交换adic代数上的导Hochschild函子
Derived Hochschild functors over commutative adic algebras
摘要：
设$\k$是一个交换环，$(A,\mfrak{a})$是一个adic环，它是一个$\k$ -代数.研究了$\k$上$A$的霍赫希尔德同调和上同调函子的完备和扭形式.为此，我们首先建立了诺特环的平基变换中某些理想的弱预正则性。接下来，我们发展了一个DG-仿射形式概形的理论，扩展了Greenlees-May对偶和MGM等价。最后，我们在这种设置中定义了完全的和挠导出的霍赫希尔德同调和上同调函子，并证明如果$\k$是诺特式的，并且$(A,\mfrak{a})$本质上是$\k$上的有限型（在adic意义上），则存在计算它们的公式，这些公式保持在诺特范畴内。在经典情形下，当$\k$是一个域时，我们证明了拓扑Hochschild上同调与离散Hochschild上同调是同构的.
英文摘要：
Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian and $(A,\mfrak{a})$ is essentially of finite type (in the adic sense) over $\k$, then there are formulas to compute them that stay inside the noetherian category. In the classical case, where $\k$ is a field, we deduce that topological Hochschild cohomology and discrete Hochschild cohomology are isomorphic.
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