Cohen structure theorem

<p>In mathematics, the Cohen structure theorem, introduced by <a href="/facts/Irvin_Cohen/s9B1JmNW">Cohen</a> (1946), describes the structure of <a href="/facts/Completion_(ring_theory)/Q0vjA9JI">complete</a> <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Local_ring/gRTdg5Qx">local rings</a>.
</p><p>Some consequences of Cohen's structure theorem include three conjectures of <a href="/facts/Wolfgang_Krull/yFtzRRIj">Krull</a>:
</p>
<ul><li>Any complete <a href="/facts/Regular_local_ring/P9s5Fvrl">regular</a> equicharacteristic Noetherian local ring is a <a href="/facts/Ring_of_formal_power_series/CNmaG0Vk">ring of formal power series</a> over a field. (Equicharacteristic means that the local ring and its <a href="/facts/Residue_field/e6nflEmB">residue field</a> have the same <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a>, and is equivalent to the local ring containing a field.)</li>
<li>Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension.</li>
<li>Any complete Noetherian local ring is the image of a complete regular Noetherian local ring.</li></ul>

Cohen structure theorem open-in-new

Cohen structure theorem