块，我们的构造等同于$ \ m athbbm {d}（a \ mymod）$。对于通用环类别的初始对象，$ \ mathbb {f} $``带有一个元素的字段''，我们获得了对称频谱的类别，以及带有Smash产品的相关稳定同型类别（$ \ Mathbb {f} $ - set是一个指向$ x $ x $ x $ o o o o o o。因此，稳定的同喻和模块链复合物的类似理论是交换环上的两个侧面的两个侧面，而且它们出现在它们相互作用的上下文中（通过健忘的函数和左伴随 - 基本变化函数）。对于``真实整数''$ a = \ z _ {\ r} $，$ \ z _ {\ r} $ - 集合包括$ \ r $ - 向量空间的对称凸子集。我们还以$ \ bigo_x $ set的派生类别的全局理论，以一种基于本地投影模型结构的方式，用于概括方案$ x $。
For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is the derived category $\mathbbm{D}(A\mymod)$, where one inverts all the quasi-isomorphisms, and it has the good description as the chain complexes made up of projective $A$-module in each dimension, and chain maps taken up to chain homotopy. We give here the analogous theory for a (commutative) generalized ring in the sense of \cite{MR3605614}. We refer to the new concept as ``$\aset$''. For an ordinary commutative ring $A$, an $A$-set is just an $A$-module in the usual meaning, and our construction will be equivalent to $\mathbbm{D}(A\mymod)$. For the initial object of the category of generalized rings $\mathbb{F}$ ``the field with one element'', we obtain the category of symmetric spectra, and the associated stable homotopy category with its smash product (an $\mathbb{F}$-set is just a pointed set, i.e. a set $X$ with a distinguish element $O_X\in X$). Thus the analogous theories of stable homotopy and of chain complexes of modules over a commutative ring appear as two sides of the same coin, and moreover, they appear in a context where they interact (via the forgetful functor and its left adjoint - the base change functor). For the ``real integers'' $A=\Z_{\R}$, the $\Z_{\R}$-sets include the symmetric convex subsets of $\R$-vector spaces. We also give the global theory of the derived category of $\bigo_X$-sets, for a generalized scheme $X$, in a way that is based on the local projective model structure.
